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    <p>This is a classic linear algebra problem, the key phrase to search on is "multiple linear regression".</p> <p>I've had to code some variation of this many times over the years. For example, code to calibrate a digitizer tablet or stylus touch-screen uses the same math. </p> <hr> <p>Here's the math:</p> <p>Let <strong>p</strong> be an input vector and <strong>q</strong> the corresponding output vector.</p> <p>The transformation you want is a 3x3 matrix; call it <strong>A</strong>.</p> <p>For a single input and output vector <strong>p</strong> and <strong>q</strong>, there is an error vector <strong>e</strong></p> <p><strong>e</strong> = <strong>q</strong> - <strong>A</strong> x <strong>p</strong></p> <p>The square of the magnitude of the error is a scalar value:</p> <p><strong>e</strong>T x <strong>e</strong> = (<strong>q</strong> - <strong>A</strong> x <strong>p</strong>)T x (<strong>q</strong> - <strong>A</strong> x <strong>p</strong>)</p> <p>(where the T operator is transpose).</p> <p>What you really want to minimize is the sum of <strong>e</strong> values over the sets:</p> <p><strong>E</strong> = sum (<strong>e</strong>)</p> <p>This minimum satisfies the matrix equation <strong>D</strong> = 0 where</p> <p><strong>D</strong>(i,j) = the partial derivative of <strong>E</strong> with respect to <strong>A</strong>(i,j)</p> <p>Say you have N input and output vectors.</p> <p>Your set of input 3-vectors is a 3xN matrix; call this matrix <strong>P</strong>. The ith column of <strong>P</strong> is the ith input vector.</p> <p>So is the set of output 3-vectors; call this matrix <strong>Q</strong>.</p> <p>When you grind thru all of the algebra, the solution is</p> <p><strong>A</strong> = <strong>Q</strong> x <strong>P</strong>T x (<strong>P</strong> x <strong>P</strong>T) ^-1</p> <p>(where ^-1 is the inverse operator -- sorry about no superscripts or subscripts)</p> <hr> <p>Here's the algorithm:</p> <p>Create the 3xN matrix <strong>P</strong> from the set of input vectors.</p> <p>Create the 3xN matrix <strong>Q</strong> from the set of output vectors.</p> <p>Matrix Multiply <strong>R</strong> = <strong>P</strong> x transpose (<strong>P</strong>)</p> <p>Compute the inverseof <strong>R</strong></p> <p>Matrix Multiply <strong>A</strong> = <strong>Q</strong> x transpose(<strong>P</strong>) x inverse (<strong>R</strong>)</p> <p>using the matrix multiplication and matrix inversion routines of your linear algebra library of choice. </p> <hr> <p><strong>However</strong>, a 3x3 affine transform matrix is capable of scaling and rotating the input vectors, but <strong>not</strong> doing any translation! This might not be general enough for your problem. It's usually a good idea to append a "1" on the end of each of the 3-vectors to make then a 4-vector, and look for the best 3x4 transform matrix that minimizes the error. This can't hurt; it can only lead to a better fit of the data.</p>
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    1. POfinding matrix through optimisation
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    1. COThis is a great answer too, but explanation were better (stackoverflow is not great for math) in the link I discovered in Kena's answer. So Kena got the bounty.
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    2. CO@shodnex: fair enough!
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