Image Analysis and Reconstruction using a Wavelet Transform Constructed from a Reducible Representation of the Euclidean Motion Group

Abstract

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Infrastructure for Retinal Image Analysis

“…The orientation score transform is a wavelet-type transform [8], in which an image is filtered by a set of rotated anisotropic wavelets, i.e. cake wavelets [4].…”

Left Invariant Evolution Equations on Gabor Transforms

“…We adapt the group theoretical approach developed for the Euclidean motion groups in the recent works [9,18,12,13,15,11], thus illustrating the scope of the methods devised for general Lie groups in [10] in signal and image processing. Reassignment will be seen to be a special case of left-invariant convection.…”

“…. , a 2d ) T and/or conductivity matrix D. We will use ideas similar to our previous work on adaptive diffusions on invertible orientation scores [17], [12], [11], [13] (where we employed evolution equations for the 2D-Euclidean motion group). We use the absolute value to adapt the diffusion and convection to avoid oscillations.…”

Nonlinear Diffusion on the 2D Euclidean Motion Group

“…For some choices of K there exists a stable inverse transformation [6], which is obtained by either convolving U (·, θ) with the mirrored conjugate kernel of K followed by integration over θ, or simply by f = 2π 0 U (x, θ)dθ. 1 The space of orientation scores of images V = {U f |f ∈ L2(R 2 )} is a vector subspace of L2(SE (2)).…”

“…A related type of data are orientation scores [5] [6], where orientation is made an explicit dimension. Orientation scores arise naturally in high angular resolution diffusion imaging, but can also be created out of an image by applying a wavelet transform [6]. Both tensor images and orientation scores have in common that they contain richer information on local orientation.…”