50 research outputs found
Mumford-Shah and Potts Regularization for Manifold-Valued Data with Applications to DTI and Q-Ball Imaging
Mumford-Shah and Potts functionals are powerful variational models for
regularization which are widely used in signal and image processing; typical
applications are edge-preserving denoising and segmentation. Being both
non-smooth and non-convex, they are computationally challenging even for scalar
data. For manifold-valued data, the problem becomes even more involved since
typical features of vector spaces are not available. In this paper, we propose
algorithms for Mumford-Shah and for Potts regularization of manifold-valued
signals and images. For the univariate problems, we derive solvers based on
dynamic programming combined with (convex) optimization techniques for
manifold-valued data. For the class of Cartan-Hadamard manifolds (which
includes the data space in diffusion tensor imaging), we show that our
algorithms compute global minimizers for any starting point. For the
multivariate Mumford-Shah and Potts problems (for image regularization) we
propose a splitting into suitable subproblems which we can solve exactly using
the techniques developed for the corresponding univariate problems. Our method
does not require any a priori restrictions on the edge set and we do not have
to discretize the data space. We apply our method to diffusion tensor imaging
(DTI) as well as Q-ball imaging. Using the DTI model, we obtain a segmentation
of the corpus callosum
Joint Image Reconstruction and Segmentation Using the Potts Model
We propose a new algorithmic approach to the non-smooth and non-convex Potts
problem (also called piecewise-constant Mumford-Shah problem) for inverse
imaging problems. We derive a suitable splitting into specific subproblems that
can all be solved efficiently. Our method does not require a priori knowledge
on the gray levels nor on the number of segments of the reconstruction.
Further, it avoids anisotropic artifacts such as geometric staircasing. We
demonstrate the suitability of our method for joint image reconstruction and
segmentation. We focus on Radon data, where we in particular consider limited
data situations. For instance, our method is able to recover all segments of
the Shepp-Logan phantom from angular views only. We illustrate the
practical applicability on a real PET dataset. As further applications, we
consider spherical Radon data as well as blurred data
Model-based learning of local image features for unsupervised texture segmentation
Features that capture well the textural patterns of a certain class of images
are crucial for the performance of texture segmentation methods. The manual
selection of features or designing new ones can be a tedious task. Therefore,
it is desirable to automatically adapt the features to a certain image or class
of images. Typically, this requires a large set of training images with similar
textures and ground truth segmentation. In this work, we propose a framework to
learn features for texture segmentation when no such training data is
available. The cost function for our learning process is constructed to match a
commonly used segmentation model, the piecewise constant Mumford-Shah model.
This means that the features are learned such that they provide an
approximately piecewise constant feature image with a small jump set. Based on
this idea, we develop a two-stage algorithm which first learns suitable
convolutional features and then performs a segmentation. We note that the
features can be learned from a small set of images, from a single image, or
even from image patches. The proposed method achieves a competitive rank in the
Prague texture segmentation benchmark, and it is effective for segmenting
histological images
Graph Spectral Image Processing
Recent advent of graph signal processing (GSP) has spurred intensive studies
of signals that live naturally on irregular data kernels described by graphs
(e.g., social networks, wireless sensor networks). Though a digital image
contains pixels that reside on a regularly sampled 2D grid, if one can design
an appropriate underlying graph connecting pixels with weights that reflect the
image structure, then one can interpret the image (or image patch) as a signal
on a graph, and apply GSP tools for processing and analysis of the signal in
graph spectral domain. In this article, we overview recent graph spectral
techniques in GSP specifically for image / video processing. The topics covered
include image compression, image restoration, image filtering and image
segmentation
Jump-sparse and sparse recovery using Potts functionals
We recover jump-sparse and sparse signals from blurred incomplete data
corrupted by (possibly non-Gaussian) noise using inverse Potts energy
functionals. We obtain analytical results (existence of minimizers, complexity)
on inverse Potts functionals and provide relations to sparsity problems. We
then propose a new optimization method for these functionals which is based on
dynamic programming and the alternating direction method of multipliers (ADMM).
A series of experiments shows that the proposed method yields very satisfactory
jump-sparse and sparse reconstructions, respectively. We highlight the
capability of the method by comparing it with classical and recent approaches
such as TV minimization (jump-sparse signals), orthogonal matching pursuit,
iterative hard thresholding, and iteratively reweighted minimization
(sparse signals)
Multiclass Data Segmentation using Diffuse Interface Methods on Graphs
We present two graph-based algorithms for multiclass segmentation of
high-dimensional data. The algorithms use a diffuse interface model based on
the Ginzburg-Landau functional, related to total variation compressed sensing
and image processing. A multiclass extension is introduced using the Gibbs
simplex, with the functional's double-well potential modified to handle the
multiclass case. The first algorithm minimizes the functional using a convex
splitting numerical scheme. The second algorithm is a uses a graph adaptation
of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates
between diffusion and thresholding. We demonstrate the performance of both
algorithms experimentally on synthetic data, grayscale and color images, and
several benchmark data sets such as MNIST, COIL and WebKB. We also make use of
fast numerical solvers for finding the eigenvectors and eigenvalues of the
graph Laplacian, and take advantage of the sparsity of the matrix. Experiments
indicate that the results are competitive with or better than the current
state-of-the-art multiclass segmentation algorithms.Comment: 14 page
Exact algorithms for -TV regularization of real-valued or circle-valued signals
We consider -TV regularization of univariate signals with values on the
real line or on the unit circle. While the real data space leads to a convex
optimization problem, the problem is non-convex for circle-valued data. In this
paper, we derive exact algorithms for both data spaces. A key ingredient is the
reduction of the infinite search spaces to a finite set of configurations,
which can be scanned by the Viterbi algorithm. To reduce the computational
complexity of the involved tabulations, we extend the technique of distance
transforms to non-uniform grids and to the circular data space. In total, the
proposed algorithms have complexity where is the length
of the signal and is the number of different values in the data set. In
particular, the complexity is for quantized data. It is the
first exact algorithm for TV regularization with circle-valued data, and it is
competitive with the state-of-the-art methods for scalar data, assuming that
the latter are quantized