{"id":16385,"date":"2024-01-24T11:50:36","date_gmt":"2024-01-24T08:20:36","guid":{"rendered":"https:\/\/rasanegar.com\/blog\/%d8%aa%d8%ba%db%8c%db%8c%d8%b1-%d8%b4%da%a9%d9%84-%d8%aa%d8%b5%d9%88%db%8c%d8%b1-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d8%a8%d8%a7-numpy%d8%8c-pillow-%d9%88-opencv\/"},"modified":"2024-01-24T11:50:36","modified_gmt":"2024-01-24T08:20:36","slug":"%d8%aa%d8%ba%db%8c%db%8c%d8%b1-%d8%b4%da%a9%d9%84-%d8%aa%d8%b5%d9%88%db%8c%d8%b1-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d8%a8%d8%a7-numpy%d8%8c-pillow-%d9%88-opencv","status":"publish","type":"post","link":"https:\/\/rasanegaar.com\/blog\/%d8%aa%d8%ba%db%8c%db%8c%d8%b1-%d8%b4%da%a9%d9%84-%d8%aa%d8%b5%d9%88%db%8c%d8%b1-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d8%a8%d8%a7-numpy%d8%8c-pillow-%d9%88-opencv\/","title":{"rendered":"\u062a\u063a\u06cc\u06cc\u0631 \u0634\u06a9\u0644 \u062a\u0635\u0648\u06cc\u0631 \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646 \u0628\u0627 Numpy\u060c Pillow \u0648 OpenCV"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\" style=\"cursor:inherit\">\u0633\u0631\u0641\u0635\u0644\u0647\u0627\u06cc \u0645\u0637\u0644\u0628<\/p>\n<\/div><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/rasanegaar.com\/blog\/%d8%aa%d8%ba%db%8c%db%8c%d8%b1-%d8%b4%da%a9%d9%84-%d8%aa%d8%b5%d9%88%db%8c%d8%b1-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d8%a8%d8%a7-numpy%d8%8c-pillow-%d9%88-opencv\/#%d8%aa%d8%a8%d8%af%db%8c%d9%84_%d8%a2%d9%81%db%8c%d9%86_%da%86%db%8c%d8%b3%d8%aa%d8%9f\" >\u062a\u0628\u062f\u06cc\u0644 \u0622\u0641\u06cc\u0646 \u0686\u06cc\u0633\u062a\u061f<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/rasanegaar.com\/blog\/%d8%aa%d8%ba%db%8c%db%8c%d8%b1-%d8%b4%da%a9%d9%84-%d8%aa%d8%b5%d9%88%db%8c%d8%b1-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d8%a8%d8%a7-numpy%d8%8c-pillow-%d9%88-opencv\/#%da%a9%d8%a7%d8%b1_%d8%a8%d8%a7_%db%8c%da%a9_%d8%aa%d8%b5%d9%88%db%8c%d8%b1\" >\u06a9\u0627\u0631 \u0628\u0627 \u06cc\u06a9 \u062a\u0635\u0648\u06cc\u0631<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/rasanegaar.com\/blog\/%d8%aa%d8%ba%db%8c%db%8c%d8%b1-%d8%b4%da%a9%d9%84-%d8%aa%d8%b5%d9%88%db%8c%d8%b1-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d8%a8%d8%a7-numpy%d8%8c-pillow-%d9%88-opencv\/#%d8%aa%d8%a8%d8%af%db%8c%d9%84_%d8%a2%d9%81%db%8c%d9%86_%d8%a8%d8%a7_%d8%a8%d8%a7%d9%84%d8%b4\" >\u062a\u0628\u062f\u06cc\u0644 \u0622\u0641\u06cc\u0646 \u0628\u0627 \u0628\u0627\u0644\u0634<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/rasanegaar.com\/blog\/%d8%aa%d8%ba%db%8c%db%8c%d8%b1-%d8%b4%da%a9%d9%84-%d8%aa%d8%b5%d9%88%db%8c%d8%b1-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d8%a8%d8%a7-numpy%d8%8c-pillow-%d9%88-opencv\/#affine_transformations_%d8%a8%d8%a7_opencv2\" >Affine Transformations \u0628\u0627 OpenCV2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/rasanegaar.com\/blog\/%d8%aa%d8%ba%db%8c%db%8c%d8%b1-%d8%b4%da%a9%d9%84-%d8%aa%d8%b5%d9%88%db%8c%d8%b1-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d8%a8%d8%a7-numpy%d8%8c-pillow-%d9%88-opencv\/#%d9%86%d8%aa%db%8c%d8%ac%d9%87\" >\u0646\u062a\u06cc\u062c\u0647<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/rasanegaar.com\/blog\/%d8%aa%d8%ba%db%8c%db%8c%d8%b1-%d8%b4%da%a9%d9%84-%d8%aa%d8%b5%d9%88%db%8c%d8%b1-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d8%a8%d8%a7-numpy%d8%8c-pillow-%d9%88-opencv\/#%d9%85%d9%86%d8%a7%d8%a8%d8%b9\" >\u0645\u0646\u0627\u0628\u0639<\/a><\/li><\/ul><\/nav><\/div>\n<span class=\"span-reading-time rt-reading-time\" style=\"display: block;\"><span class=\"rt-label rt-prefix\">\u0632\u0645\u0627\u0646 \u0644\u0627\u0632\u0645 \u0628\u0631\u0627\u06cc \u0645\u0637\u0627\u0644\u0639\u0647: <\/span> <span class=\"rt-time\"> 15<\/span> <span class=\"rt-label rt-postfix\">\u062f\u0642\u06cc\u0642\u0647<\/span><\/span><p> <br \/>\n<\/p>\n<div><noscript><\/noscript><\/p>\n<p>\u062f\u0631 \u0627\u06cc\u0646 \u0645\u0642\u0627\u0644\u0647 \u0645\u0646 \u062a\u0648\u0636\u06cc\u062d \u062e\u0648\u0627\u0647\u0645 \u062f\u0627\u062f \u06a9\u0647 \u0627\u0639\u0645\u0627\u0644 \u06cc\u06a9 \u062a\u0628\u062f\u06cc\u0644 \u0627\u0641\u06cc\u0646 \u0628\u0647 \u06cc\u06a9 \u062a\u0635\u0648\u06cc\u0631 \u0686\u06cc\u0633\u062a \u0648 \u0686\u06af\u0648\u0646\u0647 \u0622\u0646 \u0631\u0627 \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646 \u0627\u0646\u062c\u0627\u0645 \u062f\u0647\u06cc\u0645.  \u0627\u0628\u062a\u062f\u0627 \u0639\u0645\u0644\u06cc\u0627\u062a \u0633\u0637\u062d \u067e\u0627\u06cc\u06cc\u0646 \u0631\u0627 \u0646\u0634\u0627\u0646 \u062e\u0648\u0627\u0647\u0645 \u062f\u0627\u062f <a rel=\"nofollow noopener\" target=\"_blank\" href=\"http:\/\/www.numpy.org\/\">NumPy<\/a> \u0628\u0631\u0627\u06cc \u0627\u0631\u0627\u0626\u0647 \u06cc\u06a9 \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u0647\u0646\u062f\u0633\u06cc \u062f\u0642\u06cc\u0642.  \u0633\u067e\u0633 \u0622\u0646\u0647\u0627 \u0631\u0627 \u062f\u0631 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0639\u0645\u0644\u06cc \u062a\u0631 \u0627\u0632 \u067e\u0627\u06cc\u062a\u0648\u0646 \u0642\u0631\u0627\u0631 \u0645\u06cc \u062f\u0647\u0645 <a rel=\"nofollow noopener\" target=\"_blank\" href=\"https:\/\/python-pillow.org\/\" class=\"broken_link\">\u0628\u0627\u0644\u0634<\/a> \u0648 <a rel=\"nofollow noopener\" target=\"_blank\" href=\"https:\/\/opencv.org\/\">OpenCV<\/a> \u06a9\u062a\u0627\u0628\u062e\u0627\u0646\u0647 \u0647\u0627<\/p>\n<p>\u0627\u06cc\u0646 \u0645\u0642\u0627\u0644\u0647 \u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 a \u0646\u0648\u0634\u062a\u0647 \u0634\u062f\u0647 \u0627\u0633\u062a Jupyter notebook \u0648 \u0645\u0646\u0628\u0639 \u0631\u0627 \u0645\u06cc \u062a\u0648\u0627\u0646 \u062f\u0631 \u0633\u0627\u06cc\u062a \u0645\u0646 \u06cc\u0627\u0641\u062a <a rel=\"nofollow noopener\" target=\"_blank\" href=\"https:\/\/github.com\/amcquistan\/affine-image-transforms-python\">\u0645\u062e\u0632\u0646 GitHub<\/a> \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646\u060c \u0644\u0637\u0641\u0627 \u0628\u0627 \u062e\u06cc\u0627\u0644 \u0631\u0627\u062d\u062a \u0622\u0646 \u0631\u0627 \u0634\u0628\u06cc\u0647 \u0633\u0627\u0632\u06cc \u06a9\u0646\u06cc\u062f \/ \u0641\u0648\u0631\u06a9 \u06a9\u0646\u06cc\u062f \u0648 \u06a9\u062f \u0631\u0627 \u0622\u0632\u0645\u0627\u06cc\u0634 \u06a9\u0646\u06cc\u062f.<\/p>\n<h2 id=\"whatisanaffinetransformation\"><span class=\"ez-toc-section\" id=\"%d8%aa%d8%a8%d8%af%db%8c%d9%84_%d8%a2%d9%81%db%8c%d9%86_%da%86%db%8c%d8%b3%d8%aa%d8%9f\"><\/span>\u062a\u0628\u062f\u06cc\u0644 \u0622\u0641\u06cc\u0646 \u0686\u06cc\u0633\u062a\u061f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u0645\u0637\u0627\u0628\u0642 \u0628\u0627 <a rel=\"nofollow noopener\" target=\"_blank\" href=\"https:\/\/en.wikipedia.org\/wiki\/Affine_transformation\">\u0648\u06cc\u06a9\u06cc\u067e\u062f\u06cc\u0627<\/a> \u062a\u0628\u062f\u06cc\u0644 \u0627\u0641\u06cc\u0646 \u06cc\u06a9 \u0646\u0642\u0634\u0647 \u0628\u0631\u062f\u0627\u0631\u06cc \u0639\u0645\u0644\u06a9\u0631\u062f\u06cc \u0628\u06cc\u0646 \u062f\u0648 \u0641\u0636\u0627\u06cc \u0647\u0646\u062f\u0633\u06cc (\u0622\u0641\u06cc\u0646) \u0627\u0633\u062a \u06a9\u0647 \u0646\u0642\u0627\u0637\u060c \u062e\u0637\u0648\u0637 \u0645\u0633\u062a\u0642\u06cc\u0645 \u0648 \u0645\u0648\u0627\u0632\u06cc \u0648 \u0647\u0645\u0686\u0646\u06cc\u0646 \u0646\u0633\u0628\u062a \u0628\u06cc\u0646 \u0646\u0642\u0627\u0637 \u0631\u0627 \u062d\u0641\u0638 \u0645\u06cc \u06a9\u0646\u062f.  \u062a\u0645\u0627\u0645 \u0622\u0646 \u062c\u0645\u0644\u0647 \u0628\u0646\u062f\u06cc \u0627\u0646\u062a\u0632\u0627\u0639\u06cc \u0631\u06cc\u0627\u0636\u06cc \u06cc\u06a9 \u062a\u0628\u062f\u06cc\u0644 \u062e\u0637\u06cc \u0633\u0627\u062f\u0647 \u0627\u0633\u062a \u06a9\u0647 \u062d\u062f\u0627\u0642\u0644 \u062f\u0631 \u0632\u0645\u06cc\u0646\u0647 \u067e\u0631\u062f\u0627\u0632\u0634 \u062a\u0635\u0648\u06cc\u0631 \u0645\u0646\u062c\u0631 \u0628\u0647 \u06cc\u06a9 \u06cc\u0627 \u0686\u0646\u062f \u062f\u0633\u062a\u06a9\u0627\u0631\u06cc \u0645\u0627\u0646\u0646\u062f \u0686\u0631\u062e\u0634\u060c \u0686\u0631\u062e\u0634\u060c \u0645\u0642\u06cc\u0627\u0633 \u06cc\u0627 \u0628\u0631\u0634 \u0628\u0627 \u0627\u0639\u0645\u0627\u0644 \u06cc\u06a9 \u0645\u0627\u062a\u0631\u06cc\u0633 \u062a\u0628\u062f\u06cc\u0644 \u0645\u06cc \u0634\u0648\u062f.<\/p>\n<p>\u06cc\u06a9 \u0686\u06cc\u0632 \u062e\u0648\u0628 \u0627\u06cc\u0646 \u0627\u0633\u062a \u06a9\u0647 \u0627\u0632 \u0622\u0646\u062c\u0627\u06cc\u06cc \u06a9\u0647 \u0627\u06cc\u0646 \u0627\u0633\u0627\u0633\u0627\u064b \u06cc\u06a9 \u0639\u0645\u0644\u06cc\u0627\u062a \u0647\u0646\u062f\u0633\u06cc \u062f\u0648 \u0628\u0639\u062f\u06cc \u0627\u0633\u062a\u060c \u0645\u06cc \u062a\u0648\u0627\u0646\u06cc\u0645 \u0622\u0646 \u0631\u0627 \u062a\u062c\u0633\u0645 \u06a9\u0646\u06cc\u0645.  \u0627\u062c\u0627\u0632\u0647 \u062f\u0647\u06cc\u062f \u0628\u0627 \u0627\u0631\u0627\u0626\u0647 \u062c\u062f\u0648\u0644\u06cc \u0627\u0632 \u062a\u0628\u062f\u06cc\u0644 \u0647\u0627\u06cc \u0648\u0627\u0628\u0633\u062a\u0647 \u06a9\u0647 \u0647\u0631 \u0646\u0648\u0639 \u062f\u0633\u062a\u06a9\u0627\u0631\u06cc \u0647\u0646\u062f\u0633\u06cc \u0631\u0627 \u062a\u0648\u0635\u06cc\u0641 \u0645\u06cc \u06a9\u0646\u062f\u060c \u0634\u0631\u0648\u0639 \u06a9\u0646\u0645.<\/p>\n<table class=\"table\">\n<thead>\n<tr>\n<th>\u0646\u0648\u0639 \u062a\u0628\u062f\u06cc\u0644<\/th>\n<th>\u0645\u0627\u062a\u0631\u06cc\u0633 \u062a\u0628\u062f\u06cc\u0644<\/th>\n<th>\u0645\u0639\u0627\u062f\u0644\u0647 \u0646\u0642\u0634\u0647 \u0628\u0631\u062f\u0627\u0631\u06cc \u067e\u06cc\u06a9\u0633\u0644<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u0647\u0648\u06cc\u062a<\/td>\n<td>\n<p>            $$ \\begin{bmatrix} 1 &#038; 0 &#038; 0 \\\\ 0 &#038; 1 &#038; 0 \\\\ 0 &#038; 0 &#038; 1 \\end{bmatrix} $$\n            <\/td>\n<td>\n<p>                $$x^{&#8216;} = x$$ <br \/>$$y^{&#8216;} = y$$\n            <\/td>\n<\/tr>\n<tr>\n<td>\u0645\u0642\u06cc\u0627\u0633 \u0628\u0646\u062f\u06cc<\/td>\n<td>\n<p>            $$ \\begin{bmatrix} c_{x} &#038; 0 &#038; 0 \\\\ 0 &#038; c_{y} &#038; 0 \\\\ 0 &#038; 0 &#038; 1 \\end{bmatrix} $$\n            <\/td>\n<td>\n<p>                $$x^{&#8216;} = c_{x} * x$$ <br \/>$$y^{&#8216;} = c_{y} * y$$\n            <\/td>\n<\/tr>\n<tr>\n<td>\u0686\u0631\u062e\u0634*<\/td>\n<td>\n<p>            $$ \\begin{bmatrix} cos \\Theta &#038; sin \\Theta &#038; 0 \\\\ -sin \\Theta &#038; cos \\Theta &#038; 0 \\\\ 0 &#038; 0 &#038; 1 \\end{bmatrix} $$\n            <\/td>\n<td>\n<p>                $$x^{&#8216;} = x * cos \\Theta &#8211; y * sin \\Theta$$ <br \/>$$y^{&#8216;} = x * cos \\Theta + y * sin \\Theta$$\n            <\/td>\n<\/tr>\n<tr>\n<td>\u062a\u0631\u062c\u0645\u0647<\/td>\n<td>\n<p>            $$ \\begin{bmatrix} 1 &#038; 0 &#038; t_{x} \\\\ 0 &#038; 1 &#038; t_{y} \\\\ 0 &#038; 0 &#038; 1 \\end{bmatrix} $$\n            <\/td>\n<td>\n<p>                $$x^{&#8216;} = x + t_{x}$$ <br \/>$$y^{&#8216;} = y + t_{y}$$\n            <\/td>\n<\/tr>\n<tr>\n<td>\u0628\u0631\u0634\u06cc \u0627\u0641\u0642\u06cc<\/td>\n<td>\n<p>            $$ \\begin{bmatrix} 1 &#038; s_{h} &#038; \u200b\u200b0 \\\\ 0 &#038; 1 &#038; 0 \\\\ 0 &#038; 0 &#038; 1 \\end{bmatrix} $$\n            <\/td>\n<td>\n<p>                $$x^{&#8216;} = x + s_{v} * y$$ <br \/>$$y^{&#8216;} = y$$\n            <\/td>\n<\/tr>\n<tr>\n<td>\u0628\u0631\u0634 \u0639\u0645\u0648\u062f\u06cc<\/td>\n<td>\n<p>            $$ \\begin{bmatrix} 1 &#038; 0 &#038; 0 \\\\ s_{v} &#038; 1 &#038; 0 \\\\ 0 &#038; 0 &#038; 1 \\end{bmatrix} $$\n            <\/td>\n<td>\n<p>                $$x^{&#8216;} = x$$ <br \/>$$y^{&#8216;} = x * s_{h} + y$$\n            <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ul>\n<li><em>\u062a\u0628\u062f\u06cc\u0644 \u0622\u0641\u06cc\u0646 \u0627\u0632 \u0632\u0627\u0648\u06cc\u0647 \u0686\u0631\u062e\u0634 \u062f\u0631 \u062c\u0647\u062a \u0639\u0642\u0631\u0628\u0647\u200c\u0647\u0627\u06cc \u0633\u0627\u0639\u062a \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc\u200c\u06a9\u0646\u062f \u06a9\u0647 \u0628\u0631\u062e\u0644\u0627\u0641 \u062f\u0627\u06cc\u0631\u0647 \u0648\u0627\u062d\u062f \u0647\u0646\u062f\u0633\u06cc \u0645\u0639\u0645\u0648\u0644\u06cc \u0632\u0627\u0648\u06cc\u0647\u200c\u0647\u0627 \u0627\u0633\u062a \u06a9\u0647 \u062f\u0631 \u0686\u0631\u062e\u0634 \u062e\u0644\u0627\u0641 \u062c\u0647\u062a \u0639\u0642\u0631\u0628\u0647\u200c\u0647\u0627\u06cc \u0633\u0627\u0639\u062a \u0628\u0627 \u0635\u0641\u0631 \u0634\u0631\u0648\u0639 \u0627\u0632 \u0645\u062d\u0648\u0631 \u0645\u062b\u0628\u062a X \u0627\u0646\u062f\u0627\u0632\u0647\u200c\u06af\u06cc\u0631\u06cc \u0645\u06cc\u200c\u0634\u0648\u062f\u060c \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u062e\u0648\u0627\u0647\u06cc\u062f \u062f\u06cc\u062f \u06a9\u0647 \u0645\u0646\u0641\u06cc \u0632\u0627\u0648\u06cc\u0647 \u0627\u063a\u0644\u0628 \u0627\u0639\u0645\u0627\u0644 \u0645\u06cc\u200c\u0634\u0648\u062f.<\/em><\/li>\n<\/ul>\n<p><code>'<\/code>  \u0639\u0644\u0627\u0645\u062a \u06af\u0630\u0627\u0631\u06cc \u062f\u0631 \u0627\u06cc\u0646\u062c\u0627 \u0641\u0642\u0637 \u0628\u0647 \u0645\u062e\u062a\u0635\u0627\u062a \u062e\u0631\u0648\u062c\u06cc \u062a\u0628\u062f\u06cc\u0644 \u0634\u062f\u0647 x \u06cc\u0627 y \u0627\u0634\u0627\u0631\u0647 \u062f\u0627\u0631\u062f \u0646\u0647 \u0646\u0645\u0627\u062f \u062d\u0633\u0627\u0628 \u0628\u0631\u0627\u06cc \u06cc\u06a9 \u0645\u0634\u062a\u0642<\/p>\n<p>\u0628\u0631\u0627\u06cc \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0646 \u0633\u0627\u062f\u0647\u060c \u0645\u0646 \u0686\u0646\u062f \u062a\u0628\u062f\u06cc\u0644 \u0631\u0627 \u0628\u0631\u0627\u06cc \u062f\u0633\u062a\u06a9\u0627\u0631\u06cc \u0645\u062e\u062a\u0635\u0627\u062a x \u0648 y \u0646\u0642\u0627\u0637 \u0632\u06cc\u0631 \u0627\u0639\u0645\u0627\u0644 \u062e\u0648\u0627\u0647\u0645 \u06a9\u0631\u062f \u06a9\u0647 \u062f\u0627\u0631\u0627\u06cc \u0645\u0648\u0644\u0641\u0647 \u0647\u0627\u06cc \u0633\u0647 \u0628\u0639\u062f\u06cc x\u060c y \u0648 \u0634\u0627\u062e\u0635 \u06a9\u0627\u0631\u0627\u06a9\u062a\u0631 ASCII \u0647\u0633\u062a\u0646\u062f\u060c \u0634\u0628\u06cc\u0647 \u0628\u0647 \u0631\u0648\u0634\u06cc \u06a9\u0647 \u06cc\u06a9 \u067e\u06cc\u06a9\u0633\u0644 \u062a\u0635\u0648\u06cc\u0631 \u062f\u0627\u0631\u0627\u06cc \u0645\u0648\u0644\u0641\u0647 \u0647\u0627\u06cc 3 \u0628\u0639\u062f\u06cc x\u060c y \u0627\u0633\u062a. \u0648 \u0641\u0631\u06a9\u0627\u0646\u0633 (\u06cc\u0627 \u0634\u062f\u062a).<\/p>\n<p>a = (0\u060c 1\u060c 0)<br \/>b = (1\u060c 0\u060c 1)<br \/>c = (0\u060c -1\u060c 2)<br \/>d = (-1\u060c 0\u060c 3)<\/p>\n<p>\u062f\u06af\u0631\u06af\u0648\u0646\u06cc \u0647\u0627\u06cc \u0627\u06cc\u0646 \u0645\u062b\u0627\u0644 \u0628\u0647 \u0635\u0648\u0631\u062a Scaling by 2 \u062f\u0631 \u062a\u0645\u0627\u0645 \u062c\u0647\u0627\u062a \u0648 \u0686\u0631\u062e\u0634 90 \u062f\u0631\u062c\u0647 \u062f\u0631 \u062c\u0647\u062a \u0639\u0642\u0631\u0628\u0647 \u0647\u0627\u06cc \u0633\u0627\u0639\u062a \u062e\u0648\u0627\u0647\u062f \u0628\u0648\u062f.  \u0627\u0628\u062a\u062f\u0627 \u062a\u0628\u062f\u06cc\u0644 \u0647\u0627 \u0631\u0627 \u0628\u0647 \u0635\u0648\u0631\u062a \u062c\u062f\u0627\u06af\u0627\u0646\u0647 \u0627\u0646\u062c\u0627\u0645 \u0645\u06cc \u062f\u0647\u0645 \u062a\u0627 \u062a\u0623\u062b\u06cc\u0631 \u0645\u0633\u062a\u0642\u06cc\u0645 \u0647\u0631 \u06a9\u062f\u0627\u0645 \u0631\u0627 \u0646\u0634\u0627\u0646 \u062f\u0647\u0645 \u0631\u0648\u06cc \u0646\u0642\u0627\u0637 \u0631\u0627 \u0628\u0647 \u0627\u0637\u0631\u0627\u0641 \u062d\u0631\u06a9\u062a \u0645\u06cc \u062f\u0647\u0645\u060c \u0633\u067e\u0633 \u062a\u0628\u062f\u06cc\u0644 \u0647\u0627 \u0631\u0627 \u062a\u0631\u06a9\u06cc\u0628 \u0645\u06cc \u06a9\u0646\u0645 \u0648 \u0622\u0646\u0647\u0627 \u0631\u0627 \u062f\u0631 \u06cc\u06a9 \u0639\u0645\u0644 \u0627\u0639\u0645\u0627\u0644 \u0645\u06cc \u06a9\u0646\u0645.<\/p>\n<p>\u0628\u0631\u0627\u06cc \u0634\u0631\u0648\u0639 \u0645\u06cc\u200c\u062e\u0648\u0627\u0647\u0645 \u06cc\u06a9 \u0622\u0631\u0627\u06cc\u0647 NumPy \u0628\u0633\u0627\u0632\u0645 (\u0628\u0639\u0636\u06cc\u200c\u0647\u0627 \u0645\u0645\u06a9\u0646 \u0627\u0633\u062a \u0622\u0646 \u0631\u0627 \u0645\u0627\u062a\u0631\u06cc\u0633 \u0628\u062e\u0648\u0627\u0646\u0646\u062f) \u06a9\u0647 \u0647\u0631 \u0633\u0637\u0631 \u0646\u0634\u0627\u0646\u200c\u062f\u0647\u0646\u062f\u0647 \u0646\u0642\u0637\u0647\u200c\u0627\u06cc \u0627\u0633\u062a \u06a9\u0647 \u0633\u062a\u0648\u0646 \u0627\u0648\u0644 x\u060c \u0633\u062a\u0648\u0646 \u062f\u0648\u0645 y \u0648 \u0633\u0648\u0645\u06cc\u0646 \u0634\u0627\u062e\u0635 \u062d\u0631\u0641 \u0622\u0646 \u062f\u0631 \u0645\u062c\u0645\u0648\u0639\u0647 \u06a9\u0627\u0631\u0627\u06a9\u062a\u0631\u0647\u0627\u06cc ASCII \u0627\u0633\u062a. \u0645\u0634\u0627\u0628\u0647 \u062c\u062f\u0648\u0644 \u0632\u06cc\u0631  \u0628\u0639\u062f \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc\u06a9\u0646\u0645 <a rel=\"nofollow noopener\" target=\"_blank\" href=\"https:\/\/matplotlib.org\/\">Matplotlib<\/a> \u0628\u0631\u0627\u06cc \u0631\u0633\u0645 \u0646\u0642\u0627\u0637 (\u0628\u0639\u062f \u0627\u0632 \u0627\u0639\u0645\u0627\u0644 \u062a\u0628\u062f\u06cc\u0644 \u0647\u0648\u06cc\u062a \u062a\u063a\u06cc\u06cc\u0631\u0646\u0627\u067e\u0630\u06cc\u0631) \u0628\u0631\u0627\u06cc \u0627\u0631\u0627\u0626\u0647 \u062a\u0635\u0648\u06cc\u0631\u06cc \u067e\u0627\u06cc\u0647 \u0627\u0632 \u062c\u0627\u06cc\u06cc \u06a9\u0647 \u062f\u0631 \u0622\u0646 \u0627\u06cc\u0633\u062a\u0627\u062f\u0647 \u0627\u06cc\u0645.<\/p>\n<table class=\"table\">\n<thead>\n<tr>\n<th>\u0646\u0642\u0637\u0647<\/th>\n<th>x (\u0631\u062f\u06cc\u0641)<\/th>\n<th>y (\u0633\u062a\u0648\u0646)<\/th>\n<th>\u0634\u0627\u062e\u0635 \u0622\u0633\u06a9\u06cc<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u0622<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>\u0628<\/td>\n<td>1<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>\u062c<\/td>\n<td>0<\/td>\n<td>-1<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>\u062f<\/td>\n<td>-1<\/td>\n<td>0<\/td>\n<td>3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<pre><code class=\"hljs\"><span class=\"hljs-keyword\">import<\/span> matplotlib.pyplot <span class=\"hljs-keyword\">as<\/span> plt\n<span class=\"hljs-keyword\">import<\/span> numpy <span class=\"hljs-keyword\">as<\/span> np\n<span class=\"hljs-keyword\">import<\/span> string\n\n\na, b, c, d = (<span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">0<\/span>), (<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">1<\/span>), (<span class=\"hljs-number\">0<\/span>, -<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">2<\/span>), (-<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">3<\/span>)\n\n\nA = np.array((a, b, c, d))\n\n\nI = np.eye(<span class=\"hljs-number\">3<\/span>)\n<\/code><\/pre>\n<pre><code class=\"hljs\">color_lut = <span class=\"hljs-string\">'rgbc'<\/span>\nfig = plt.figure()\nax = plt.gca()\nxs = ()\nys = ()\n<span class=\"hljs-keyword\">for<\/span> row <span class=\"hljs-keyword\">in<\/span> A:\n    output_row = I @ row\n    x, y, i = output_row\n    xs.append(x)\n    ys.append(y)\n    i = <span class=\"hljs-built_in\">int<\/span>(i) \n    c = color_lut(i)\n    plt.scatter(x, y, color=c)\n    plt.text(x + <span class=\"hljs-number\">0.15<\/span>, y, <span class=\"hljs-string\">f\"<span class=\"hljs-subst\">{string.ascii_letters(i)}<\/span>\"<\/span>)\nxs.append(xs(<span class=\"hljs-number\">0<\/span>))\nys.append(ys(<span class=\"hljs-number\">0<\/span>))\nplt.plot(xs, ys, color=<span class=\"hljs-string\">\"gray\"<\/span>, linestyle=<span class=\"hljs-string\">'dotted'<\/span>)\nax.set_xticks(np.arange(-<span class=\"hljs-number\">2.5<\/span>, <span class=\"hljs-number\">3<\/span>, <span class=\"hljs-number\">0.5<\/span>))\nax.set_yticks(np.arange(-<span class=\"hljs-number\">2.5<\/span>, <span class=\"hljs-number\">3<\/span>, <span class=\"hljs-number\">0.5<\/span>))\nplt.grid()\nplt.show()\n<\/code><\/pre>\n<p><!-- --><br \/>\n<img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/2019-03-grid1-2.png\" alt=\"png\" title=\"\"><\/p>\n<p>\u0633\u0647 \u0646\u0642\u0637\u0647 a\u060c b \u0648 c \u0631\u0633\u0645 \u0634\u062f\u0647 \u0627\u0633\u062a \u0631\u0648\u06cc \u06cc\u06a9 \u0634\u0628\u06a9\u0647 \u067e\u0633 \u0627\u0632 \u0627\u0639\u0645\u0627\u0644 \u062a\u0628\u062f\u06cc\u0644 Identity \u0628\u0647 \u0622\u0646\u0647\u0627 \u0627\u0632 \u0637\u0631\u06cc\u0642 \u06cc\u06a9 \u0645\u0627\u062a\u0631\u06cc\u0633 \u0628\u0631\u062f\u0627\u0631\u06cc \u0633\u0627\u062f\u0647 \u062d\u0627\u0635\u0644 \u0636\u0631\u0628 \u0646\u0642\u0637\u0647 \u0627\u06cc \u0622\u0646\u0647\u0627 \u0631\u0627 \u0628\u062f\u0648\u0646 \u062a\u063a\u06cc\u06cc\u0631 \u0631\u0647\u0627 \u0645\u06cc \u06a9\u0646\u062f.<\/p>\n<p>\u0627\u06a9\u0646\u0648\u0646 \u062d\u0631\u06a9\u062a \u062e\u0648\u0627\u0647\u0645 \u06a9\u0631\u062f \u0631\u0648\u06cc \u0628\u0631\u0627\u06cc \u0627\u06cc\u062c\u0627\u062f \u06cc\u06a9 \u0645\u0627\u062a\u0631\u06cc\u0633 \u062a\u063a\u06cc\u06cc\u0631 \u0645\u0642\u06cc\u0627\u0633 \\(T_s\\) \u060c \u0647\u0645\u0627\u0646\u0637\u0648\u0631 \u06a9\u0647 \u062f\u0631 \u0632\u06cc\u0631 \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0647 \u0634\u062f\u0647 \u0627\u0633\u062a\u060c \u06a9\u0647 \u0642\u0631\u0627\u0631\u06af\u06cc\u0631\u06cc \u0646\u0642\u0627\u0637 \u0631\u0627 \u062f\u0631 \u0647\u0645\u0647 \u062c\u0647\u0627\u062a \u0645\u0642\u06cc\u0627\u0633 \u0645\u06cc \u06a9\u0646\u062f.<\/p>\n<p>$$ T_s = \\begin{bmatrix} 2 &#038; 0 &#038; 0 \\\\ 0 &#038; 2 &#038; 0 \\\\ 0 &#038; 0 &#038; 1 \\end{bmatrix} $$\n<\/p>\n<p>\u062d\u0627\u0644\u0627 \u0645\u0646 \u062d\u0631\u06a9\u062a \u062e\u0648\u0627\u0647\u0645 \u06a9\u0631\u062f \u0631\u0648\u06cc \u0628\u0631\u0627\u06cc \u062a\u0631\u0633\u06cc\u0645 \u0646\u0642\u0627\u0637 \u062a\u0628\u062f\u06cc\u0644 \u0634\u062f\u0647 \u0645\u0634\u0627\u0628\u0647 \u0622\u0646\u0686\u0647 \u0628\u0627 \u0646\u0642\u0627\u0637 \u0627\u0635\u0644\u06cc \u0628\u062f\u0648\u0646 \u062a\u063a\u06cc\u06cc\u0631 \u0628\u0627 \u062a\u0628\u062f\u06cc\u0644 \u0647\u0648\u06cc\u062a \u0627\u0646\u062c\u0627\u0645 \u0634\u062f\u060c \u0627\u0645\u0627\u060c \u0627\u06cc\u0646 \u0628\u0627\u0631 \u0645\u0627\u062a\u0631\u06cc\u0633 \u062a\u0628\u062f\u06cc\u0644 \u0645\u0642\u06cc\u0627\u0633 \u0628\u0646\u062f\u06cc \u062a\u0639\u0631\u06cc\u0641 \u0634\u062f\u0647 \u062f\u0631 \u0628\u0627\u0644\u0627 \u0631\u0627 \u0627\u0639\u0645\u0627\u0644 \u0645\u06cc \u06a9\u0646\u0645.  \u0628\u0631\u0627\u06cc \u062a\u062c\u0633\u0645 \u0628\u0647\u062a\u0631\u060c \u0645\u0646 \u06cc\u06a9 \u062e\u0637 \u0646\u0642\u0637\u0647 \u0686\u06cc\u0646 \u0631\u0627 \u0631\u0633\u0645 \u0645\u06cc \u06a9\u0646\u0645 \u06a9\u0647 \u0646\u0642\u0627\u0637 \u0631\u0627 \u0628\u0647 \u0647\u0645 \u0645\u062a\u0635\u0644 \u0645\u06cc \u06a9\u0646\u062f.<\/p>\n<pre><code class=\"hljs\">\nT_s = np.array(((<span class=\"hljs-number\">2<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>), (<span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">2<\/span>, <span class=\"hljs-number\">0<\/span>), (<span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">1<\/span>)))\n\nfig = plt.figure()\nax = plt.gca()\nxs_s = ()\nys_s = ()\n<span class=\"hljs-keyword\">for<\/span> row <span class=\"hljs-keyword\">in<\/span> A:\n    output_row = T_s @ row\n    x, y, i = row\n    x_s, y_s, i_s = output_row\n    xs_s.append(x_s)\n    ys_s.append(y_s)\n    i, i_s = <span class=\"hljs-built_in\">int<\/span>(i), <span class=\"hljs-built_in\">int<\/span>(i_s) \n    c, c_s = color_lut(i), color_lut(i_s) \n    plt.scatter(x, y, color=c)\n    plt.scatter(x_s, y_s, color=c_s)\n    plt.text(x + <span class=\"hljs-number\">0.15<\/span>, y, <span class=\"hljs-string\">f\"<span class=\"hljs-subst\">{string.ascii_letters(<span class=\"hljs-built_in\">int<\/span>(i))}<\/span>\"<\/span>)\n    plt.text(x_s + <span class=\"hljs-number\">0.15<\/span>, y_s, <span class=\"hljs-string\">f\"<span class=\"hljs-subst\">{string.ascii_letters(<span class=\"hljs-built_in\">int<\/span>(i_s))}<\/span>'\"<\/span>)\n\nxs_s.append(xs_s(<span class=\"hljs-number\">0<\/span>))\nys_s.append(ys_s(<span class=\"hljs-number\">0<\/span>))\nplt.plot(xs, ys, color=<span class=\"hljs-string\">\"gray\"<\/span>, linestyle=<span class=\"hljs-string\">'dotted'<\/span>)\nplt.plot(xs_s, ys_s, color=<span class=\"hljs-string\">\"gray\"<\/span>, linestyle=<span class=\"hljs-string\">'dotted'<\/span>)\nax.set_xticks(np.arange(-<span class=\"hljs-number\">2.5<\/span>, <span class=\"hljs-number\">3<\/span>, <span class=\"hljs-number\">0.5<\/span>))\nax.set_yticks(np.arange(-<span class=\"hljs-number\">2.5<\/span>, <span class=\"hljs-number\">3<\/span>, <span class=\"hljs-number\">0.5<\/span>))\nplt.grid()\nplt.show()\n<\/code><\/pre>\n<p><!-- --><br \/>\n<img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/2019-03-grid2_2Xscale.png\" alt=\"png\" title=\"\"><\/p>\n<p>\u0627\u0632 \u0646\u0645\u0648\u062f\u0627\u0631 \u0628\u0627\u0644\u0627 \u0628\u0627\u06cc\u062f \u0628\u0633\u06cc\u0627\u0631 \u0648\u0627\u0636\u062d \u0628\u0627\u0634\u062f \u06a9\u0647 \u0627\u0628\u0639\u0627\u062f x \u0648 y \u0628\u0647 \u0633\u0627\u062f\u06af\u06cc \u0628\u0627 \u0636\u0631\u06cc\u0628 \u062f\u0648 \u0628\u0632\u0631\u06af\u200c\u062a\u0631 \u0634\u062f\u0647\u200c\u0627\u0646\u062f \u062f\u0631 \u062d\u0627\u0644\u06cc \u06a9\u0647 \u0628\u0639\u062f \u0633\u0648\u0645 \u0645\u0633\u0626\u0648\u0644 \u0634\u0627\u062e\u0635 \u062d\u0631\u0648\u0641 ASCII \u0628\u062f\u0648\u0646 \u062a\u063a\u06cc\u06cc\u0631 \u0628\u0627\u0642\u06cc \u0645\u0627\u0646\u062f\u0647 \u0627\u0633\u062a.  \u062f\u0631 \u0648\u0627\u0642\u0639\u060c \u06a9\u0633\u0627\u0646\u06cc \u06a9\u0647 \u0628\u0627 \u062c\u0628\u0631 \u0645\u0627\u062a\u0631\u06cc\u0633\u06cc \u0622\u0634\u0646\u0627 \u0647\u0633\u062a\u0646\u062f \u0645\u062a\u0648\u062c\u0647 \u062e\u0648\u0627\u0647\u0646\u062f \u0634\u062f \u06a9\u0647 \u0628\u0631\u0627\u06cc \u0647\u0645\u0647 \u062a\u0628\u062f\u06cc\u0644\u200c\u0647\u0627\u06cc \u0648\u0627\u0628\u0633\u062a\u0647 \u0641\u0647\u0631\u0633\u062a \u0634\u062f\u0647 \u062f\u0631 \u062c\u062f\u0648\u0644 \u0627\u0648\u0644\u060c \u0645\u0642\u062f\u0627\u0631 \u0646\u0634\u0627\u0646\u200c\u062f\u0627\u062f\u0647\u200c\u0634\u062f\u0647 \u062f\u0631 \u0628\u0639\u062f \u0633\u0648\u0645 \u0647\u0645\u06cc\u0634\u0647 \u0628\u062f\u0648\u0646 \u062a\u063a\u06cc\u06cc\u0631 \u0628\u0627\u0642\u06cc \u0645\u06cc\u200c\u0645\u0627\u0646\u062f\u060c \u0647\u0645\u0627\u0646\u0637\u0648\u0631 \u06a9\u0647 \u0628\u0627 \u0647\u0645\u0647 \u0635\u0641\u0631\u0647\u0627 \u0648 \u06cc\u06a9 \u0645\u0642\u062f\u0627\u0631 \u062a\u0646\u0647\u0627 \u062f\u0631 \u0634\u0627\u062e\u0635 \u0628\u0639\u062f \u0633\u0648\u0645 \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0647 \u0645\u06cc\u200c\u0634\u0648\u062f. \u0622\u062e\u0631\u06cc\u0646 \u0633\u062a\u0648\u0646<\/p>\n<p>\u062d\u0627\u0644\u0627 \u0627\u062c\u0627\u0632\u0647 \u062f\u0647\u06cc\u062f \u0631\u0648\u0634 \u062a\u0641\u0633\u06cc\u0631 \u062a\u0628\u062f\u06cc\u0644 \u0686\u0631\u062e\u0634 \u0631\u0627 \u062a\u0648\u0636\u06cc\u062d \u062f\u0647\u0645.  \u0645\u0646 \u0628\u0627 \u062d\u0644 \u062f\u0648 \u062a\u0627\u0628\u0639 \u0645\u062b\u0644\u062b\u0627\u062a\u06cc \u0628\u0631\u0627\u06cc \u0632\u0627\u0648\u06cc\u0647 \u0686\u0631\u062e\u0634 \u0645\u0648\u0631\u062f \u0646\u0638\u0631 90 \u062f\u0631\u062c\u0647 \u0634\u0631\u0648\u0639 \u0645\u06cc \u06a9\u0646\u0645\u060c \u0633\u067e\u0633 \u0622\u0646\u0647\u0627 \u0631\u0627 \u0628\u0647 \u0633\u0627\u062f\u06af\u06cc \u0628\u0647 \u0645\u0627\u062a\u0631\u06cc\u0633 \u062a\u0628\u062f\u06cc\u0644 \u0686\u0631\u062e\u0634 \u0644\u06cc\u0633\u062a \u0634\u062f\u0647 \u062f\u0631 \u062c\u062f\u0648\u0644 \u0642\u0628\u0644\u06cc \u0645\u062a\u0635\u0644 \u0645\u06cc \u06a9\u0646\u0645.<\/p>\n<p>$$<br \/>\u06af\u0646\u0627\u0647 (90^{o}) = 1<br \/>$$<\/p>\n<p>$$<br \/>cos (90^{o}) = 0<br \/>$$<\/p>\n<p>$$ T_r = \\begin{bmatrix} 0 &#038; 1 &#038; 0 \\\\ -1 &#038; 0 &#038; 0 \\\\ 0 &#038; 0 &#038; 1 \\end{bmatrix} $$\n<\/p>\n<p>\u0627\u06a9\u0646\u0648\u0646 \u062a\u0646\u0647\u0627 \u06a9\u0627\u0631\u06cc \u06a9\u0647 \u0628\u0627\u06cc\u062f \u0627\u0646\u062c\u0627\u0645 \u062f\u0647\u0645 \u0627\u06cc\u0646 \u0627\u0633\u062a \u06a9\u0647 \u0647\u0645\u0627\u0646 \u0645\u0646\u0637\u0642 \u0631\u0627 \u0628\u0631\u0627\u06cc \u062a\u0628\u062f\u06cc\u0644 \u0648 \u0631\u0633\u0645 \u0646\u0642\u0627\u0637 \u0645\u0627\u0646\u0646\u062f \u0632\u06cc\u0631 \u0627\u0639\u0645\u0627\u0644 \u06a9\u0646\u0645:<\/p>\n<pre><code class=\"hljs\">\nT_r = np.array(((<span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">0<\/span>), (-<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>), (<span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">1<\/span>)))\n\nfig = plt.figure()\nax = plt.gca()\n<span class=\"hljs-keyword\">for<\/span> row <span class=\"hljs-keyword\">in<\/span> A:\n    output_row = T_r @ row\n    x_r, y_r, i_r = output_row\n    i_r = <span class=\"hljs-built_in\">int<\/span>(i_r) \n    c_r = color_lut(i_r) \n    letter_r = string.ascii_letters(i_r)\n    plt.scatter(x_r, y_r, color=c_r)\n    plt.text(x_r + <span class=\"hljs-number\">0.15<\/span>, y_r, <span class=\"hljs-string\">f\"<span class=\"hljs-subst\">{letter_r}<\/span>'\"<\/span>)\n\nplt.plot(xs, ys, color=<span class=\"hljs-string\">\"gray\"<\/span>, linestyle=<span class=\"hljs-string\">'dotted'<\/span>)\nax.set_xticks(np.arange(-<span class=\"hljs-number\">2.5<\/span>, <span class=\"hljs-number\">3<\/span>, <span class=\"hljs-number\">0.5<\/span>))\nax.set_yticks(np.arange(-<span class=\"hljs-number\">2.5<\/span>, <span class=\"hljs-number\">3<\/span>, <span class=\"hljs-number\">0.5<\/span>))\nplt.grid()\nplt.show()\n<\/code><\/pre>\n<p><!-- --><br \/>\n<img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/2019-03-grid3_rotate.png\" alt=\"png\" title=\"\"><\/p>\n<p>\u0627\u0645\u06cc\u062f\u0648\u0627\u0631\u06cc\u0645 \u0628\u062a\u0648\u0627\u0646\u06cc\u062f \u0627\u0632 \u0637\u0631\u062d \u0645\u062a\u0648\u062c\u0647 \u0634\u0648\u06cc\u062f \u06a9\u0647 \u062a\u0645\u0627\u0645 \u0646\u0642\u0627\u0637 90 \u062f\u0631\u062c\u0647 \u062d\u0648\u0644 \u06cc\u06a9 \u0645\u062d\u0648\u0631 \u0686\u0631\u062e\u0634 \u062f\u0631 \u0645\u0628\u062f\u0627 \u0686\u0631\u062e\u06cc\u062f\u0647 \u0627\u0646\u062f.<\/p>\n<p>\u0646\u06a9\u062a\u0647 \u062f\u0642\u06cc\u0642 \u062f\u0631 \u0645\u0648\u0631\u062f \u062a\u0628\u062f\u06cc\u0644\u200c\u0647\u0627\u06cc affine \u06a9\u0647 \u0627\u0633\u0627\u0633\u0627\u064b \u062a\u0628\u062f\u06cc\u0644\u200c\u0647\u0627\u06cc \u062e\u0637\u06cc \u0647\u0633\u062a\u0646\u062f \u0627\u06cc\u0646 \u0627\u0633\u062a \u06a9\u0647 \u0645\u06cc\u200c\u062a\u0648\u0627\u0646\u06cc\u062f \u062a\u0628\u062f\u06cc\u0644\u200c\u0647\u0627 \u0631\u0627 \u062a\u0631\u06a9\u06cc\u0628 \u06a9\u0631\u062f\u0647 \u0648 \u0622\u0646\u0647\u0627 \u0631\u0627 \u062f\u0631 \u06cc\u06a9 \u0645\u0631\u062d\u0644\u0647 \u0627\u0639\u0645\u0627\u0644 \u06a9\u0646\u06cc\u062f.  \u0628\u0631\u0627\u06cc \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0646 \u0627\u06cc\u0646\u060c \u0645\u0646 \u062d\u0627\u0635\u0644 \u0636\u0631\u0628 \u0646\u0642\u0637\u0647 \u0627\u06cc (\u0636\u0631\u0628 \u0645\u0627\u062a\u0631\u06cc\u0633) \u062f\u0648 \u0645\u0627\u062a\u0631\u06cc\u0633 \u062a\u0628\u062f\u06cc\u0644 \u062e\u0648\u062f \u0631\u0627 \u0627\u0639\u0645\u0627\u0644 \u0645\u06cc \u06a9\u0646\u0645\u060c \u0645\u0627\u0646\u0646\u062f:<\/p>\n<p>$$ T_{comb} = \\begin{bmatrix} 0 &#038; 1 &#038; 0 \\\\ -1 &#038; 0 &#038; 0 \\\\ 0 &#038; 0 &#038; 1 \\end{bmatrix} * \\begin{bmatrix} 2 &#038; 0 &#038; 0 \\\\ 0 \u0648 2 \u0648 0 \\\\ 0 \u0648 0 \u0648 1 \\end{bmatrix} = \\begin{bmatrix} 0 &#038; 2 &#038; 0 \\\\ -2 &#038; 0 &#038; 0 \\\\ 0 &#038; 0 &#038; 1 \\end{bmatrix} $$\n<\/p>\n<p>\u0627\u06a9\u0646\u0648\u0646 \u0645\u06cc \u062a\u0648\u0627\u0646\u0645 \u0627\u06cc\u0646 \u0645\u0627\u062a\u0631\u06cc\u0633 \u062a\u0628\u062f\u06cc\u0644 \u062a\u0631\u06a9\u06cc\u0628\u06cc \u0631\u0627 \u0631\u0648\u06cc \u0646\u0642\u0627\u0637 \u0627\u0639\u0645\u0627\u0644 \u06a9\u0646\u0645 \u0648 \u062f\u0648\u0628\u0627\u0631\u0647 \u0622\u0646\u0647\u0627 \u0631\u0627 \u0631\u0633\u0645 \u06a9\u0646\u0645 \u062a\u0627 \u062a\u0631\u06a9\u06cc\u0628\u06cc \u0627\u0632 \u0645\u0642\u06cc\u0627\u0633 \u062f\u0648 \u0648 \u0686\u0631\u062e\u0634 90 \u062f\u0631\u062c\u0647 \u0631\u0627 \u0646\u0634\u0627\u0646 \u062f\u0647\u0645.<\/p>\n<pre><code class=\"hljs\">\nT = T_s @ T_r\n\nfig = plt.figure()\nax = plt.gca()\n\nxs_comb = ()\nys_comb = ()\n<span class=\"hljs-keyword\">for<\/span> row <span class=\"hljs-keyword\">in<\/span> A:\n    output_row = T @ row\n    x, y, i = row\n    x_comb, y_comb, i_comb = output_row\n    xs_comb.append(x_comb)\n    ys_comb.append(y_comb)\n    i, i_comb = <span class=\"hljs-built_in\">int<\/span>(i), <span class=\"hljs-built_in\">int<\/span>(i_comb) \n    c, c_comb = color_lut(i), color_lut(i_comb) \n    letter, letter_comb = string.ascii_letters(i), string.ascii_letters(i_comb)\n    plt.scatter(x, y, color=c)\n    plt.scatter(x_comb, y_comb, color=c_comb)\n    plt.text(x + <span class=\"hljs-number\">0.15<\/span> , y, <span class=\"hljs-string\">f\"<span class=\"hljs-subst\">{letter}<\/span>\"<\/span>)\n    plt.text(x_comb + <span class=\"hljs-number\">0.15<\/span>, y_comb, <span class=\"hljs-string\">f\"<span class=\"hljs-subst\">{letter_comb}<\/span>'\"<\/span>)\nxs_comb.append(xs_comb(<span class=\"hljs-number\">0<\/span>))\nys_comb.append(ys_comb(<span class=\"hljs-number\">0<\/span>))\nplt.plot(xs, ys, color=<span class=\"hljs-string\">\"gray\"<\/span>, linestyle=<span class=\"hljs-string\">'dotted'<\/span>)\nplt.plot(xs_comb, ys_comb, color=<span class=\"hljs-string\">\"gray\"<\/span>, linestyle=<span class=\"hljs-string\">'dotted'<\/span>)\nax.set_xticks(np.arange(-<span class=\"hljs-number\">2.5<\/span>, <span class=\"hljs-number\">3<\/span>, <span class=\"hljs-number\">0.5<\/span>))\nax.set_yticks(np.arange(-<span class=\"hljs-number\">2.5<\/span>, <span class=\"hljs-number\">3<\/span>, <span class=\"hljs-number\">0.5<\/span>))\nplt.grid()\nplt.show()\n<\/code><\/pre>\n<p><!-- --><br \/>\n<img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/2019-03-grid4_scale_rotate.png\" alt=\"png\" title=\"\"><\/p>\n<h2 id=\"workingwithanimage\"><span class=\"ez-toc-section\" id=\"%da%a9%d8%a7%d8%b1_%d8%a8%d8%a7_%db%8c%da%a9_%d8%aa%d8%b5%d9%88%db%8c%d8%b1\"><\/span>\u06a9\u0627\u0631 \u0628\u0627 \u06cc\u06a9 \u062a\u0635\u0648\u06cc\u0631<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u062a\u0627 \u0628\u0647 \u062d\u0627\u0644 \u0627\u0645\u06cc\u062f\u0648\u0627\u0631\u0645 \u062a\u0648\u0627\u0646\u0633\u062a\u0647 \u0628\u0627\u0634\u0645 \u0634\u0647\u0648\u062f\u06cc \u062f\u0631 \u0645\u0648\u0631\u062f \u0631\u0648\u0634 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u062a\u0628\u062f\u06cc\u0644 \u0647\u0627\u06cc \u0648\u0627\u0628\u0633\u062a\u0647 \u0628\u0631\u0627\u06cc \u062d\u0631\u06a9\u062a \u0628\u0647 \u0633\u0627\u062f\u06af\u06cc \u062f\u0631 \u0627\u0637\u0631\u0627\u0641 \u0646\u0642\u0627\u0637 \u062f\u0631 \u0641\u0636\u0627\u06cc \u062f\u0648\u0628\u0639\u062f\u06cc \u0627\u06cc\u062c\u0627\u062f \u06a9\u0631\u062f\u0647 \u0628\u0627\u0634\u0645\u060c \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0645\u06cc \u062e\u0648\u0627\u0647\u0645 \u0628\u0627 \u0628\u0631\u062e\u06cc \u0627\u0632 \u062f\u0627\u062f\u0647 \u0647\u0627\u06cc \u062a\u0635\u0648\u06cc\u0631 \u0648\u0627\u0642\u0639\u06cc \u06a9\u0627\u0631 \u0631\u0627 \u0634\u0631\u0648\u0639 \u06a9\u0646\u0645. \u0646\u0645\u0627\u06cc\u0634 \u062f\u0642\u06cc\u0642 \u062a\u0631\u06cc \u0627\u0632 \u0631\u0648\u0634 \u0639\u0645\u0644\u06a9\u0631\u062f \u0647\u0645\u0647 \u0627\u06cc\u0646\u0647\u0627 \u0627\u0631\u0627\u0626\u0647 \u062f\u0647\u06cc\u062f.<\/p>\n<p>\u0627\u06cc\u0646 \u0647\u0645\u0686\u0646\u06cc\u0646 \u0628\u0647 \u0645\u0646 \u0627\u062c\u0627\u0632\u0647 \u0645\u06cc \u062f\u0647\u062f \u062a\u0627 \u0645\u0648\u0636\u0648\u0639 \u0645\u0647\u0645 \u062f\u06cc\u06af\u0631\u06cc \u0627\u0632 \u062a\u0628\u062f\u06cc\u0644 \u0647\u0627\u06cc \u0648\u0627\u0628\u0633\u062a\u0647 \u0631\u0627 \u06a9\u0647 \u0628\u0647 \u0628\u0639\u062f \u0633\u0648\u0645 \u0645\u06cc \u067e\u0631\u062f\u0627\u0632\u062f\u060c \u067e\u0648\u0634\u0634 \u062f\u0647\u0645.  \u0628\u0639\u062f \u0633\u0648\u0645 \u062f\u0627\u062f\u0647 \u0647\u0627 \u062f\u0631 \u06cc\u06a9 \u062a\u0635\u0648\u06cc\u0631 \u0646\u0634\u0627\u0646 \u062f\u0647\u0646\u062f\u0647 \u0645\u0642\u062f\u0627\u0631 \u0648\u0627\u0642\u0639\u06cc \u067e\u06cc\u06a9\u0633\u0644 \u0627\u0633\u062a\u060c \u06cc\u0627 \u06af\u0627\u0647\u06cc \u0627\u0648\u0642\u0627\u062a \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u062f\u0627\u0645\u0646\u0647 \u0634\u062f\u062a \u0628\u0647 \u0622\u0646 \u0627\u0634\u0627\u0631\u0647 \u0645\u06cc \u0634\u0648\u062f\u060c \u062f\u0631 \u062d\u0627\u0644\u06cc \u06a9\u0647 \u0645\u06a9\u0627\u0646 \u0641\u06cc\u0632\u06cc\u06a9\u06cc \u062f\u0648 \u0628\u0639\u062f\u06cc \u067e\u06cc\u06a9\u0633\u0644 \u0647\u0627 \u062f\u0631 \u062f\u0648 \u0628\u0639\u062f \u062f\u06cc\u06af\u0631 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u062d\u0648\u0632\u0647 \u0641\u0636\u0627\u06cc\u06cc \u0646\u0627\u0645\u06cc\u062f\u0647 \u0645\u06cc \u0634\u0648\u062f.<\/p>\n<p>\u0628\u0631\u0627\u06cc \u0634\u0631\u0648\u0639 \u0645\u0646 \u06cc\u06a9 \u062a\u0635\u0648\u06cc\u0631 \u0631\u0627 \u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 matplotlib \u0645\u06cc \u062e\u0648\u0627\u0646\u0645 \u0648 \u0646\u0645\u0627\u06cc\u0634 \u0645\u06cc \u062f\u0647\u0645 \u06a9\u0647 \u0628\u0647 \u0633\u0627\u062f\u06af\u06cc \u06cc\u06a9 \u062d\u0631\u0641 \u0628\u0632\u0631\u06af \u0628\u0632\u0631\u06af R \u0627\u0633\u062a.<\/p>\n<pre><code class=\"hljs\">img = plt.imread(<span class=\"hljs-string\">'letterR.jpg'<\/span>)\nimg.shape \n<\/code><\/pre>\n<p>\u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 <code>imread(...)<\/code> \u0631\u0648\u0634\u06cc \u06a9\u0647 \u0645\u0646 \u0645\u06cc \u062a\u0648\u0627\u0646\u0645 \u062f\u0631 \u062a\u0635\u0648\u06cc\u0631 JPG \u0628\u062e\u0648\u0627\u0646\u0645 \u06a9\u0647 \u0646\u0634\u0627\u0646 \u062f\u0647\u0646\u062f\u0647 \u062d\u0631\u0641 \u0628\u0632\u0631\u06af R \u062f\u0631 NumPy \u0627\u0633\u062a <code>ndarray<\/code>.  \u0633\u067e\u0633 \u0627\u0628\u0639\u0627\u062f \u0622\u0631\u0627\u06cc\u0647 \u0631\u0627 \u06a9\u0647 1000 \u0633\u0637\u0631 \u062f\u0631 1000 \u0633\u062a\u0648\u0646 \u0627\u0633\u062a \u0646\u0645\u0627\u06cc\u0634 \u0645\u06cc \u062f\u0647\u0645 \u06a9\u0647 \u0628\u0627 \u0647\u0645 \u0645\u06a9\u0627\u0646 \u0647\u0627\u06cc 1000000 \u067e\u06cc\u06a9\u0633\u0644\u06cc \u0631\u0627 \u062f\u0631 \u062d\u0648\u0632\u0647 \u0641\u0636\u0627\u06cc\u06cc \u062a\u0634\u06a9\u06cc\u0644 \u0645\u06cc \u062f\u0647\u0646\u062f.  \u0633\u067e\u0633 \u062f\u0627\u062f\u0647\u200c\u0647\u0627\u06cc \u067e\u06cc\u06a9\u0633\u0644 \u062c\u062f\u0627\u06af\u0627\u0646\u0647 \u0628\u0647 \u0634\u06a9\u0644 \u0622\u0631\u0627\u06cc\u0647\u200c\u0627\u06cc \u0627\u0632 4 \u0639\u062f\u062f \u0635\u062d\u06cc\u062d \u0628\u062f\u0648\u0646 \u0639\u0644\u0627\u0645\u062a \u0627\u0633\u062a \u06a9\u0647 \u0646\u0634\u0627\u0646\u200c\u062f\u0647\u0646\u062f\u0647 \u06cc\u06a9 \u06a9\u0627\u0646\u0627\u0644 \u0642\u0631\u0645\u0632\u060c \u0633\u0628\u0632\u060c \u0622\u0628\u06cc \u0648 \u0622\u0644\u0641\u0627 (\u06cc\u0627 \u0646\u0645\u0648\u0646\u0647) \u0627\u0633\u062a \u06a9\u0647 \u0628\u0627 \u0647\u0645 \u062f\u0627\u062f\u0647\u200c\u0647\u0627\u06cc \u0634\u062f\u062a \u0647\u0631 \u067e\u06cc\u06a9\u0633\u0644 \u0631\u0627 \u0627\u0631\u0627\u0626\u0647 \u0645\u06cc\u200c\u06a9\u0646\u0646\u062f.<\/p>\n<pre><code class=\"hljs\">plt.figure(figsize=(<span class=\"hljs-number\">5<\/span>, <span class=\"hljs-number\">5<\/span>))\nplt.imshow(img)\n<\/code><\/pre>\n<p><!-- --><br \/>\n<img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/2019-03-letterR.png\" alt=\"png\" title=\"\"><\/p>\n<p>\u062f\u0631 \u0645\u0631\u062d\u0644\u0647 \u0628\u0639\u062f\u060c \u0645\u06cc\u200c\u062e\u0648\u0627\u0647\u0645 \u0645\u0642\u06cc\u0627\u0633 \u0648 \u0686\u0631\u062e\u0634 \u0642\u0628\u0644\u06cc \u0631\u0627 \u062f\u0631 \u062d\u0648\u0632\u0647 \u0641\u0636\u0627\u06cc\u06cc \u062f\u0627\u062f\u0647\u200c\u0647\u0627\u06cc \u062a\u0635\u0648\u06cc\u0631 \u0627\u0639\u0645\u0627\u0644 \u06a9\u0646\u0645\u060c \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0645\u06a9\u0627\u0646\u200c\u0647\u0627\u06cc \u067e\u06cc\u06a9\u0633\u0644 \u0631\u0627 \u0645\u0634\u0627\u0628\u0647 \u0622\u0646\u0686\u0647 \u0642\u0628\u0644\u0627\u064b \u0628\u0627 \u062f\u0627\u062f\u0647\u200c\u0647\u0627\u06cc \u0646\u0642\u0627\u0637 \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0645 \u062a\u063a\u06cc\u06cc\u0631 \u0645\u06cc\u200c\u062f\u0647\u062f.  \u0628\u0627 \u0627\u06cc\u0646 \u062d\u0627\u0644\u060c \u0645\u0646 \u0628\u0627\u06cc\u062f \u0631\u0648\u06cc\u06a9\u0631\u062f \u0645\u062a\u0641\u0627\u0648\u062a\u06cc \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u0645 \u0632\u06cc\u0631\u0627 \u062f\u0627\u062f\u0647 \u0647\u0627\u06cc \u062a\u0635\u0648\u06cc\u0631 \u0628\u0647 \u0631\u0648\u0634\u06cc \u0645\u062a\u0641\u0627\u0648\u062a \u0627\u0632 \u0631\u062f\u06cc\u0641 \u0647\u0627\u06cc \u0646\u0642\u0627\u0637 \u062f\u0627\u062f\u0647 \u0627\u06cc \u06a9\u0647 \u0642\u0628\u0644\u0627\u064b \u0628\u0627 \u0622\u0646\u0647\u0627 \u06a9\u0627\u0631 \u06a9\u0631\u062f\u0645 \u0633\u0627\u0632\u0645\u0627\u0646\u062f\u0647\u06cc \u0634\u062f\u0647 \u0627\u0646\u062f.  \u0628\u0627 \u062f\u0627\u062f\u0647\u200c\u0647\u0627\u06cc \u062a\u0635\u0648\u06cc\u0631\u060c \u0628\u0627\u06cc\u062f \u0634\u0627\u062e\u0635\u200c\u0647\u0627\u06cc \u0647\u0631 \u067e\u06cc\u06a9\u0633\u0644 \u0627\u0632 \u062f\u0627\u062f\u0647\u200c\u0647\u0627\u06cc \u0648\u0631\u0648\u062f\u06cc \u0631\u0627 \u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u0645\u0627\u062a\u0631\u06cc\u0633 \u062a\u0628\u062f\u06cc\u0644 T\u060c \u06a9\u0647 \u0642\u0628\u0644\u0627\u064b \u062a\u0639\u0631\u06cc\u0641 \u0634\u062f\u060c \u0628\u0647 \u0634\u0627\u062e\u0635\u200c\u0647\u0627\u06cc \u062e\u0631\u0648\u062c\u06cc \u062a\u0628\u062f\u06cc\u0644\u200c\u0634\u062f\u0647 \u0646\u06af\u0627\u0634\u062a \u06a9\u0646\u0645.<\/p>\n<pre><code class=\"hljs\">\nimg_transformed = np.empty((<span class=\"hljs-number\">2000<\/span>, <span class=\"hljs-number\">2000<\/span>, <span class=\"hljs-number\">4<\/span>), dtype=np.uint8)\n<span class=\"hljs-keyword\">for<\/span> i, row <span class=\"hljs-keyword\">in<\/span> <span class=\"hljs-built_in\">enumerate<\/span>(img):\n    <span class=\"hljs-keyword\">for<\/span> j, col <span class=\"hljs-keyword\">in<\/span> <span class=\"hljs-built_in\">enumerate<\/span>(row):\n        pixel_data = img(i, j, :)\n        input_coords = np.array((i, j, <span class=\"hljs-number\">1<\/span>))\n        i_out, j_out, _ = T @ input_coords\n        img_transformed(i_out, j_out, :) = pixel_data\n\nplt.figure(figsize=(<span class=\"hljs-number\">5<\/span>, <span class=\"hljs-number\">5<\/span>))\nplt.imshow(img_transformed)\n<\/code><\/pre>\n<p><!-- --><br \/>\n<img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/2019-03-letterR_pixelated.png\" alt=\"png\" title=\"\"><\/p>\n<p>\u0631\u0633\u0645 \u062a\u0635\u0648\u06cc\u0631 \u067e\u0633 \u0627\u0632 \u0627\u0639\u0645\u0627\u0644 \u062a\u0628\u062f\u06cc\u0644 \u0628\u0647 \u0648\u0636\u0648\u062d \u0646\u0634\u0627\u0646 \u0645\u06cc \u062f\u0647\u062f \u06a9\u0647 \u062a\u0635\u0648\u06cc\u0631 \u0627\u0635\u0644\u06cc 90 \u062f\u0631\u062c\u0647 \u062f\u0631 \u062c\u0647\u062a \u0639\u0642\u0631\u0628\u0647 \u0647\u0627\u06cc \u0633\u0627\u0639\u062a \u0686\u0631\u062e\u06cc\u062f\u0647 \u0648 2X \u0628\u0632\u0631\u06af \u0634\u062f\u0647 \u0627\u0633\u062a.  \u0628\u0627 \u0627\u06cc\u0646 \u062d\u0627\u0644\u060c \u0646\u062a\u06cc\u062c\u0647 \u0627\u06a9\u0646\u0648\u0646 \u0628\u0647 \u0648\u0636\u0648\u062d \u06a9\u0627\u0647\u0634 \u06cc\u0627\u0641\u062a\u0647 \u0627\u0633\u062a \u0632\u06cc\u0631\u0627 \u0628\u0647 \u0631\u0627\u062d\u062a\u06cc \u0645\u06cc \u062a\u0648\u0627\u0646\u06cc\u062f \u0646\u0627\u067e\u06cc\u0648\u0633\u062a\u06af\u06cc \u062f\u0631 \u0634\u062f\u062a \u067e\u06cc\u06a9\u0633\u0644 \u0647\u0627 \u0631\u0627 \u0645\u0634\u0627\u0647\u062f\u0647 \u06a9\u0646\u06cc\u062f.<\/p>\n<p>\u0628\u0631\u0627\u06cc \u062f\u0631\u06a9 \u062f\u0644\u06cc\u0644 \u0627\u06cc\u0646 \u0645\u0648\u0636\u0648\u0639\u060c \u062f\u0648\u0628\u0627\u0631\u0647 \u0627\u0632 \u06cc\u06a9 \u0646\u0645\u0648\u062f\u0627\u0631 \u0634\u0628\u06a9\u0647 \u0633\u0627\u062f\u0647 \u0628\u0631\u0627\u06cc \u0646\u0645\u0627\u06cc\u0634 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u06a9\u0646\u0645.  \u0637\u0631\u062d\u06cc \u0627\u0632 4 \u0645\u0631\u0628\u0639 \u0631\u0627 \u062f\u0631 \u06cc\u06a9 \u0634\u0628\u06a9\u0647 2&#215;2 \u0645\u0634\u0627\u0628\u0647 \u062d\u0648\u0632\u0647 \u0641\u0636\u0627\u06cc\u06cc \u06cc\u06a9 \u062a\u0635\u0648\u06cc\u0631 2&#215;2 \u062f\u0631 \u0646\u0638\u0631 \u0628\u06af\u06cc\u0631\u06cc\u062f.<\/p>\n<pre><code class=\"hljs\"><span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">plot_box<\/span>(<span class=\"hljs-params\">plt, x0, y0, txt, w=<span class=\"hljs-number\">1<\/span>, h=<span class=\"hljs-number\">1<\/span><\/span>):<\/span>\n    plt.scatter(x0, y0)\n    plt.scatter(x0, y0 + h)\n    plt.scatter(x0 + w, y0 + h)\n    plt.scatter(x0 + w, y0)\n    plt.plot((x0, x0, x0 + w, x0 + w, x0), (y0, y0 + h, y0 + h, y0, y0), color=<span class=\"hljs-string\">\"gray\"<\/span>, linestyle=<span class=\"hljs-string\">'dotted'<\/span>)\n    plt.text(x0 + (<span class=\"hljs-number\">.33<\/span> * w), y0 + (<span class=\"hljs-number\">.5<\/span> * h), txt)\n\n\na = np.array((<span class=\"hljs-number\">0<\/span>,  <span class=\"hljs-number\">1<\/span>,  <span class=\"hljs-number\">0<\/span>))\nb = np.array((<span class=\"hljs-number\">1<\/span>,  <span class=\"hljs-number\">1<\/span>,  <span class=\"hljs-number\">1<\/span>))\nc = np.array((<span class=\"hljs-number\">0<\/span>,  <span class=\"hljs-number\">0<\/span>,  <span class=\"hljs-number\">2<\/span>))\nd = np.array((<span class=\"hljs-number\">1<\/span>,  <span class=\"hljs-number\">0<\/span>,  <span class=\"hljs-number\">3<\/span>))\n\nA = np.array((a, b, c, d))\nfig = plt.figure()\nax = plt.gca()\n<span class=\"hljs-keyword\">for<\/span> pt <span class=\"hljs-keyword\">in<\/span> A:\n    x0, y0, i = I @ pt\n    x0, y0, i = <span class=\"hljs-built_in\">int<\/span>(x0), <span class=\"hljs-built_in\">int<\/span>(y0), <span class=\"hljs-built_in\">int<\/span>(i)\n    plot_box(plt, x0, y0, <span class=\"hljs-string\">f\"<span class=\"hljs-subst\">{string.ascii_letters(<span class=\"hljs-built_in\">int<\/span>(i))}<\/span> (<span class=\"hljs-subst\">{x0}<\/span>, <span class=\"hljs-subst\">{y0}<\/span>)\"<\/span>)\n\nax.set_xticks(np.arange(-<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">5<\/span>, <span class=\"hljs-number\">1<\/span>))\nax.set_yticks(np.arange(-<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">5<\/span>, <span class=\"hljs-number\">1<\/span>))\nplt.grid()\nplt.show()\n<\/code><\/pre>\n<p><!-- --><br \/>\n<img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/2019-03-grid5_2x2.png\" alt=\"png\" title=\"\"><\/p>\n<p>\u062d\u0627\u0644\u0627 \u0628\u0628\u06cc\u0646\u06cc\u062f \u0686\u0647 \u0627\u062a\u0641\u0627\u0642\u06cc \u0645\u06cc\u200c\u0627\u0641\u062a\u062f \u0648\u0642\u062a\u06cc \u06cc\u06a9 \u062a\u0628\u062f\u06cc\u0644 \u0645\u0642\u06cc\u0627\u0633\u200c\u0628\u0646\u062f\u06cc 2X \u0631\u0627 \u0647\u0645\u0627\u0646\u0637\u0648\u0631 \u06a9\u0647 \u062f\u0631 \u0632\u06cc\u0631 \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0647 \u0634\u062f\u0647 \u0627\u0639\u0645\u0627\u0644 \u0645\u06cc\u200c\u06a9\u0646\u0645.  \u0628\u0647 \u06cc\u0627\u062f \u0628\u06cc\u0627\u0648\u0631\u06cc\u062f \u06a9\u0647:<\/p>\n<p>$$ T_s = \\begin{bmatrix} 2 &#038; 0 &#038; 0 \\\\ 0 &#038; 2 &#038; 0 \\\\ 0 &#038; 0 &#038; 1 \\end{bmatrix} $$\n<\/p>\n<p>\u0645\u062a\u0648\u062c\u0647 \u062e\u0648\u0627\u0647\u06cc\u062f \u0634\u062f \u06a9\u0647 \u0686\u0646\u06cc\u0646 \u062f\u06af\u0631\u06af\u0648\u0646\u06cc \u0641\u0636\u0627\u06cc\u06cc \u0645\u0646\u062c\u0631 \u0628\u0647 &#8230; \u062e\u0628\u060c \u0628\u0647 \u0628\u06cc\u0627\u0646 \u0633\u0627\u062f\u0647\u060c &#8220;\u0634\u06a9\u0627\u0641&#8221; \u0645\u06cc \u0634\u0648\u062f\u060c \u06a9\u0647 \u0645\u0646 \u0628\u0627 \u062a\u0631\u0633\u06cc\u0645 \u0639\u0644\u0627\u0645\u062a \u0647\u0627\u06cc \u0633\u0648\u0627\u0644 \u0647\u0645\u0631\u0627\u0647 \u0628\u0627 \u0645\u062e\u062a\u0635\u0627\u062a \u0622\u0634\u06a9\u0627\u0631 \u06a9\u0631\u062f\u0647 \u0627\u0645.  \u0634\u0628\u06a9\u0647 2&#215;2 \u0628\u0647 \u06cc\u06a9 \u0634\u0628\u06a9\u0647 3&#215;3 \u062a\u0628\u062f\u06cc\u0644 \u0645\u06cc \u0634\u0648\u062f \u06a9\u0647 \u0645\u0631\u0628\u0639 \u0647\u0627\u06cc \u0627\u0635\u0644\u06cc \u0628\u0631 \u0627\u0633\u0627\u0633 \u062a\u063a\u06cc\u06cc\u0631 \u0645\u06a9\u0627\u0646 \u0645\u06cc \u06cc\u0627\u0628\u0646\u062f. \u0631\u0648\u06cc \u062a\u0628\u062f\u06cc\u0644 \u062e\u0637\u06cc \u0627\u0639\u0645\u0627\u0644 \u0634\u062f\u0647  \u0627\u06cc\u0646 \u0628\u062f\u0627\u0646 \u0645\u0639\u0646\u06cc \u0627\u0633\u062a \u06a9\u0647 (0,0) * \\(T_s\\) (0,0) \u0628\u0647 \u062f\u0644\u06cc\u0644 \u0648\u06cc\u0698\u06af\u06cc\u200c\u0647\u0627\u06cc \u0622\u0646 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u06cc\u06a9 \u0628\u0631\u062f\u0627\u0631 0 \u0628\u0627\u0642\u06cc \u0645\u06cc\u200c\u0645\u0627\u0646\u062f\u060c \u0627\u0645\u0627 \u0628\u0642\u06cc\u0647 \u0628\u0627 \u062f\u0648 \u0645\u0642\u06cc\u0627\u0633 \u0645\u06cc\u200c\u0634\u0648\u0646\u062f\u060c \u0645\u0627\u0646\u0646\u062f (1,1) * \\(T_s\\) -> (2\u060c2).<\/p>\n<pre><code class=\"hljs\">fig = plt.figure()\nax = plt.gca()\n<span class=\"hljs-keyword\">for<\/span> pt <span class=\"hljs-keyword\">in<\/span> A:\n    xt, yt, i = T_s @ pt\n    xt, yt, i = <span class=\"hljs-built_in\">int<\/span>(xt), <span class=\"hljs-built_in\">int<\/span>(yt), <span class=\"hljs-built_in\">int<\/span>(i)\n    plot_box(plt, xt, yt, <span class=\"hljs-string\">f\"<span class=\"hljs-subst\">{string.ascii_letters(i)}<\/span>' (<span class=\"hljs-subst\">{xt}<\/span>, <span class=\"hljs-subst\">{yt}<\/span>)\"<\/span>)\n\ndelta_w, delta_h = <span class=\"hljs-number\">0.33<\/span>, <span class=\"hljs-number\">0.5<\/span>\nplt.text(<span class=\"hljs-number\">0<\/span> + delta_w, <span class=\"hljs-number\">1<\/span> + delta_h, <span class=\"hljs-string\">\"? (0, 1)\"<\/span>)\nplt.text(<span class=\"hljs-number\">1<\/span> + delta_w, <span class=\"hljs-number\">0<\/span> + delta_h, <span class=\"hljs-string\">\"? (1, 0)\"<\/span>)\nplt.text(<span class=\"hljs-number\">1<\/span> + delta_w, <span class=\"hljs-number\">1<\/span> + delta_h, <span class=\"hljs-string\">\"? (1, 1)\"<\/span>)\nplt.text(<span class=\"hljs-number\">1<\/span> + delta_w, <span class=\"hljs-number\">2<\/span> + delta_h, <span class=\"hljs-string\">\"? (1, 2)\"<\/span>)\nplt.text(<span class=\"hljs-number\">2<\/span> + delta_w, <span class=\"hljs-number\">1<\/span> + delta_h, <span class=\"hljs-string\">\"? (2, 1)\"<\/span>)\n\nax.set_xticks(np.arange(-<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">5<\/span>, <span class=\"hljs-number\">1<\/span>))\nax.set_yticks(np.arange(-<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">5<\/span>, <span class=\"hljs-number\">1<\/span>))\nplt.grid()\nplt.show()\n<\/code><\/pre>\n<p><!-- --><br \/>\n<img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/2019-03-grid6_3x3_gappy.png\" alt=\"png\" title=\"\"><\/p>\n<p>\u0627\u06cc\u0646 \u0633\u0648\u0627\u0644 \u0628\u0627\u0642\u06cc \u0645\u06cc \u0645\u0627\u0646\u062f \u06a9\u0647 \u0628\u0627 \u0622\u0646 \u0634\u06a9\u0627\u0641 \u0647\u0627\u06cc\u06cc \u06a9\u0647 \u0645\u0639\u0631\u0641\u06cc \u0634\u062f\u0647 \u0627\u0646\u062f \u0686\u0647 \u0628\u0627\u06cc\u062f \u06a9\u0631\u062f\u061f  \u06cc\u06a9 \u0641\u06a9\u0631 \u0634\u0647\u0648\u062f\u06cc \u0627\u06cc\u0646 \u0627\u0633\u062a \u06a9\u0647 \u0628\u0647 \u0633\u0627\u062f\u06af\u06cc \u0628\u0647 \u062a\u0635\u0648\u06cc\u0631 \u0627\u0635\u0644\u06cc \u0628\u0631\u0627\u06cc \u067e\u0627\u0633\u062e \u0646\u06af\u0627\u0647 \u06a9\u0646\u06cc\u0645.  \u0627\u062a\u0641\u0627\u0642\u0627\u064b \u0627\u06af\u0631 \u0645\u0639\u06a9\u0648\u0633 \u062a\u0628\u062f\u06cc\u0644 \u0631\u0627 \u0628\u0647 \u0645\u062e\u062a\u0635\u0627\u062a\u06cc \u062f\u0631 \u062e\u0631\u0648\u062c\u06cc \u0627\u0639\u0645\u0627\u0644 \u06a9\u0646\u06cc\u0645\u060c \u0645\u062d\u0644 \u0645\u0631\u0628\u0648\u0637 \u0628\u0647 \u0648\u0631\u0648\u062f\u06cc \u0627\u0635\u0644\u06cc \u0631\u0627 \u0628\u0647 \u062f\u0633\u062a \u0645\u06cc\u200c\u0622\u0648\u0631\u0645.<\/p>\n<p>\u062f\u0631 \u0639\u0645\u0644\u06cc\u0627\u062a \u0645\u0627\u062a\u0631\u06cc\u0633\u06cc \u0645\u0627\u0646\u0646\u062f \u0646\u06af\u0627\u0634\u062a \u0628\u0647 \u0639\u0642\u0628 \u0628\u0647 \u0635\u0648\u0631\u062a \u0632\u06cc\u0631 \u0627\u0633\u062a:<\/p>\n<p>$$ (x, y, 1) = T_s^{-1} * (x&#8217; y&#8217; 1) $$\n<\/p>\n<p>\u06a9\u0647 \u062f\u0631 \u0622\u0646 x&#8217;\u060c y&#8217; \u0645\u062e\u062a\u0635\u0627\u062a \u062f\u0631 \u0634\u0628\u06a9\u0647 3&#215;3 \u062a\u0628\u062f\u06cc\u0644 \u0634\u062f\u0647 \u0641\u0648\u0642 \u0647\u0633\u062a\u0646\u062f\u060c \u0628\u0647 \u0637\u0648\u0631 \u062e\u0627\u0635 \u0645\u06a9\u0627\u0646 \u06af\u0645\u0634\u062f\u0647\u060c \u0645\u0627\u0646\u0646\u062f (2\u060c 1)\u060c \\(T_s^{-1}\\) (\u0645\u0642\u0627\u062f\u06cc\u0631 \u0648\u0627\u0642\u0639\u06cc \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0647 \u0634\u062f\u0647 \u062f\u0631 \u0632\u06cc\u0631) \u0645\u0639\u06a9\u0648\u0633 \u0627\u0633\u062a \u0645\u0627\u062a\u0631\u06cc\u0633 \u0645\u0642\u06cc\u0627\u0633 2x \\(T_s\\) \u0648 x, y \u0645\u062e\u062a\u0635\u0627\u062a\u06cc \u0647\u0633\u062a\u0646\u062f \u06a9\u0647 \u062f\u0631 \u0634\u0628\u06a9\u0647 \u0627\u0635\u0644\u06cc 2&#215;2 \u06cc\u0627\u0641\u062a \u0645\u06cc \u0634\u0648\u0646\u062f.<\/p>\n<p>$$ T_s^{-1} = \\begin{bmatrix} 1\/2 &#038; 0 &#038; 0 \\\\ 0 &#038; 1\/2 &#038; 0 \\\\ 0 &#038; 0 &#038; 1 \\end{bmatrix}^{-1} $$\n<\/p>\n<p>\u0628\u0627 \u0627\u06cc\u0646 \u062d\u0627\u0644\u060c \u0628\u0647 \u0632\u0648\u062f\u06cc \u0645\u062a\u0648\u062c\u0647 \u062e\u0648\u0627\u0647\u06cc\u062f \u0634\u062f \u06a9\u0647 \u0645\u0634\u06a9\u0644\u06cc \u0648\u062c\u0648\u062f \u062f\u0627\u0631\u062f \u06a9\u0647 \u0647\u0646\u0648\u0632 \u0628\u0627\u06cc\u062f \u0628\u0631\u0637\u0631\u0641 \u0634\u0648\u062f\u060c \u0632\u06cc\u0631\u0627 \u0647\u0631 \u06cc\u06a9 \u0627\u0632 \u0645\u062e\u062a\u0635\u0627\u062a \u0634\u06a9\u0627\u0641 \u0628\u0647 \u0645\u0642\u0627\u062f\u06cc\u0631 \u06a9\u0633\u0631\u06cc \u0633\u06cc\u0633\u062a\u0645 \u0645\u062e\u062a\u0635\u0627\u062a 2&#215;2 \u0628\u0631\u0645\u06cc \u06af\u0631\u062f\u062f.  \u062f\u0631 \u0645\u0648\u0631\u062f \u062f\u0627\u062f\u0647 \u0647\u0627\u06cc \u062a\u0635\u0648\u06cc\u0631\u060c \u0634\u0645\u0627 \u0648\u0627\u0642\u0639\u0627 \u0646\u0645\u06cc \u062a\u0648\u0627\u0646\u06cc\u062f \u06a9\u0633\u0631\u06cc \u0627\u0632 \u067e\u06cc\u06a9\u0633\u0644 \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u06cc\u062f.  \u0627\u06cc\u0646 \u0628\u0627 \u0645\u062b\u0627\u0644\u06cc \u0627\u0632 \u0646\u06af\u0627\u0634\u062a \u0634\u06a9\u0627\u0641 (2\u060c 1) \u0628\u0647 \u0641\u0636\u0627\u06cc \u0627\u0635\u0644\u06cc 2&#215;2 \u0648\u0627\u0636\u062d \u062a\u0631 \u062e\u0648\u0627\u0647\u062f \u0628\u0648\u062f\u060c \u0645\u0627\u0646\u0646\u062f:<\/p>\n<p>$$ T_s^{-1} * (2\u060c 1\u060c 1) = (1\u060c 1\/2\u060c 1) $$\n<\/p>\n<p>\u062f\u0631 \u0627\u06cc\u0646 \u062d\u0627\u0644\u062a \u0645\u0646 y&#8217; = 1\/2 \u0631\u0627 \u0628\u0647 0 \u06af\u0631\u062f \u0645\u06cc \u06a9\u0646\u0645 \u0648 \u0645\u06cc \u06af\u0648\u06cc\u0645 \u06a9\u0647 \u0628\u0647 (1, 0) \u0646\u06af\u0627\u0634\u062a \u0645\u06cc \u0634\u0648\u062f.  \u062f\u0631 \u0645\u0641\u0647\u0648\u0645 \u06a9\u0644\u06cc\u060c \u0627\u06cc\u0646 \u0631\u0648\u0634 \u0627\u0646\u062a\u062e\u0627\u0628 \u06cc\u06a9 \u0645\u0642\u062f\u0627\u0631 \u062f\u0631 \u0634\u0628\u06a9\u0647 \u0627\u0635\u0644\u06cc 2&#215;2 \u0628\u0631\u0627\u06cc \u0642\u0631\u0627\u0631 \u062f\u0627\u062f\u0646 \u062f\u0631 \u0634\u06a9\u0627\u0641 \u0647\u0627\u06cc \u0634\u0628\u06a9\u0647 3&#215;3 \u062a\u0628\u062f\u06cc\u0644 \u0634\u062f\u0647 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u062f\u0631\u0648\u0646 \u06cc\u0627\u0628\u06cc \u0634\u0646\u0627\u062e\u062a\u0647 \u0645\u06cc \u0634\u0648\u062f\u060c \u0648 \u062f\u0631 \u0627\u06cc\u0646 \u0645\u062b\u0627\u0644 \u062e\u0627\u0635 \u0645\u0646 \u0627\u0632 \u06cc\u06a9 \u0646\u0633\u062e\u0647 \u0633\u0627\u062f\u0647 \u0634\u062f\u0647 \u0627\u0632 \u0631\u0648\u0634 \u062f\u0631\u0648\u0646\u06cc\u0627\u0628\u06cc \u0646\u0632\u062f\u06cc\u06a9\u062a\u0631\u06cc\u0646 \u0647\u0645\u0633\u0627\u06cc\u0647 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u06a9\u0646\u0645.<\/p>\n<p>\u062e\u0648\u0628\u060c \u062d\u0627\u0644\u0627 \u0628\u0647 \u062f\u0627\u062f\u0647 \u0647\u0627\u06cc \u062a\u0635\u0648\u06cc\u0631 \u0628\u0631\u06af\u0631\u062f\u06cc\u0645.  \u0628\u0627\u06cc\u062f \u06a9\u0627\u0645\u0644\u0627\u064b \u0648\u0627\u0636\u062d \u0628\u0627\u0634\u062f \u06a9\u0647 \u0627\u06a9\u0646\u0648\u0646 \u0628\u0631\u0627\u06cc \u0631\u0641\u0639 \u0627\u06cc\u0646 \u0634\u06a9\u0627\u0641 \u0647\u0627 \u062f\u0631 \u0646\u0633\u062e\u0647 \u0645\u0642\u06cc\u0627\u0633 \u0634\u062f\u0647 \u0648 \u0686\u0631\u062e\u0627\u0646\u062f\u0647 \u0634\u062f\u0647 \u062d\u0631\u0641 R \u0686\u0647 \u0628\u0627\u06cc\u062f \u06a9\u0631\u062f. \u0645\u062e\u062a\u0635\u0627\u062a \u067e\u06cc\u06a9\u0633\u0644 \u062f\u0631 \u062a\u0635\u0648\u06cc\u0631 \u062a\u0628\u062f\u06cc\u0644 \u0634\u062f\u0647 \u0628\u0631\u0627\u06cc \u06cc\u0627\u0641\u062a\u0646 \u0645\u0637\u0627\u0628\u0642\u062a \u062f\u0642\u06cc\u0642 \u06cc\u0627 \u0646\u0632\u062f\u06cc\u06a9\u062a\u0631\u06cc\u0646 \u0647\u0645\u0633\u0627\u06cc\u0647 \u062f\u0631 \u062a\u0635\u0648\u06cc\u0631 \u0627\u0635\u0644\u06cc.<\/p>\n<pre><code class=\"hljs\">T_inv = np.linalg.inv(T)\n\n\n<span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">nearest_neighbors<\/span>(<span class=\"hljs-params\">i, j, M, T_inv<\/span>):<\/span>\n    x_max, y_max = M.shape(<span class=\"hljs-number\">0<\/span>) - <span class=\"hljs-number\">1<\/span>, M.shape(<span class=\"hljs-number\">1<\/span>) - <span class=\"hljs-number\">1<\/span>\n    x, y, _ = T_inv @ np.array((i, j, <span class=\"hljs-number\">1<\/span>))\n    <span class=\"hljs-keyword\">if<\/span> np.floor(x) == x <span class=\"hljs-keyword\">and<\/span> np.floor(y) == y:\n        x, y = <span class=\"hljs-built_in\">int<\/span>(x), <span class=\"hljs-built_in\">int<\/span>(y)\n        <span class=\"hljs-keyword\">return<\/span> M(x, y)\n    <span class=\"hljs-keyword\">if<\/span> np.<span class=\"hljs-built_in\">abs<\/span>(np.floor(x) - x) &lt; np.<span class=\"hljs-built_in\">abs<\/span>(np.ceil(x) - x):\n        x = <span class=\"hljs-built_in\">int<\/span>(np.floor(x))\n    <span class=\"hljs-keyword\">else<\/span>:\n        x = <span class=\"hljs-built_in\">int<\/span>(np.ceil(x))\n    <span class=\"hljs-keyword\">if<\/span> np.<span class=\"hljs-built_in\">abs<\/span>(np.floor(y) - y) &lt; np.<span class=\"hljs-built_in\">abs<\/span>(np.ceil(y) - y):\n        y = <span class=\"hljs-built_in\">int<\/span>(np.floor(y))\n    <span class=\"hljs-keyword\">else<\/span>:\n        y = <span class=\"hljs-built_in\">int<\/span>(np.ceil(y))\n    <span class=\"hljs-keyword\">if<\/span> x &gt; x_max:\n        x = x_max\n    <span class=\"hljs-keyword\">if<\/span> y &gt; y_max:\n        y = y_max\n    <span class=\"hljs-keyword\">return<\/span> M(x, y,)\n\nimg_nn = np.empty((<span class=\"hljs-number\">2000<\/span>, <span class=\"hljs-number\">2000<\/span>, <span class=\"hljs-number\">4<\/span>), dtype=np.uint8)\n<span class=\"hljs-keyword\">for<\/span> i, row <span class=\"hljs-keyword\">in<\/span> <span class=\"hljs-built_in\">enumerate<\/span>(img_transformed):\n    <span class=\"hljs-keyword\">for<\/span> j, col <span class=\"hljs-keyword\">in<\/span> <span class=\"hljs-built_in\">enumerate<\/span>(row):\n        img_nn(i, j, :) = nearest_neighbors(i, j, img, T_inv)\n\nplt.figure(figsize=(<span class=\"hljs-number\">5<\/span>, <span class=\"hljs-number\">5<\/span>))\nplt.imshow(img_nn)\n<\/code><\/pre>\n<p><!-- --><br \/>\n<img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/2019-03-letterR_numpy_transform.png\" alt=\"png\" title=\"\"><\/p>\n<p>\u062e\u06cc\u0644\u06cc \u06a9\u062b\u06cc\u0641 \u0646\u06cc\u0633\u062a \u062f\u0631\u0633\u062a\u0647\u061f<\/p>\n<p>\u0628\u0627\u06cc\u062f \u062a\u0648\u062c\u0647 \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u0645 \u06a9\u0647 \u062f\u0631 \u0627\u06a9\u062b\u0631 \u0645\u0648\u0627\u0631\u062f \u0631\u0648\u0634 \u0646\u0632\u062f\u06cc\u06a9\u062a\u0631\u06cc\u0646 \u0647\u0645\u0633\u0627\u06cc\u0647 \u06a9\u0627\u0641\u06cc \u0646\u062e\u0648\u0627\u0647\u062f \u0628\u0648\u062f.  \u062f\u0648 \u0631\u0648\u0634 \u0645\u062a\u062f\u0627\u0648\u0644 \u062f\u06cc\u06af\u0631 \u062f\u0631\u0648\u0646 \u06cc\u0627\u0628\u06cc \u0648\u062c\u0648\u062f \u062f\u0627\u0631\u062f \u06a9\u0647 \u0628\u0647 \u0646\u0627\u0645 \u0647\u0627\u06cc \u062f\u0631\u0648\u0646 \u06cc\u0627\u0628\u06cc \u062f\u0648 \u062e\u0637\u06cc \u0648 \u062f\u0648 \u0645\u06a9\u0639\u0628\u06cc \u0634\u0646\u0627\u062e\u062a\u0647 \u0645\u06cc \u0634\u0648\u0646\u062f \u06a9\u0647 \u0628\u0647 \u0637\u0648\u0631 \u06a9\u0644\u06cc \u0646\u062a\u0627\u06cc\u062c \u0628\u0633\u06cc\u0627\u0631 \u0628\u0647\u062a\u0631\u06cc \u0627\u0631\u0627\u0626\u0647 \u0645\u06cc \u062f\u0647\u0646\u062f.  \u0647\u0646\u06af\u0627\u0645 \u0645\u0639\u0631\u0641\u06cc \u06a9\u062a\u0627\u0628\u062e\u0627\u0646\u0647 \u0647\u0627\u06cc Pillow \u0648 OpenCV \u062f\u0631 \u0628\u062e\u0634 \u0647\u0627\u06cc \u0628\u0639\u062f\u06cc\u060c \u062f\u0631 \u0645\u0648\u0631\u062f \u0627\u06cc\u0646 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0647\u0627\u06cc \u062f\u0631\u0648\u0646 \u06cc\u0627\u0628\u06cc \u062f\u06cc\u06af\u0631 \u0628\u06cc\u0634\u062a\u0631 \u0635\u062d\u0628\u062a \u062e\u0648\u0627\u0647\u0645 \u06a9\u0631\u062f.  \u0647\u062f\u0641 \u0627\u06cc\u0646 \u0628\u062e\u0634 \u0641\u0642\u0637 \u0627\u06cc\u062c\u0627\u062f \u06cc\u06a9 \u062f\u0631\u06a9 \u0634\u0647\u0648\u062f\u06cc \u0627\u0632 \u0631\u0648\u0634 \u06a9\u0627\u0631 \u06a9\u0631\u062f\u0646 \u0686\u06cc\u0632\u0647\u0627 \u0627\u0633\u062a.<\/p>\n<h2 id=\"affinetransformationswithpillow\"><span class=\"ez-toc-section\" id=\"%d8%aa%d8%a8%d8%af%db%8c%d9%84_%d8%a2%d9%81%db%8c%d9%86_%d8%a8%d8%a7_%d8%a8%d8%a7%d9%84%d8%b4\"><\/span>\u062a\u0628\u062f\u06cc\u0644 \u0622\u0641\u06cc\u0646 \u0628\u0627 \u0628\u0627\u0644\u0634<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u062f\u0631 \u0627\u06cc\u0646 \u0628\u062e\u0634 \u0628\u0647 \u0637\u0648\u0631 \u0645\u062e\u062a\u0635\u0631 \u0631\u0648\u0634 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u06a9\u062a\u0627\u0628\u062e\u0627\u0646\u0647 \u067e\u0631\u062f\u0627\u0632\u0634 \u062a\u0635\u0648\u06cc\u0631 \u0639\u0627\u0644\u06cc \u067e\u0627\u06cc\u062a\u0648\u0646 Pillow \u0631\u0627 \u0628\u0631\u0627\u06cc \u0627\u0646\u062c\u0627\u0645 \u062a\u0628\u062f\u06cc\u0644\u200c\u0647\u0627\u06cc \u0648\u0627\u0628\u0633\u062a\u0647 \u062a\u0648\u0636\u06cc\u062d \u062e\u0648\u0627\u0647\u0645 \u062f\u0627\u062f.<\/p>\n<p>\u0627\u0648\u0644 \u0627\u0632 \u0647\u0645\u0647\u060c Pillow \u0628\u0627\u06cc\u062f \u0646\u0635\u0628 \u0634\u0648\u062f.  \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u06a9\u0631\u062f\u0645 pip \u0628\u0631\u0627\u06cc \u0627\u0646\u062c\u0627\u0645 \u0627\u06cc\u0646 \u06a9\u0627\u0631\u060c \u0645\u0627\u0646\u0646\u062f \u0627\u06cc\u0646 \u0645\u0648\u0627\u0631\u062f:<\/p>\n<pre><code class=\"hljs\"><span class=\"hljs-meta\">$<\/span><span class=\"bash\"> pip install pillow<\/span>\n<\/code><\/pre>\n<p>\u0627\u06a9\u0646\u0648\u0646 \u0627\u0648\u0644\u06cc\u0646 \u0642\u062f\u0645 \u0627\u06cc\u0646 \u0627\u0633\u062a \u06a9\u0647 import \u0631\u0627 <code>Image<\/code> \u06a9\u0644\u0627\u0633 \u0627\u0632 \u0645\u0627\u0698\u0648\u0644 PIL (PIL \u0646\u0627\u0645 \u0645\u0627\u0698\u0648\u0644 \u067e\u0627\u06cc\u062a\u0648\u0646 \u0645\u0631\u062a\u0628\u0637 \u0628\u0627 Pillow \u0627\u0633\u062a) \u0648 \u062f\u0631 \u062a\u0635\u0648\u06cc\u0631 \u0645\u0646 \u0628\u062e\u0648\u0627\u0646\u06cc\u062f.<\/p>\n<pre><code class=\"hljs\"><span class=\"hljs-keyword\">from<\/span> PIL <span class=\"hljs-keyword\">import<\/span> Image\n<\/code><\/pre>\n<p>\u0628\u0631\u0627\u06cc \u062e\u0648\u0627\u0646\u062f\u0646 \u062f\u0631 \u0646\u0627\u0645 \u0641\u0627\u06cc\u0644 \u062a\u0635\u0648\u06cc\u0631\u06cc \u0646\u0645\u0648\u0646\u0647 <code>letterR.jpg<\/code> \u0645\u0646 \u0631\u0648\u0634 \u06a9\u0644\u0627\u0633 \u0631\u0627 \u0635\u062f\u0627 \u0645\u06cc \u0632\u0646\u0645 <code>Image.open(...)<\/code>\u060c \u0628\u0627 \u0627\u0631\u0633\u0627\u0644 \u0646\u0627\u0645 \u0641\u0627\u06cc\u0644 \u0628\u0647 \u0622\u0646\u060c \u06a9\u0647 \u0646\u0645\u0648\u0646\u0647 \u0627\u06cc \u0627\u0632 \u0641\u0627\u06cc\u0644 \u0631\u0627 \u0628\u0631\u0645\u06cc \u06af\u0631\u062f\u0627\u0646\u062f <code>Image<\/code> \u06a9\u0644\u0627\u0633\u060c \u06a9\u0647 \u0633\u067e\u0633 \u0622\u0646 \u0631\u0627 \u0628\u0647 \u06cc\u06a9 \u0622\u0631\u0627\u06cc\u0647 numpy \u062a\u0628\u062f\u06cc\u0644 \u0645\u06cc \u06a9\u0646\u0645 \u0648 \u0628\u0627 matplotlib \u0646\u0645\u0627\u06cc\u0634 \u0645\u06cc \u062f\u0647\u0645.<\/p>\n<pre><code class=\"hljs\">img = Image.<span class=\"hljs-built_in\">open<\/span>(<span class=\"hljs-string\">'letterR.jpg'<\/span>)\nplt.figure(figsize=(<span class=\"hljs-number\">5<\/span>, <span class=\"hljs-number\">5<\/span>))\nplt.imshow(np.asarray(img))\n<\/code><\/pre>\n<p><!-- --><br \/>\n<img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/2019-03-letterR_pillow.png\" alt=\"png\" title=\"\"><\/p>\n<p>\u0628\u0627\u0644\u0634\u062a <code>Image<\/code> \u06a9\u0644\u0627\u0633 \u06cc\u06a9 \u0645\u062a\u062f \u0645\u0641\u06cc\u062f \u0628\u0647 \u0646\u0627\u0645 \u062f\u0627\u0631\u062f <code>transform(...)<\/code> \u06a9\u0647 \u0628\u0647 \u0634\u0645\u0627 \u0627\u0645\u06a9\u0627\u0646 \u0645\u06cc\u200c\u062f\u0647\u062f \u062a\u0628\u062f\u06cc\u0644\u200c\u0647\u0627\u06cc \u067e\u06cc\u0648\u0646\u062f\u06cc \u0631\u06cc\u0632\u062f\u0627\u0646\u0647 \u0631\u0627 \u0627\u0646\u062c\u0627\u0645 \u062f\u0647\u06cc\u062f\u060c \u0627\u0645\u0627 \u0686\u0646\u062f \u0645\u0648\u0631\u062f \u0639\u062c\u06cc\u0628 \u0648 \u063a\u0631\u06cc\u0628 \u0648\u062c\u0648\u062f \u062f\u0627\u0631\u062f \u06a9\u0647 \u0642\u0628\u0644 \u0627\u0632 \u0627\u06cc\u0646\u06a9\u0647 \u0628\u0647 \u0646\u0645\u0627\u06cc\u0634 \u0622\u0646 \u0628\u067e\u0631\u062f\u0627\u0632\u0645\u060c \u0627\u0628\u062a\u062f\u0627 \u0628\u0627\u06cc\u062f \u062f\u0631 \u0645\u0648\u0631\u062f \u0622\u0646\u0647\u0627 \u0635\u062d\u0628\u062a \u06a9\u0646\u0645.  \u0627\u06cc\u0646 <code>transform(...)<\/code> \u0631\u0648\u0634 \u0628\u0627 \u0646\u0645\u0627\u06cc\u0634 \u062f\u0648 \u067e\u0627\u0631\u0627\u0645\u062a\u0631 \u0645\u0648\u0631\u062f \u0646\u06cc\u0627\u0632 \u0622\u063a\u0627\u0632 \u0645\u06cc \u0634\u0648\u062f <code>size<\/code> \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u06cc\u06a9 \u062a\u0627\u067e\u0644 \u0627\u0632 \u0627\u0631\u062a\u0641\u0627\u0639 \u0648 \u0639\u0631\u0636\u060c \u0648 \u0628\u0647 \u062f\u0646\u0628\u0627\u0644 \u0622\u0646 <code>method<\/code> \u0627\u0632 \u062a\u0628\u062f\u06cc\u0644 \u0627\u0639\u0645\u0627\u0644 \u062e\u0648\u0627\u0647\u062f \u0634\u062f\u060c \u06a9\u0647 \u062e\u0648\u0627\u0647\u062f \u0628\u0648\u062f <code>Image.AFFINE<\/code> \u062f\u0631 \u0627\u06cc\u0646 \u0645\u0648\u0631\u062f.<\/p>\n<p>\u067e\u0627\u0631\u0627\u0645\u062a\u0631\u0647\u0627\u06cc \u0628\u0627\u0642\u06cc\u0645\u0627\u0646\u062f\u0647 \u0622\u0631\u06af\u0648\u0645\u0627\u0646 \u0647\u0627\u06cc \u06a9\u0644\u06cc\u062f\u0648\u0627\u0698\u0647 \u0627\u062e\u062a\u06cc\u0627\u0631\u06cc \u0647\u0633\u062a\u0646\u062f \u06a9\u0647 \u0631\u0648\u0634 \u0627\u0646\u062c\u0627\u0645 \u062a\u0628\u062f\u06cc\u0644 \u0631\u0627 \u06a9\u0646\u062a\u0631\u0644 \u0645\u06cc \u06a9\u0646\u0646\u062f.  \u062f\u0631 \u0645\u0648\u0631\u062f \u0627\u06cc\u0646 \u0645\u062b\u0627\u0644 \u0645\u0646 \u0627\u0632 \u0622\u0646 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u062e\u0648\u0627\u0647\u0645 \u06a9\u0631\u062f <code>data<\/code> \u067e\u0627\u0631\u0627\u0645\u062a\u0631\u060c \u06a9\u0647 \u062f\u0648 \u0631\u062f\u06cc\u0641 \u0627\u0648\u0644 \u06cc\u06a9 \u0645\u0627\u062a\u0631\u06cc\u0633 \u062a\u0628\u062f\u06cc\u0644 \u0627\u0641\u06cc\u0646 \u0631\u0627 \u0645\u06cc \u06af\u06cc\u0631\u062f.<\/p>\n<p>\u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u0645\u062b\u0627\u0644\u060c \u0645\u0627\u062a\u0631\u06cc\u0633 \u062a\u0628\u062f\u06cc\u0644 \u0645\u0642\u06cc\u0627\u0633 2 \u0628\u0631\u0627\u0628\u0631\u06cc \u06a9\u0647 \u0645\u0646 \u0628\u0627 \u0622\u0646 \u06a9\u0627\u0631 \u06a9\u0631\u062f\u0647\u200c\u0627\u0645 \u06a9\u0647 \u0641\u0642\u0637 \u0628\u0647 \u062f\u0648 \u0631\u062f\u06cc\u0641 \u0627\u0648\u0644 \u0628\u0631\u06cc\u062f\u0647 \u0634\u062f\u0647 \u0627\u0633\u062a \u0628\u0647 \u0627\u06cc\u0646 \u0635\u0648\u0631\u062a \u0627\u0633\u062a:<\/p>\n<p>$$ T_s = \\begin{bmatrix} 2 &#038; 0 &#038; 0 \\\\ 0 &#038; 2 &#038; 0 \\end{bmatrix} $$\n<\/p>\n<p>\u0622\u062e\u0631\u06cc\u0646 \u067e\u0627\u0631\u0627\u0645\u062a\u0631\u06cc \u06a9\u0647 \u0628\u0627 \u0622\u0646 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u062e\u0648\u0627\u0647\u0645 \u06a9\u0631\u062f <code>transform(...)<\/code> \u0631\u0648\u0634 \u0627\u0633\u062a <code>resample<\/code>\u060c \u06a9\u0647 \u0628\u0631\u0627\u06cc \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0646 \u0646\u0648\u0639 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u062f\u0631\u0648\u0646\u06cc\u0627\u0628\u06cc \u067e\u06cc\u06a9\u0633\u0644\u06cc \u0628\u0631\u0627\u06cc \u0627\u0639\u0645\u0627\u0644 \u062e\u0627\u0631\u062c \u0627\u0632 \u06af\u0632\u06cc\u0646\u0647 \u0647\u0627\u06cc \u0645\u0645\u06a9\u0646 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u0634\u0648\u062f. <code>Image.NEAREST<\/code> (\u0646\u0632\u062f\u06cc\u06a9\u062a\u0631\u06cc\u0646 \u0647\u0645\u0633\u0627\u06cc\u0647)\u060c <code>Image.BILINEAR<\/code>\u060c \u06cc\u0627 <code>Image.BICUBIC<\/code>.  \u0627\u06cc\u0646 \u0627\u0646\u062a\u062e\u0627\u0628 \u0627\u063a\u0644\u0628 \u0628\u0633\u062a\u0647 \u0628\u0647 \u0622\u0646 \u0645\u062a\u0641\u0627\u0648\u062a \u062e\u0648\u0627\u0647\u062f \u0628\u0648\u062f \u0631\u0648\u06cc \u062a\u062d\u0648\u0644 \u062f\u0631 \u062d\u0627\u0644 \u0627\u0639\u0645\u0627\u0644  \u0628\u0627 \u0627\u06cc\u0646 \u062d\u0627\u0644\u060c \u062f\u0648 \u062e\u0637\u06cc \u0648 \u062f\u0648 \u0645\u06a9\u0639\u0628\u06cc \u0628\u0647 \u0637\u0648\u0631 \u06a9\u0644\u06cc \u0646\u062a\u0627\u06cc\u062c \u0628\u0647\u062a\u0631\u06cc \u0646\u0633\u0628\u062a \u0628\u0647 \u0646\u0632\u062f\u06cc\u06a9\u062a\u0631\u06cc\u0646 \u0647\u0645\u0633\u0627\u06cc\u0647 \u0645\u06cc \u062f\u0647\u0646\u062f\u060c \u0627\u0645\u0627 \u0647\u0645\u0627\u0646\u0637\u0648\u0631 \u06a9\u0647 \u0642\u0628\u0644\u0627\u064b \u062f\u0631 \u0627\u06cc\u0646 \u0645\u062b\u0627\u0644 \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0647 \u0634\u062f \u0646\u0632\u062f\u06cc\u06a9\u062a\u0631\u06cc\u0646 \u0647\u0645\u0633\u0627\u06cc\u0647 \u06a9\u0627\u0645\u0644\u0627\u064b \u062e\u0648\u0628 \u0639\u0645\u0644 \u0645\u06cc \u06a9\u0646\u062f.<\/p>\n<p>\u0686\u0646\u062f \u0648\u06cc\u0698\u06af\u06cc \u0648\u062c\u0648\u062f \u062f\u0627\u0631\u062f \u06a9\u0647 \u0627\u0648\u0644\u06cc\u0646 \u0628\u0627\u0631\u06cc \u06a9\u0647 \u0627\u0632 \u0622\u0646 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u06a9\u0631\u062f\u0645 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u06cc\u06a9 \u06af\u0648\u0686\u0627 \u0648\u0627\u0642\u0639\u06cc \u0628\u0631\u0627\u06cc \u0645\u0646 \u0639\u0645\u0644 \u06a9\u0631\u062f <code>Image.transform(...)<\/code> \u0631\u0648\u0634\u060c \u0628\u0647 \u0648\u06cc\u0698\u0647 \u062f\u0631 \u0645\u0648\u0631\u062f \u0633\u0627\u062e\u062a \u0645\u0627\u062a\u0631\u06cc\u0633 \u062a\u0628\u062f\u06cc\u0644 \u0627\u0641\u06cc\u0646 \u0628\u0627 \u0631\u062f\u06cc\u0641 \u0622\u062e\u0631 \u06a9\u0648\u062a\u0627\u0647 \u0634\u062f\u0647 \u0627\u0633\u062a.  \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646\u060c \u0645\u0646 \u0645\u06cc\u200c\u062e\u0648\u0627\u0647\u0645 \u0645\u062f\u062a\u06cc \u0631\u0627 \u0635\u0631\u0641 \u0628\u0631\u0631\u0633\u06cc \u0627\u06cc\u0646 \u0645\u0648\u0636\u0648\u0639 \u06a9\u0646\u0645 \u06a9\u0647 \u0686\u0631\u0627 \u06a9\u0627\u0631\u0647\u0627 \u0628\u0647 \u0647\u0645\u0627\u0646 \u0634\u06a9\u0644 \u0627\u0646\u062c\u0627\u0645 \u0645\u06cc\u200c\u0634\u0648\u0646\u062f\u060c \u0632\u06cc\u0631\u0627 \u0627\u06cc\u0646 \u06a9\u0645\u06cc \u0645\u0634\u06a9\u0644 \u0627\u0633\u062a process.<\/p>\n<p>\u0627\u0648\u0644\u06cc\u0646 \u0686\u06cc\u0632\u06cc \u06a9\u0647 \u0628\u0627\u06cc\u062f \u0627\u062a\u0641\u0627\u0642 \u0628\u06cc\u0641\u062a\u062f \u0627\u06cc\u0646 \u0627\u0633\u062a \u06a9\u0647 \u062a\u0635\u0648\u06cc\u0631 \u0628\u0627\u06cc\u062f \u0637\u0648\u0631\u06cc \u062a\u0631\u062c\u0645\u0647 \u0634\u0648\u062f \u06a9\u0647 \u0645\u0628\u062f\u0627 (0, 0) \u062f\u0631 \u0648\u0633\u0637 \u062a\u0635\u0648\u06cc\u0631 \u0628\u0627\u0634\u062f.  \u062f\u0631 \u0645\u0648\u0631\u062f \u062a\u0635\u0648\u06cc\u0631 1000 x 1000 \u062d\u0631\u0641 R \u062f\u0631 \u0627\u06cc\u0646 \u0645\u062b\u0627\u0644 \u0628\u0647 \u0645\u0639\u0646\u0627\u06cc \u062a\u0631\u062c\u0645\u0647 500- \u062f\u0631 x \u0648 y \u0627\u0633\u062a.<\/p>\n<p>\u062f\u0631 \u0632\u06cc\u0631 \u0645\u0627\u062a\u0631\u06cc\u0633 \u062a\u0628\u062f\u06cc\u0644 \u0639\u0645\u0648\u0645\u06cc \u062a\u0631\u062c\u0645\u0647 \\(T_{translate}\\) \u0648 \u0645\u0627\u062a\u0631\u06cc\u0633 \u06a9\u0647 \u062f\u0631 \u0645\u062b\u0627\u0644 \\(T_{neg500}\\) \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u062e\u0648\u0627\u0647\u0645 \u06a9\u0631\u062f \u0631\u0627 \u0646\u0634\u0627\u0646 \u0645\u06cc\u200c\u062f\u0647\u0645.<\/p>\n<p>$$ T_{translate} = \\begin{bmatrix} 1 &#038; 0 &#038; t_x \\\\ 0 &#038; 1 &#038; t_y \\\\ 0 &#038; 0 &#038; 1 \\end{bmatrix} $$\n<\/p>\n<p>$$<br \/>T_{neg500} = \\begin{bmatrix}<br \/>1 \u0648 0 \u0648 -500 \\<br \/>0 \u0648 1 \u0648 -500 \\<br \/>0 \u0648 0 \u0648 1<br \/>\\end{bmatrix}<br \/>$$<\/p>\n<p>\u0633\u067e\u0633 \u0645\u0627\u062a\u0631\u06cc\u0633 \u0647\u0627\u06cc 2X \u0645\u0642\u06cc\u0627\u0633 \\(T_{scale}\\) \u0648 \u0686\u0631\u062e\u0634 90 \u062f\u0631\u062c\u0647 \\(T_{rotate}\\) \u0627\u0632 \u0642\u0628\u0644 \u0648\u062c\u0648\u062f \u062f\u0627\u0631\u0646\u062f.  \u0628\u0627 \u0627\u06cc\u0646 \u062d\u0627\u0644\u060c \u06a9\u062a\u0627\u0628\u062e\u0627\u0646\u0647 Pillow \u062f\u0631 \u0648\u0627\u0642\u0639 \u062a\u0635\u0645\u06cc\u0645 \u06af\u0631\u0641\u062a \u0627\u0632 \u0632\u0648\u0627\u06cc\u0627\u06cc \u0647\u0646\u062f\u0633\u06cc \u0627\u0633\u062a\u0627\u0646\u062f\u0627\u0631\u062f (\u06cc\u0639\u0646\u06cc \u062f\u0631 \u062e\u0644\u0627\u0641 \u062c\u0647\u062a \u0639\u0642\u0631\u0628\u0647\u200c\u0647\u0627\u06cc \u0633\u0627\u0639\u062a) \u0628\u0647 \u062c\u0627\u06cc \u0686\u0631\u062e\u0634 \u062f\u0631 \u062c\u0647\u062a \u0639\u0642\u0631\u0628\u0647\u200c\u0647\u0627\u06cc \u0633\u0627\u0639\u062a \u06a9\u0647 \u0642\u0628\u0644\u0627\u064b \u062a\u0648\u0636\u06cc\u062d \u062f\u0627\u062f\u0645\u060c \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u06a9\u0646\u062f. \u0631\u0648\u06cc \u062a\u0648\u0627\u0628\u0639 \u06af\u0646\u0627\u0647 \u062a\u0644\u0646\u06af\u0631.  \u062f\u0631 \u0632\u06cc\u0631 \u0645\u0627\u062a\u0631\u06cc\u0633 \u0647\u0627\u06cc \u062a\u0628\u062f\u06cc\u0644 \u0641\u0631\u062f\u06cc \u062d\u0627\u0635\u0644 \u0622\u0645\u062f\u0647 \u0627\u0633\u062a.<\/p>\n<p>$$ T_{rotate} = \\begin{bmatrix} 0 &#038; -1 &#038; 0 \\\\ 1 &#038; 0 &#038; 0 \\\\ 0 &#038; 0 &#038; 1 \\end{bmatrix} $$\n<\/p>\n<p>$$<br \/>T_{scale} = \\begin{bmatrix}<br \/>2 \u0648 0 \u0648 0 \\<br \/>0 \u0648 2 \u0648 0 \\<br \/>0 \u0648 0 \u0648 1<br \/>\\end{bmatrix}<br \/>$$<\/p>\n<p>\u062f\u0631 \u0645\u0631\u062d\u0644\u0647 \u0628\u0639\u062f\u060c \u0645\u0627\u062a\u0631\u06cc\u0633 \u062a\u0631\u062c\u0645\u0647 \u062f\u06cc\u06af\u0631\u06cc \u0628\u0627\u06cc\u062f \u0627\u0639\u0645\u0627\u0644 \u0634\u0648\u062f \u06a9\u0647 \u0628\u0631\u0627\u06cc \u062a\u063a\u06cc\u06cc\u0631 \u0645\u06a9\u0627\u0646 \u062f\u0627\u0645\u0646\u0647 \u0641\u0636\u0627\u06cc\u06cc \u067e\u06cc\u06a9\u0633\u0644 \u0647\u0627 \u0639\u0645\u0644 \u0645\u06cc \u06a9\u0646\u062f \u0648 \u0627\u0633\u0627\u0633\u0627\u064b \u0627\u0648\u0644\u06cc\u0646 \u0645\u0648\u0631\u062f\u06cc \u0631\u0627 \u06a9\u0647 \u0645\u0628\u062f\u0627 \u0631\u0627 \u062f\u0631 \u0645\u0631\u06a9\u0632 \u0642\u0631\u0627\u0631 \u062f\u0627\u062f\u0647 \u0627\u0633\u062a\u060c \u0646\u0641\u06cc \u0645\u06cc \u06a9\u0646\u062f.  \u062f\u0631 \u0627\u06cc\u0646 \u0645\u0648\u0631\u062f\u060c \u0645\u0646 \u0628\u0647 \u06cc\u06a9 \u062a\u0631\u062c\u0645\u0647 \u0645\u062b\u0628\u062a 1000 \u062f\u0631 x \u0648 y \u0646\u06cc\u0627\u0632 \u062f\u0627\u0631\u0645\u060c \u06a9\u0647 \u062f\u0631 \u0622\u0646 1000 \u0627\u0632 \u062f\u0648 \u0628\u0631\u0627\u0628\u0631 \u0646\u0633\u062e\u0647 \u0627\u0635\u0644\u06cc \u0645\u06cc \u0622\u06cc\u062f\u060c \u0632\u06cc\u0631\u0627 \u0622\u0646 \u0631\u0627 \u062f\u0648 \u0628\u0631\u0627\u0628\u0631 \u06a9\u0631\u062f\u0647 \u0627\u0633\u062a.<\/p>\n<p>$$ T_{pos1000} = \\begin{bmatrix} 1 &#038; 0 &#038; 1000 \\\\ 0 &#038; 1 &#038; 1000 \\\\ 0 &#038; 0 &#038; 1 \\end{bmatrix} $$\n<\/p>\n<p>\u0627\u06cc\u0646\u0647\u0627 \u0645\u0631\u0627\u062d\u0644 \u062a\u0628\u062f\u06cc\u0644 \u0641\u0631\u062f\u06cc \u0645\u0648\u0631\u062f \u0646\u06cc\u0627\u0632 \u0631\u0627 \u062a\u0634\u06a9\u06cc\u0644 \u0645\u06cc\u200c\u062f\u0647\u0646\u062f\u060c \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u062a\u0646\u0647\u0627 \u0686\u06cc\u0632\u06cc \u06a9\u0647 \u0628\u0627\u0642\u06cc \u0645\u06cc\u200c\u0645\u0627\u0646\u062f \u0636\u0631\u0628 \u06a9\u0631\u062f\u0646 \u0645\u0627\u062a\u0631\u06cc\u0633\u200c\u0647\u0627 \u0628\u0647 \u062a\u0631\u062a\u06cc\u0628 (\u06cc\u0639\u0646\u06cc \u0627\u0632 \u0631\u0627\u0633\u062a \u0628\u0647 \u0686\u067e) \u0627\u0633\u062a\u060c \u0645\u0627\u0646\u0646\u062f \u0632\u06cc\u0631:<\/p>\n<p>$$ T = T_{pos1000} * T_{rotate} * T_{scale} * T_{neg500} $$\n<\/p>\n<p>\u062e\u0648\u0628\u060c \u067e\u0633 \u062f\u0631 \u0648\u0627\u0642\u0639 \u0622\u062e\u0631\u06cc\u0646 \u0645\u0648\u0631\u062f \u0639\u062c\u06cc\u0628 \u0648\u062c\u0648\u062f \u062f\u0627\u0631\u062f.  \u0627\u06cc\u0646 <code>Image.transform(...)<\/code> \u0631\u0648\u0634 \u062f\u0631 \u0648\u0627\u0642\u0639 \u0645\u0633\u062a\u0644\u0632\u0645 \u0622\u0646 \u0627\u0633\u062a \u06a9\u0647 \u0645\u0639\u06a9\u0648\u0633 \u0645\u0627\u062a\u0631\u06cc\u0633 \u062a\u0628\u062f\u06cc\u0644 \u0628\u0647 \u0622\u0646 \u0627\u0631\u0627\u0626\u0647 \u0634\u0648\u062f <code>data<\/code> \u067e\u0627\u0631\u0627\u0645\u062a\u0631 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u06cc\u06a9 \u0622\u0631\u0627\u06cc\u0647 \u0645\u0633\u0637\u062d (\u06cc\u0627 \u0686\u0646\u062f \u062a\u0627\u06cc\u06cc) \u0628\u0647 \u0627\u0633\u062a\u062b\u0646\u0627\u06cc \u0622\u062e\u0631\u06cc\u0646 \u0631\u062f\u06cc\u0641.<\/p>\n<p>$$ T_{inv} = T^{-1} $$\n<\/p>\n<p>\u062f\u0631 \u06a9\u062f \u0627\u06cc\u0646 \u0647\u0645\u0647 \u0628\u0647 \u0635\u0648\u0631\u062a \u0632\u06cc\u0631 \u06a9\u0627\u0631 \u0645\u06cc \u06a9\u0646\u062f:<\/p>\n<pre><code class=\"hljs\">\n\nT_pos1000 = np.array((\n    (<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">1000<\/span>),\n    (<span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">1000<\/span>),\n    (<span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">1<\/span>)))\n\nT_rotate = np.array((\n    (<span class=\"hljs-number\">0<\/span>, -<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">0<\/span>),\n    (<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>),\n    (<span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">1<\/span>)))\n\nT_scale = np.array((\n    (<span class=\"hljs-number\">2<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>),\n    (<span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">2<\/span>, <span class=\"hljs-number\">0<\/span>),\n    (<span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">1<\/span>)))\n\nT_neg500 = np.array((\n    (<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">0<\/span>, -<span class=\"hljs-number\">500<\/span>),\n    (<span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">1<\/span>, -<span class=\"hljs-number\">500<\/span>),\n    (<span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">1<\/span>)))\nT = T_pos1000 @ T_rotate @ T_scale @ T_neg500\nT_inv = np.linalg.inv(T)\n<\/code><\/pre>\n<pre><code class=\"hljs\">img_transformed = img.transform((<span class=\"hljs-number\">2000<\/span>, <span class=\"hljs-number\">2000<\/span>), Image.AFFINE, data=T_inv.flatten()(:<span class=\"hljs-number\">6<\/span>), resample=Image.NEAREST)\nplt.imshow(np.asarray(img_transformed))\n<\/code><\/pre>\n<p><!-- --><br \/>\n<img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/2019-03-letterR_pillow_transformed.png\" alt=\"png\" title=\"\"><\/p>\n<h2 id=\"affinetransformationswithopencv2\"><span class=\"ez-toc-section\" id=\"affine_transformations_%d8%a8%d8%a7_opencv2\"><\/span>Affine Transformations \u0628\u0627 OpenCV2<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u0627\u062f\u0627\u0645\u0647 \u062f\u0627\u0631\u062f \u0631\u0648\u06cc \u0645\u0646 \u0645\u0627\u06cc\u0644\u0645 \u0628\u0647 \u0637\u0648\u0631 \u062e\u0644\u0627\u0635\u0647 \u0631\u0648\u0634 \u0627\u0646\u062c\u0627\u0645 \u0627\u06cc\u0646 \u062f\u06af\u0631\u06af\u0648\u0646\u06cc \u0647\u0627\u06cc \u0648\u0627\u0628\u0633\u062a\u0647 \u0631\u0627 \u0628\u0627 \u06a9\u062a\u0627\u0628\u062e\u0627\u0646\u0647 \u0645\u062d\u0628\u0648\u0628 \u067e\u0631\u062f\u0627\u0632\u0634 \u062a\u0635\u0648\u06cc\u0631 \u0648 \u0628\u06cc\u0646\u0627\u06cc\u06cc \u06a9\u0627\u0645\u067e\u06cc\u0648\u062a\u0631\u06cc OpenCV \u0634\u0631\u062d \u062f\u0647\u0645.  \u0645\u0646 \u062f\u0631 \u0627\u06cc\u0646\u062c\u0627 \u0627\u0632 \u06a9\u0644\u0645\u0647 \u0645\u062e\u062a\u0635\u0631 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u06a9\u0646\u0645 \u0632\u06cc\u0631\u0627 \u062a\u0627 \u062d\u062f \u0632\u06cc\u0627\u062f\u06cc \u0647\u0645\u0627\u0646 \u0686\u06cc\u0632\u06cc \u0627\u0633\u062a \u06a9\u0647 \u062f\u0631 \u0646\u0645\u0627\u06cc\u0634 \u0642\u0628\u0644\u06cc \u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 Pillow \u0645\u0648\u0631\u062f \u0646\u06cc\u0627\u0632 \u0627\u0633\u062a.<\/p>\n<p>\u0627\u0648\u0644 \u0627\u0632 \u0647\u0645\u0647\u060c \u0634\u0645\u0627 \u0628\u0627\u06cc\u062f \u0628\u0647 \u0627\u06cc\u0646 \u0635\u0648\u0631\u062a \u0646\u0635\u0628 \u06a9\u0646\u06cc\u062f:<\/p>\n<pre><code class=\"hljs\"><span class=\"hljs-meta\">$<\/span><span class=\"bash\"> pip install opencv-python<\/span>\n<\/code><\/pre>\n<p>\u0647\u0645\u0627\u0646\u0637\u0648\u0631 \u06a9\u0647 \u062f\u0631 \u0628\u0627\u0644\u0627 \u0630\u06a9\u0631 \u06a9\u0631\u062f\u0645\u060c \u0647\u0645\u067e\u0648\u0634\u0627\u0646\u06cc \u0642\u0627\u0628\u0644 \u062a\u0648\u062c\u0647\u06cc \u062f\u0631 \u0631\u0648\u0634 \u0634\u0646\u0627\u0633\u06cc \u0628\u06cc\u0646 \u0631\u0648\u06cc\u06a9\u0631\u062f Pillow \u0648 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 OpenCV \u0648\u062c\u0648\u062f \u062f\u0627\u0631\u062f.  \u0628\u0631\u0627\u06cc \u0645\u062b\u0627\u0644\u060c \u0634\u0645\u0627 \u0647\u0645\u0686\u0646\u0627\u0646 \u06cc\u06a9 \u0645\u0627\u062a\u0631\u06cc\u0633 \u062a\u0628\u062f\u06cc\u0644 \u0627\u06cc\u062c\u0627\u062f \u0645\u06cc\u200c\u06a9\u0646\u06cc\u062f \u06a9\u0647 \u0627\u0628\u062a\u062f\u0627 \u0622\u0631\u0627\u06cc\u0647 \u067e\u06cc\u06a9\u0633\u0644\u200c\u0647\u0627 \u0631\u0627 \u0631\u0648\u06cc \u0645\u0628\u062f\u0627 \u0645\u062a\u0645\u0631\u06a9\u0632 \u0645\u06cc\u200c\u06a9\u0646\u062f \u0648 \u0641\u0642\u0637 \u0627\u0632 \u062f\u0648 \u0631\u062f\u06cc\u0641 \u0627\u0648\u0644 \u0645\u0627\u062a\u0631\u06cc\u0633 \u062a\u0628\u062f\u06cc\u0644 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc\u200c\u06a9\u0646\u06cc\u062f.  \u062a\u0641\u0627\u0648\u062a \u0639\u0645\u062f\u0647 \u0627\u06cc\u0646 \u0627\u0633\u062a \u06a9\u0647 \u0628\u0627 OpenCV \u0628\u0647 \u062c\u0627\u06cc \u0645\u0627\u062a\u0631\u06cc\u0633 \u0645\u0639\u06a9\u0648\u0633\u060c \u0645\u0627\u062a\u0631\u06cc\u0633 \u0627\u0633\u062a\u0627\u0646\u062f\u0627\u0631\u062f \u0631\u0627 \u0628\u0647 \u0622\u0646 \u0645\u06cc \u062f\u0647\u06cc\u062f.<\/p>\n<p>\u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646\u060c \u0628\u0627 \u062f\u0631\u06a9 \u0627\u06cc\u0646 \u0645\u0648\u0636\u0648\u0639\u060c \u0645\u0646 \u0648\u0627\u0631\u062f \u06a9\u062f \u0645\u06cc\u200c\u0634\u0648\u0645 \u06a9\u0647 \u0628\u0627 \u0648\u0627\u0631\u062f \u06a9\u0631\u062f\u0646 \u06a9\u062f \u0634\u0631\u0648\u0639 \u0645\u06cc\u200c\u0634\u0648\u062f <code>opencv-python<\/code> \u0645\u0627\u0698\u0648\u0644 \u06a9\u0647 \u0646\u0627\u0645\u06af\u0630\u0627\u0631\u06cc \u0634\u062f\u0647 \u0627\u0633\u062a <code>cv2<\/code>.<\/p>\n<pre><code class=\"hljs\"><span class=\"hljs-keyword\">import<\/span> cv2\n<\/code><\/pre>\n<p>\u062e\u0648\u0627\u0646\u062f\u0646 \u062a\u0635\u0648\u06cc\u0631 \u0628\u0647 \u0633\u0627\u062f\u06af\u06cc \u0641\u0631\u0627\u062e\u0648\u0627\u0646\u06cc \u0627\u0633\u062a <code>cv2.imread(...)<\/code> \u0631\u0648\u0634\u060c \u0627\u0631\u0633\u0627\u0644 \u0646\u0627\u0645 \u0641\u0627\u06cc\u0644 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u0622\u0631\u06af\u0648\u0645\u0627\u0646.  \u0627\u06cc\u0646 \u062f\u0627\u062f\u0647\u200c\u0647\u0627\u06cc \u062a\u0635\u0648\u06cc\u0631 \u0631\u0627 \u0628\u0647 \u0634\u06a9\u0644 \u06cc\u06a9 \u0622\u0631\u0627\u06cc\u0647 3 \u0628\u0639\u062f\u06cc numpy \u0628\u0631\u0645\u06cc\u200c\u06af\u0631\u062f\u0627\u0646\u062f\u060c \u0634\u0628\u06cc\u0647 \u0628\u0647 \u0631\u0648\u0634 \u0639\u0645\u0644\u06a9\u0631\u062f matplotlib\u060c \u0627\u0645\u0627 \u062f\u0627\u062f\u0647\u200c\u0647\u0627\u06cc \u067e\u06cc\u06a9\u0633\u0644 \u062f\u0631 \u0628\u0639\u062f \u0633\u0648\u0645 \u0627\u0632 \u0622\u0631\u0627\u06cc\u0647\u200c\u0627\u06cc \u0627\u0632 \u06a9\u0627\u0646\u0627\u0644\u200c\u0647\u0627 \u0628\u0647 \u062a\u0631\u062a\u06cc\u0628 \u0622\u0628\u06cc\u060c \u0633\u0628\u0632\u060c \u0642\u0631\u0645\u0632 \u0628\u0647 \u062c\u0627\u06cc \u0642\u0631\u0645\u0632\u060c \u0633\u0628\u0632 \u062a\u0634\u06a9\u06cc\u0644 \u0634\u062f\u0647 \u0627\u0633\u062a. \u0622\u0628\u06cc\u060c \u0622\u0644\u0641\u0627 \u0647\u0645\u0627\u0646\u0637\u0648\u0631 \u06a9\u0647 \u062f\u0631 \u0645\u0648\u0631\u062f \u062e\u0648\u0627\u0646\u062f\u0646 \u0628\u0627 matplotlib \u0628\u0648\u062f.<\/p>\n<p>\u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646\u060c \u0628\u0631\u0627\u06cc \u0631\u0633\u0645 \u062f\u0627\u062f\u0647\u200c\u0647\u0627\u06cc \u062a\u0635\u0648\u06cc\u0631\u06cc \u0646\u0627\u0686\u06cc\u0632 \u06a9\u0647 \u0627\u0632 \u06a9\u062a\u0627\u0628\u062e\u0627\u0646\u0647 OpenCV \u0646\u0634\u0627\u062a \u0645\u06cc\u200c\u06af\u06cc\u0631\u062f\u060c \u0628\u0627\u06cc\u062f \u062a\u0631\u062a\u06cc\u0628 \u06a9\u0627\u0646\u0627\u0644\u200c\u0647\u0627\u06cc \u067e\u06cc\u06a9\u0633\u0644 \u0631\u0627 \u0645\u0639\u06a9\u0648\u0633 \u06a9\u0631\u062f.  \u062e\u0648\u0634\u0628\u062e\u062a\u0627\u0646\u0647\u060c OpenCV \u06cc\u06a9 \u0631\u0648\u0634 \u0631\u0627\u062d\u062a \u0627\u0631\u0627\u0626\u0647 \u0645\u06cc \u062f\u0647\u062f <code>cvtColor(...)<\/code> \u0647\u0645\u0627\u0646\u0637\u0648\u0631 \u06a9\u0647 \u062f\u0631 \u0632\u06cc\u0631 \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0647 \u0634\u062f\u0647 \u0627\u0633\u062a \u0645\u06cc \u062a\u0648\u0627\u0646 \u0627\u0632 \u0622\u0646 \u0628\u0631\u0627\u06cc \u0627\u0646\u062c\u0627\u0645 \u0627\u06cc\u0646 \u06a9\u0627\u0631 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u06a9\u0631\u062f (\u0627\u06af\u0631\u0686\u0647 \u0627\u0641\u0631\u0627\u062f \u0646\u0627\u0628\u06a9\u0627\u0631 \u0627\u062d\u062a\u0645\u0627\u0644\u0627\u064b \u0627\u06cc\u0646 \u0631\u0627 \u0645\u06cc \u062f\u0627\u0646\u0646\u062f <code>img(:,:,::-1)<\/code> \u0647\u0645\u06cc\u0646 \u06a9\u0627\u0631 \u0631\u0627 \u062e\u0648\u0627\u0647\u062f \u06a9\u0631\u062f).<\/p>\n<pre><code class=\"hljs\">img = cv2.imread(<span class=\"hljs-string\">'letterR.jpg'<\/span>)\nplt.imshow(cv2.cvtColor(img, cv2.COLOR_BGR2RGB))\n<\/code><\/pre>\n<p><!-- --><br \/>\n<img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/2019-03-letterR_opencv.png\" alt=\"png\" title=\"\"><\/p>\n<p>\u0686\u0646\u062f \u0645\u0648\u0631\u062f \u0622\u062e\u0631 \u06a9\u0647 \u0628\u0627\u06cc\u062f \u0630\u06a9\u0631 \u0634\u0648\u062f \u0627\u06cc\u0646 \u0627\u0633\u062a \u06a9\u0647 OpenCV \u0646\u06cc\u0627\u0632 \u062f\u0627\u0631\u062f \u06a9\u0647 \u062f\u0627\u062f\u0647\u200c\u0647\u0627\u06cc \u0645\u0648\u062c\u0648\u062f \u062f\u0631 \u0645\u0627\u062a\u0631\u06cc\u0633 \u062a\u0628\u062f\u06cc\u0644 \u0627\u0632 \u0646\u0648\u0639 \u0634\u0646\u0627\u0648\u0631 \u06f3\u06f2 \u0628\u06cc\u062a\u06cc \u0628\u0647 \u062c\u0627\u06cc \u0634\u0646\u0627\u0648\u0631 \u06f6\u06f4 \u0628\u06cc\u062a\u06cc \u067e\u06cc\u0634\u200c\u0641\u0631\u0636 \u0628\u0627\u0634\u062f\u060c \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0645\u0637\u0645\u0626\u0646 \u0634\u0648\u06cc\u062f \u06a9\u0647 \u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u0622\u0646 \u0628\u0647 \u06f3\u06f2 \u0628\u06cc\u062a \u062a\u0628\u062f\u06cc\u0644 \u06a9\u0646\u06cc\u062f. <code>numpy.float32(...)<\/code>.  \u0647\u0645\u0686\u0646\u06cc\u0646\u060c API \u0628\u0647 <code>cv2.warpAffine(...)<\/code> \u062a\u0648\u0627\u0646\u0627\u06cc\u06cc \u062a\u0639\u06cc\u06cc\u0646 \u0646\u0648\u0639 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u062f\u0631\u0648\u0646\u06cc\u0627\u0628\u06cc \u067e\u06cc\u06a9\u0633\u0644\u06cc \u0631\u0627 \u0627\u0631\u0627\u0626\u0647 \u0646\u0645\u06cc \u062f\u0647\u062f \u0648 \u0645\u0646 \u0646\u062a\u0648\u0627\u0646\u0633\u062a\u0645 \u0627\u0632 \u0631\u0648\u06cc \u0627\u0633\u0646\u0627\u062f \u062a\u0639\u06cc\u06cc\u0646 \u06a9\u0646\u0645 \u06a9\u0647 \u0686\u0647 \u0686\u06cc\u0632\u06cc \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u0634\u0648\u062f.  \u0627\u06af\u0631 \u0645\u06cc \u062f\u0627\u0646\u06cc\u062f \u06cc\u0627 \u0645\u06cc \u062f\u0627\u0646\u06cc\u062f \u0644\u0637\u0641\u0627 \u062f\u0631 \u0646\u0638\u0631\u0627\u062a \u0632\u06cc\u0631 \u0627\u0631\u0633\u0627\u0644 \u06a9\u0646\u06cc\u062f.<\/p>\n<pre><code class=\"hljs\">T_opencv = np.float32(T.flatten()(:<span class=\"hljs-number\">6<\/span>).reshape(<span class=\"hljs-number\">2<\/span>,<span class=\"hljs-number\">3<\/span>))\nimg_transformed = cv2.warpAffine(img, T_opencv, (<span class=\"hljs-number\">2000<\/span>, <span class=\"hljs-number\">2000<\/span>))\nplt.imshow(cv2.cvtColor(img_transformed, cv2.COLOR_BGR2RGB))\n<\/code><\/pre>\n<p><!-- --><br \/>\n<img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/2019-03-letterR_opencv_transformed.png\" alt=\"png\" title=\"\"><\/p>\n<h2 id=\"conclusion\"><span class=\"ez-toc-section\" id=\"%d9%86%d8%aa%db%8c%d8%ac%d9%87\"><\/span>\u0646\u062a\u06cc\u062c\u0647<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u062f\u0631 \u0627\u06cc\u0646 \u0645\u0642\u0627\u0644\u0647 \u0645\u0646 \u062a\u0648\u0636\u06cc\u062d \u062f\u0627\u062f\u0647\u200c\u0627\u0645 \u06a9\u0647 \u062a\u0628\u062f\u06cc\u0644 \u0627\u0641\u06cc\u0646 \u0686\u06cc\u0633\u062a \u0648 \u0686\u06af\u0648\u0646\u0647 \u0645\u06cc\u200c\u062a\u0648\u0627\u0646 \u0622\u0646 \u0631\u0627 \u0628\u0631\u0627\u06cc \u067e\u0631\u062f\u0627\u0632\u0634 \u062a\u0635\u0648\u06cc\u0631 \u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u067e\u0627\u06cc\u062a\u0648\u0646 \u0627\u0639\u0645\u0627\u0644 \u06a9\u0631\u062f.  Numpy \u062e\u0627\u0644\u0635 \u0648 matplotlib \u0628\u0631\u0627\u06cc \u0627\u0631\u0627\u0626\u0647 \u06cc\u06a9 \u062a\u0648\u0635\u06cc\u0641 \u0628\u0635\u0631\u06cc \u0633\u0637\u062d \u067e\u0627\u06cc\u06cc\u0646 \u0627\u0632 \u0631\u0648\u0634 \u0639\u0645\u0644\u06a9\u0631\u062f \u062a\u0628\u062f\u06cc\u0644\u200c\u0647\u0627\u06cc affine \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0634\u062f.  \u0645\u0646 \u0628\u0627 \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0646 \u0627\u06cc\u0646\u06a9\u0647 \u0686\u06af\u0648\u0646\u0647 \u0645\u06cc \u062a\u0648\u0627\u0646 \u0647\u0645\u06cc\u0646 \u06a9\u0627\u0631 \u0631\u0627 \u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u062f\u0648 \u06a9\u062a\u0627\u0628\u062e\u0627\u0646\u0647 \u0645\u062d\u0628\u0648\u0628 \u067e\u0627\u06cc\u062a\u0648\u0646 Pillow \u0648 OpenCV \u0628\u0647 \u067e\u0627\u06cc\u0627\u0646 \u0631\u0633\u0627\u0646\u062f.<\/p>\n<p>\u0645\u0645\u0646\u0648\u0646 \u06a9\u0647 \u062e\u0648\u0627\u0646\u062f\u06cc\u062f \u0648 \u0645\u062b\u0644 \u0647\u0645\u06cc\u0634\u0647 \u0627\u0632 \u0646\u0638\u0631 \u062f\u0627\u062f\u0646 \u06cc\u0627 \u0627\u0646\u062a\u0642\u0627\u062f \u062f\u0631 \u0632\u06cc\u0631 \u062e\u062c\u0627\u0644\u062a \u0646\u06a9\u0634\u06cc\u062f.<\/p>\n<h2 id=\"resources\"><span class=\"ez-toc-section\" id=\"%d9%85%d9%86%d8%a7%d8%a8%d8%b9\"><\/span>\u0645\u0646\u0627\u0628\u0639<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<\/div>\n<p><script>\n                        !function(f,b,e,v,n,t,s)\n                        {if(f.fbq)return;n=f.fbq=function(){n.callMethod?\n                        n.callMethod.apply(n,arguments):n.queue.push(arguments)};\n                        if(!f._fbq)f._fbq=n;n.push=n;n.loaded=!0;n.version='2.0';\n                        n.queue=();t=b.createElement(e);t.async=!0;\n                        t.src=v;s=b.getElementsByTagName(e)(0);\n                        s.parentNode.insertBefore(t,s)}(window, document,'script',\n                        'https:\/\/connect.facebook.net\/en_US\/fbevents.js');\n                        fbq('init', '525232124909042');\n                        fbq('track', 'PageView');\n                    <\/script>    (\u0628\u0631\u0686\u0633\u0628\u200c\u0647\u0627 \u0628\u0647 \u062a\u0631\u062c\u0645\u0647)# python<br \/>\n<br \/><br \/>\n<br \/>\u0645\u0646\u062a\u0634\u0631 \u0634\u062f\u0647 \u062f\u0631 1403-01-24 11:50:03<br \/>\n<\/p>\n\n\n<div class=\"kk-star-ratings kksr-auto kksr-align-center kksr-valign-bottom\"\n    data-payload='{&quot;align&quot;:&quot;center&quot;,&quot;id&quot;:&quot;16385&quot;,&quot;slug&quot;:&quot;default&quot;,&quot;valign&quot;:&quot;bottom&quot;,&quot;ignore&quot;:&quot;&quot;,&quot;reference&quot;:&quot;auto&quot;,&quot;class&quot;:&quot;&quot;,&quot;count&quot;:&quot;0&quot;,&quot;legendonly&quot;:&quot;&quot;,&quot;readonly&quot;:&quot;&quot;,&quot;score&quot;:&quot;0&quot;,&quot;starsonly&quot;:&quot;&quot;,&quot;best&quot;:&quot;5&quot;,&quot;gap&quot;:&quot;5&quot;,&quot;greet&quot;:&quot;\u0627\u0645\u062a\u06cc\u0627\u0632 \u0634\u0645\u0627 \u0628\u0647 \u0627\u06cc\u0646 \u0645\u0637\u0644\u0628&quot;,&quot;legend&quot;:&quot;0\\\/5 (0 \u0631\u0627\u06cc)&quot;,&quot;size&quot;:&quot;30&quot;,&quot;title&quot;:&quot;\u062a\u063a\u06cc\u06cc\u0631 \u0634\u06a9\u0644 \u062a\u0635\u0648\u06cc\u0631 \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646 \u0628\u0627 Numpy\u060c Pillow \u0648 OpenCV&quot;,&quot;width&quot;:&quot;0&quot;,&quot;_legend&quot;:&quot;{score}\\\/{best} ({count} \u0631\u0627\u06cc)&quot;,&quot;font_factor&quot;:&quot;1.25&quot;}'>\n            \n<div class=\"kksr-stars\">\n    \n<div class=\"kksr-stars-inactive\">\n            <div class=\"kksr-star\" data-star=\"1\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; 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height: 30px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; height: 30px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; height: 30px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; height: 30px;\"><\/div>\n        <\/div>\n    <\/div>\n<\/div>\n                \n\n<div class=\"kksr-legend\" style=\"font-size: 24px;\">\n            <span class=\"kksr-muted\">\u0627\u0645\u062a\u06cc\u0627\u0632 \u0634\u0645\u0627 \u0628\u0647 \u0627\u06cc\u0646 \u0645\u0637\u0644\u0628<\/span>\n    <\/div>\n    <\/div>\n","protected":false},"excerpt":{"rendered":"<p><span class=\"span-reading-time rt-reading-time\" style=\"display: block;\"><span class=\"rt-label rt-prefix\">\u0632\u0645\u0627\u0646 \u0644\u0627\u0632\u0645 \u0628\u0631\u0627\u06cc \u0645\u0637\u0627\u0644\u0639\u0647: <\/span> <span class=\"rt-time\"> 15<\/span> <span class=\"rt-label rt-postfix\">\u062f\u0642\u06cc\u0642\u0647<\/span><\/span>\u062f\u0631 \u0627\u06cc\u0646 \u0645\u0642\u0627\u0644\u0647 \u0645\u0646 \u062a\u0648\u0636\u06cc\u062d \u062e\u0648\u0627\u0647\u0645 \u062f\u0627\u062f \u06a9\u0647 \u0627\u0639\u0645\u0627\u0644 \u06cc\u06a9 \u062a\u0628\u062f\u06cc\u0644 \u0627\u0641\u06cc\u0646 \u0628\u0647 \u06cc\u06a9 \u062a\u0635\u0648\u06cc\u0631 \u0686\u06cc\u0633\u062a \u0648 \u0686\u06af\u0648\u0646\u0647 \u0622\u0646 \u0631\u0627 \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646 \u0627\u0646\u062c\u0627\u0645 \u062f\u0647\u06cc\u0645. \u0627\u0628\u062a\u062f\u0627 \u0639\u0645\u0644\u06cc\u0627\u062a \u0633\u0637\u062d \u067e\u0627\u06cc\u06cc\u0646 \u0631\u0627 \u0646\u0634\u0627\u0646 \u062e\u0648\u0627\u0647\u0645 \u062f\u0627\u062f NumPy \u0628\u0631\u0627\u06cc \u0627\u0631\u0627\u0626\u0647 \u06cc\u06a9 \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u0647\u0646\u062f\u0633\u06cc \u062f\u0642\u06cc\u0642. \u0633\u067e\u0633 \u0622\u0646\u0647\u0627 \u0631\u0627 \u062f\u0631 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0639\u0645\u0644\u06cc \u062a\u0631 \u0627\u0632 \u067e\u0627\u06cc\u062a\u0648\u0646 \u0642\u0631\u0627\u0631 \u0645\u06cc \u062f\u0647\u0645 \u0628\u0627\u0644\u0634 \u0648 OpenCV [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":16386,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1743,620],"tags":[],"class_list":["post-16385","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-python","category-programming"],"acf":[],"_links":{"self":[{"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/posts\/16385","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/comments?post=16385"}],"version-history":[{"count":0,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/posts\/16385\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/media\/16386"}],"wp:attachment":[{"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/media?parent=16385"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/categories?post=16385"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/tags?post=16385"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}