Iterative Basis Pursuit for Image Sequence Denoising

An iterative method is purposed in this paper using the basis pursuit algorithm for spatial denoising, coupled with temporal wavelet denoising to result in a denoised video signal. Introduction Several new techniques have been developed recently for the purposes of denoising images. The most promising of these techniques have been the curvelet and undecimated wavelet transforms. Using a basis pursuit algorithm, the advantages of each algorithm can be taken advantage of to produce a spatially denoised image. When working with image sequences, the correlation between frames can be used to denoise images. Considering each pixel as a time-domain signal across multiple frames, this signal can be denoised using hard thresholding of wavelets. This paper purposes a iterative method of basis pursuit spatial denoising combined with temporal denoising using wavelets to produce a denoised image sequence signal. Curvelet Algorithm The Curvelet transform (original purposed in [4]) consists of an overcomplete representation of an image using a series of L2 energy measurements ranging across scale, orientation, and position. Each curvelet consists of a tight frame constrained over a slice of the fourier domain. In the spatial domain, the curvelet is a scaled and rotated gabor signal along the width and is a scaled and rotated gaussian signal along the length. One of the most important properties of the curvelet is length = width^2 (length = 2^-J, width = 2^-(2*J), J = scale). This allows for the curvelet to act like a needle at fine scale representations. A brief overview of the mathematical framework from [2] is now presented to give the reader a formal representation of curvelets. Each curvelet is defined by three parameters (J, K, L). J = Scale L = orientation K = location Brian Eriksson – Iterative Basis Pursuit for Image Sequence Denoising 2 Parabolic Scaling Matrix: ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = J j