The Construction of Multiwavelet Bi-Frames and Applications to Variational Image Denoising

Abstract

We remove noise from images by solving a parameter depending variational problem. The choice of the parameter is essential for the success of the approach, and in order to compute a solution, the problem must be discretized. It is commonly known that the parameter choice according to the H-curve criterion performs well in combination with discretizations derived from a dyadic orthonormal wavelet basis. However, the concept of orthonormal wavelet bases is restrictive and bears limitations. In order to have a more flexible tool, we construct new nondyadic wavelet bi-frames by convolving scalar wavelets with wavelet vectors. We discretize the variational problem by these new bi-frames, and we verify that the H-curve method performs well for this much more flexible discretization technique.

Random Tight Frames

“…Frames are basis-like systems that span a vector space but allow for linear dependency, which can be used to reduce noise, find sparse representations, or obtain other desirable features unavailable with orthonormal bases. They have proven useful in fields like spherical codes, compressed sensing, signal processing, and wavelet analysis [6,7,8,10,11,12,14,15,17,16,19,25]. Tight frames even provide a Parseval type formula similar to orthonormal bases.…”

Learning an Adaptive Dictionary Using a Projected Gradient Method and Its Application on Image De-Noising

“…Since then, many competitive algorithms are reported in the literature. [5][6][7][8] Although both variation-based and wavelet-based methods have their strength in image de-noising, their weakness are also obvious. The main weakness of wavelet-based methods is that wavelet transforms cannot take advantage of the geometrical regularity of image structures while the main weakness of variation-based methods is that there is no practically satisfying way to choose the regularization parameter in the energy functionals.…”

Frame theory in directional statistics

“…To predict which multidirectional patterns cause such misclassifications, we merge directional statistics with frame theory. Frames have proven useful in fields like spherical codes, compressed sensing, signal processing, and wavelet analysis [5,8,9,10,11,12,14,15,16,25]. A frame is a basis-like system that spans a vector space but allows for linear dependency, which can be used to reduce noise, find sparse representations, or obtain other desirable features unavailable with orthonormal bases.…”