# Rotationally-Invariant 2D Discrete Transforms

I'm interested in 2D discrete transforms (such as discrete wavelet transforms, Curvelets, Ridgelets, Beamlets etc.) that operate on a discrete unit disk and:

1. Are invariant to rotations only
2. Output a transformed signal with relatively low informational entropy
3. Are computationally relatively efficient (in terms of computational complexity)

In other words, I'm interested in 2D discrete transforms that output the same transformation for arbitrary 2D rotations of the input, but that are not invariant to any other changes of the input. Ideally, these transforms should compress the input as much as possible in terms of information entropy (i.e. necessary bits to represent the output), and be "computable" in a practical sense.

As additional context to my question, I am planning on using such transforms in the domain of computer vision to train a classifier on instances of objects that might appear rotated arbitrarily around the image center point.

Update: From what I have learned in the past few days, Zernike polynomials are orthogonal and complete on the unit disk, and the absolute value of the Zernike coefficients are apparently invariant to rotation of the input. However, there are two problems with this, as far as I can tell:

1. The original Zernike transform is defined on the continuous disk, for which the spectrum is not bounded, and infinitely many coefficients may be needed to reverse the transform; i.e. it is not really a discrete transform. In discrete transforms, such as the Discrete Cosine Transform (DCT) or the Discrete Fourier Transform (DTFT), the spectrum is bounded and discrete, and the number of harmonics is given by the dimensionality of the transform domain. Also, the Zernike polynomials do not form a basis on the sampled disk (See "Wave-front interpretation with Zernike Polynomials" by J. Y. Wang and D. E. Silva Applied Optics, Vol. 19, Issue 9, pp. 1510-1518 (1980)).

2. The absolute value of the Zernike coefficients may not be injective beyond rotation (i.e. perturbations of the input function, other than rotations, may be transformed to the same coefficients).

Thank you

• Rotations about what? If you pick a basepoint in advance, then averaging the $O(2)$ orbit compresses the signal A LOT. – Igor Rivin Jan 9 '11 at 2:06
• It seems to me that "2D discrete transform" does not go well with "arbitraty 2D rotation". Do you assume that your objects to be rotated are matrices in some sense? Then only rotation about 90° make sense (without interpolation)... – Dirk Jan 9 '11 at 12:06
• @Igor, I updated the question. We can assume rotations around the center point of a disk that holds the 2D signal. Would averaging the $O(2)$ orbit be invariant to rotations only? – user12103 Jan 9 '11 at 14:06
• @Dirk, the "standard" representation of a 2D image in computer vision is a matrix, but for our purposes we can assume that the input signal is a disk of known radius. – user12103 Jan 9 '11 at 14:08
• @AV80: In what sense is the signal "discrete" then? – Noah Stein Jan 9 '11 at 14:28