PDEs for Tensor Image Processing

Weickert, Joachim; Feddern, Christian; Welk, Martin; Burgeth, Bernhard; Brox, Thomas

doi:10.1007/3-540-31272-2_25

Cited by 10 publications

(9 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In various applications in image processing and computer vision the functions of interest take values in a Riemannian manifold. One example is diffusion tensor imaging where the data is given on the Hadamard manifold of positive definite matrices; see, e.g., [9,21,20,22,62,69,78,80]. In the following we are interested in generalizations of ROF-like functionals to manifold-valued settings, more precisely to data having values in symmetric Hadamard manifolds.…”

Section: Introductionmentioning

confidence: 99%

A Parallel Douglas–Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds

Bergmann¹,

Persch²,

Steidl³

2016

SIAM J. Imaging Sci.

View full text Add to dashboard Cite

We are interested in restoring images having values in a symmetric Hadamard manifold by minimizing a functional with a quadratic data term and a total variation like regularizing term. To solve the convex minimization problem, we extend the Douglas-Rachford algorithm and its parallel version to symmetric Hadamard manifolds. The core of the Douglas-Rachford algorithm are reflections of the functions involved in the functional to be minimized. In the Euclidean setting the reflections of convex lower semicontinuous functions are nonexpansive. As a consequence, convergence results for Krasnoselski-Mann iterations imply the convergence of the Douglas-Rachford algorithm. Unfortunately, this general results does not carry over to Hadamard manifolds, where proper convex lower semicontinuous functions can have expansive reflections. However, splitting our restoration functional in an appropriate way, we have only to deal with special functions namely, several distance-like functions and an indicator functions of a special convex sets. We prove that the reflections of certain distance-like functions on Hadamard manifolds are nonexpansive which is an interesting result on its own. Furthermore, the reflection of the involved indicator function is nonexpansive on Hadamard manifolds with constant curvature so that the Douglas-Rachford algorithm converges here.Several numerical examples demonstrate the advantageous performance of the suggested algorithm compared to other existing methods as the cyclic proximal point algorithm or half-quadratic minimization. Numerical convergence is also observed in our experiments on the Hadamard manifold of symmetric positive definite matrices with the affine invariant metric which does not have a constant curvature.

show abstract

Section: Introductionmentioning

confidence: 99%

A Parallel Douglas–Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds

Bergmann¹,

Persch²,

Steidl³

2016

SIAM J. Imaging Sci.

View full text Add to dashboard Cite

show abstract

“…The explication of this result, according to the mathematical literature, is that under curvature motion convex lines remain convex where as non-convex ones become convex. Furthermore, in finite time, these lines vanish by approximating circular shapes while converging to points [11].…”

Section: Divergence Pde-based Approachmentioning

confidence: 98%

OCR Accuracy Improvement through a PDE-Based Approach

Drira

Lebourgeois

Emptoz

2007

Ninth International Conference on Document Analysis and Recognition (ICDAR 2007) Vol 2

View full text Add to dashboard Cite

This paper focuses on improving the Optical Character Recognition (OCR) system's accuracy by restoring damaged character through a PDE (Partial Differential Equation)-based approach. This approach, proposed by D. Tschumperlé, is an anisotropic diffusion approach driven by local tensors fields. Actually, such approach has many useful properties that are relevant for use in character restoration. For instance, this approach is very appropriate for the processing of oriented patterns which are major characteristics of textual documents. It incorporates both edge enhancing diffusion that tends to preserve local structures during smoothing and coherence-enhancing diffusion that processes oriented structures by smoothing along the flow direction. Furthermore, this tensor diffusion-based approach compared to the existing sate of the art requires neither segmentation nor training steps. Some experiments, done on degraded document images, illustrate the performance of this PDE-based approach in improving both of the visual quality and the OCR accuracy rates for degraded document images.

show abstract

“…Klein and Ertl track vector field critical points through discretely sampled scales, to better filter out noise effects [34]. In the image processing community, non-linear filtering of tensor fields has been studied [13, 65, 9], which is conceptually related to scale-space by its mathematical connection to earlier work by Perona and Malik on nonlinear scale-space [55], but there has been relatively little study of the scale-space features in tensor fields. Florack and Astola argue that a scale-space treatment of diffusion tensor fields should respect commutativity of blurring and tensor inversion, and describe the mathematical components of the resulting non-linear scale-space [19].…”

Section: Related Workmentioning

confidence: 99%

Sampling and Visualizing Creases with Scale-Space Particles

Kindlmann

Estépar

Smith

et al. 2009

IEEE Trans. Visual. Comput. Graphics

View full text Add to dashboard Cite

Particle systems have gained importance as a methodology for sampling implicit surfaces and segmented objects to improve mesh generation and shape analysis. We propose that particle systems have a significantly more general role in sampling structure from unsegmented data. We describe a particle system that computes samplings of crease features (i.e. ridges and valleys, as lines or surfaces) that effectively represent many anatomical structures in scanned medical data. Because structure naturally exists at a range of sizes relative to the image resolution, computer vision has developed the theory of scale-space, which considers an n-D image as an (n + 1)-D stack of images at different blurring levels. Our scale-space particles move through continuous four-dimensional scale-space according to spatial constraints imposed by the crease features, a particle-image energy that draws particles towards scales of maximal feature strength, and an inter-particle energy that controls sampling density in space and scale. To make scale-space practical for large three-dimensional data, we present a spline-based interpolation across scale from a small number of pre-computed blurrings at optimally selected scales. The configuration of the particle system is visualized with tensor glyphs that display information about the local Hessian of the image, and the scale of the particle. We use scale-space particles to sample the complex three-dimensional branching structure of airways in lung CT, and the major white matter structures in brain DTI.

show abstract

PDEs for Tensor Image Processing

Cited by 10 publications

References 34 publications

A Parallel Douglas–Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds

A Parallel Douglas–Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds

OCR Accuracy Improvement through a PDE-Based Approach

Sampling and Visualizing Creases with Scale-Space Particles

Contact Info

Product

Resources

About