19 research outputs found
Colour image segmentation by the vector-valued Allen-Cahn phase-field model: a multigrid solution
We propose a new method for the numerical solution of a PDE-driven model for
colour image segmentation and give numerical examples of the results. The
method combines the vector-valued Allen-Cahn phase field equation with initial
data fitting terms. This method is known to be closely related to the
Mumford-Shah problem and the level set segmentation by Chan and Vese. Our
numerical solution is performed using a multigrid splitting of a finite element
space, thereby producing an efficient and robust method for the segmentation of
large images.Comment: 17 pages, 9 figure
A stochastic-variational model for soft Mumford-Shah segmentation
In contemporary image and vision analysis, stochastic approaches demonstrate
great flexibility in representing and modeling complex phenomena, while
variational-PDE methods gain enormous computational advantages over Monte-Carlo
or other stochastic algorithms. In combination, the two can lead to much more
powerful novel models and efficient algorithms. In the current work, we propose
a stochastic-variational model for soft (or fuzzy) Mumford-Shah segmentation of
mixture image patterns. Unlike the classical hard Mumford-Shah segmentation,
the new model allows each pixel to belong to each image pattern with some
probability. We show that soft segmentation leads to hard segmentation, and
hence is more general. The modeling procedure, mathematical analysis, and
computational implementation of the new model are explored in detail, and
numerical examples of synthetic and natural images are presented.Comment: 22 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
Efficient Denoising and Sharpening of Color Images through Numerical Solution of Nonlinear Diffusion Equations
The purpose of this project is to enhance color images through denoising and sharpening, two important branches of image processing, by mathematically modeling the images. Modifications are made to two existing nonlinear diffusion image processing models to adapt them to color images. This is done by treating the red, green, and blue (RGB) channels of color images independently, contrary to the conventional idea that the channels should not be treated independently. A new numerical method is needed to solve our models for high resolution images since current methods are impractical. To produce an efficient method, the solution is represented as a linear combination of sines and cosines for easier numerical treatment and then computed by a combination of Krylov subspace spectral (KSS) methods and exponential propagation iterative (EPI) methods. Numerical experiments demonstrate that the proposed approach for image processing is effective for denoising and sharpening
A Second Order Non-Smooth Variational Model for Restoring Manifold-Valued Images
We introduce a new non-smooth variational model for the restoration of
manifold-valued data which includes second order differences in the
regularization term. While such models were successfully applied for
real-valued images, we introduce the second order difference and the
corresponding variational models for manifold data, which up to now only
existed for cyclic data. The approach requires a combination of techniques from
numerical analysis, convex optimization and differential geometry. First, we
establish a suitable definition of absolute second order differences for
signals and images with values in a manifold. Employing this definition, we
introduce a variational denoising model based on first and second order
differences in the manifold setup. In order to minimize the corresponding
functional, we develop an algorithm using an inexact cyclic proximal point
algorithm. We propose an efficient strategy for the computation of the
corresponding proximal mappings in symmetric spaces utilizing the machinery of
Jacobi fields. For the n-sphere and the manifold of symmetric positive definite
matrices, we demonstrate the performance of our algorithm in practice. We prove
the convergence of the proposed exact and inexact variant of the cyclic
proximal point algorithm in Hadamard spaces. These results which are of
interest on its own include, e.g., the manifold of symmetric positive definite
matrices
Total variation regularization for manifold-valued data
We consider total variation minimization for manifold valued data. We propose
a cyclic proximal point algorithm and a parallel proximal point algorithm to
minimize TV functionals with -type data terms in the manifold case.
These algorithms are based on iterative geodesic averaging which makes them
easily applicable to a large class of data manifolds. As an application, we
consider denoising images which take their values in a manifold. We apply our
algorithms to diffusion tensor images, interferometric SAR images as well as
sphere and cylinder valued images. For the class of Cartan-Hadamard manifolds
(which includes the data space in diffusion tensor imaging) we show the
convergence of the proposed TV minimizing algorithms to a global minimizer