12,601 research outputs found
Left-invariant evolutions of wavelet transforms on the Similitude Group
Enhancement of multiple-scale elongated structures in noisy image data is
relevant for many biomedical applications but commonly used PDE-based
enhancement techniques often fail at crossings in an image. To get an overview
of how an image is composed of local multiple-scale elongated structures we
construct a multiple scale orientation score, which is a continuous wavelet
transform on the similitude group, SIM(2). Our unitary transform maps the space
of images onto a reproducing kernel space defined on SIM(2), allowing us to
robustly relate Euclidean (and scaling) invariant operators on images to
left-invariant operators on the corresponding continuous wavelet transform.
Rather than often used wavelet (soft-)thresholding techniques, we employ the
group structure in the wavelet domain to arrive at left-invariant evolutions
and flows (diffusion), for contextual crossing preserving enhancement of
multiple scale elongated structures in noisy images. We present experiments
that display benefits of our work compared to recent PDE techniques acting
directly on the images and to our previous work on left-invariant diffusions on
orientation scores defined on Euclidean motion group.Comment: 40 page
A Regularization Term Based on a Discrete Total Variation for Mathematical Image Processing
In this paper, a new regularization term is proposed to solve mathematical
image problems. By using difference operators in the four directions;
horizontal, vertical and two diagonal directions, an estimation of derivative
amplitude is found. Based on the new obtained estimation, a new regularization
term will be defined, which can be viewed as a new discretized total variation
(TVprn) model. By improving TVprn, a more effective regularization term is
introduced. By finding conjugate of TVprn and producing vector fields with
special constraints, a new discretized TV for two dimensional discrete
functions is proposed (TVnew). The capability of the new TV model to solve
mathematical image problems is examined in some numerical experiments. It is
shown that the new proposed TV model can reconstruct the edges and corners of
the noisy images better than other TVs. Moreover, two test experiments of
resolution enhancement problem are solved and compared with some other
different TVs
Multiscale Representations for Manifold-Valued Data
We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere , the special orthogonal group , the positive definite matrices , and the Grassmann manifolds . The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the and maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as , , , where the and maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper
- …