Why usually consider unitary matrices to define image transforms?
Unitary Matrix for Image Processing
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image-processing
2 Answers
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We can transform back and forth without any matrix inversion, since for any unitary matrix A, $A^{-1}=A^*$ where * is the conjugate transpose.
Unitary matrices represent an orthogonal basis, which is useful in image processing.
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0"Unitary matrices represent an orthogonal basis, which is useful in image processing". For example? – 2012-11-26
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0For example, you know that every coefficient in the transform domain represents only the specific feature that its in charge of. In the Fourier transform, you know that once you get some value $a$ as the coef. for the freq. $f$, this is the value for this frequency only and not a combination of others. – 2012-11-26
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1Another important property of orthogonality is that the transform is produced by just a weighted sum of the original vector, and the weights are just an inner product of some sore, where for a non orthogonal, two sums will be needed, which complicates computations. – 2012-11-29
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0Thanks, by this comments, please any numeric example? – 2012-11-30
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0If you have an orthonormal (complete) basis, you can generally represent every signal in by the basis coefficients as: $x(t) = \sum
e_i$ for the basis functions $e_i$, where $< , >$ is the inner product of the signal with each basis function. If this were not an orthogonal/orthonormal basis, you couldn't have done that. – 2012-11-30
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Because, on a inner product space (i.e. Euclidean space, or Hilbert space in more generality) unitaries are precisely the linear isometric bijections. You want a bijection if you want not to lose information, and you want isometric if you want the distances in your image to be preserved. The linearity, together with the isometry, imply that orthogonality is preserved.
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0and Why image transforms need orthogonality? – 2012-11-26
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0I guess the answer to that is not obvious. But if the most common transforms are translation and rotation, which preserve orthogonality. It depends on what kind of transformations you want to allow; if you allow random reallocation of pixels, then any bijection does, with no other restriction. – 2012-11-26