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    <p>To apply a perspective transformation you first have to know four points in a plane A that will be mapped to four points in a plane B. With those points, you can derive the homographic transform. By doing this, you obtain your 8 coefficients and the transformation can take place.</p> <p>The site <a href="http://xenia.media.mit.edu/~cwren/interpolator/" rel="nofollow noreferrer">http://xenia.media.mit.edu/~cwren/interpolator/</a> (mirror: <a href="https://web.archive.org/web/20150222120106/xenia.media.mit.edu/~cwren/interpolator/" rel="nofollow noreferrer">WebArchive</a>), as well as many other texts, describes how those coefficients can be determined. To make things easy, here is a direct implementation according from the mentioned link:</p> <pre><code>import numpy def find_coeffs(pa, pb): matrix = [] for p1, p2 in zip(pa, pb): matrix.append([p1[0], p1[1], 1, 0, 0, 0, -p2[0]*p1[0], -p2[0]*p1[1]]) matrix.append([0, 0, 0, p1[0], p1[1], 1, -p2[1]*p1[0], -p2[1]*p1[1]]) A = numpy.matrix(matrix, dtype=numpy.float) B = numpy.array(pb).reshape(8) res = numpy.dot(numpy.linalg.inv(A.T * A) * A.T, B) return numpy.array(res).reshape(8) </code></pre> <p>where <code>pb</code> is the four vertices in the current plane, and <code>pa</code> contains four vertices in the resulting plane.</p> <p>So, suppose we transform an image as in:</p> <pre><code>import sys from PIL import Image img = Image.open(sys.argv[1]) width, height = img.size m = -0.5 xshift = abs(m) * width new_width = width + int(round(xshift)) img = img.transform((new_width, height), Image.AFFINE, (1, m, -xshift if m &gt; 0 else 0, 0, 1, 0), Image.BICUBIC) img.save(sys.argv[2]) </code></pre> <p>Here is a sample input and output with the code above:</p> <p><img src="https://i.stack.imgur.com/dHGcB.png" alt="enter image description here"> <img src="https://i.stack.imgur.com/EOwht.png" alt="enter image description here"></p> <p>We can continue on the last code and perform a perspective transformation to revert the shear:</p> <pre><code>coeffs = find_coeffs( [(0, 0), (256, 0), (256, 256), (0, 256)], [(0, 0), (256, 0), (new_width, height), (xshift, height)]) img.transform((width, height), Image.PERSPECTIVE, coeffs, Image.BICUBIC).save(sys.argv[3]) </code></pre> <p>Resulting in:</p> <p><img src="https://i.stack.imgur.com/wY6iQ.png" alt="enter image description here"></p> <p>You can also have some fun with the destination points:</p> <p><img src="https://i.stack.imgur.com/GicNm.png" alt="enter image description here"> <img src="https://i.stack.imgur.com/tYwvt.png" alt="enter image description here"></p>
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    1. COYour answer is very helpful and clear. Thank you. Are you aware of any pure-python implementation of `def find_coeffs(pa, pb)`? I'm hoping to avoid adding a numpy dependency for a non-central part of my system. I guess I can work it out myself but I'm hoping it is out there somewhere already.
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