285 research outputs found
Piecewise rigid curve deformation via a Finsler steepest descent
This paper introduces a novel steepest descent flow in Banach spaces. This
extends previous works on generalized gradient descent, notably the work of
Charpiat et al., to the setting of Finsler metrics. Such a generalized gradient
allows one to take into account a prior on deformations (e.g., piecewise rigid)
in order to favor some specific evolutions. We define a Finsler gradient
descent method to minimize a functional defined on a Banach space and we prove
a convergence theorem for such a method. In particular, we show that the use of
non-Hilbertian norms on Banach spaces is useful to study non-convex
optimization problems where the geometry of the space might play a crucial role
to avoid poor local minima. We show some applications to the curve matching
problem. In particular, we characterize piecewise rigid deformations on the
space of curves and we study several models to perform piecewise rigid
evolution of curves
Optical Flow on Moving Manifolds
Optical flow is a powerful tool for the study and analysis of motion in a
sequence of images. In this article we study a Horn-Schunck type
spatio-temporal regularization functional for image sequences that have a
non-Euclidean, time varying image domain. To that end we construct a Riemannian
metric that describes the deformation and structure of this evolving surface.
The resulting functional can be seen as natural geometric generalization of
previous work by Weickert and Schn\"orr (2001) and Lef\`evre and Baillet (2008)
for static image domains. In this work we show the existence and wellposedness
of the corresponding optical flow problem and derive necessary and sufficient
optimality conditions. We demonstrate the functionality of our approach in a
series of experiments using both synthetic and real data.Comment: 26 pages, 6 figure
Time Discrete Geodesic Paths in the Space of Images
In this paper the space of images is considered as a Riemannian manifold
using the metamorphosis approach, where the underlying Riemannian metric
simultaneously measures the cost of image transport and intensity variation. A
robust and effective variational time discretization of geodesics paths is
proposed. This requires to minimize a discrete path energy consisting of a sum
of consecutive image matching functionals over a set of image intensity maps
and pairwise matching deformations. For square-integrable input images the
existence of discrete, connecting geodesic paths defined as minimizers of this
variational problem is shown. Furthermore, -convergence of the
underlying discrete path energy to the continuous path energy is proved. This
includes a diffeomorphism property for the induced transport and the existence
of a square-integrable weak material derivative in space and time. A spatial
discretization via finite elements combined with an alternating descent scheme
in the set of image intensity maps and the set of matching deformations is
presented to approximate discrete geodesic paths numerically. Computational
results underline the efficiency of the proposed approach and demonstrate
important qualitative properties.Comment: 27 pages, 7 figure
Differential geometric regularization for supervised learning of classifiers
We study the problem of supervised learning for both binary and multiclass classification from a unified geometric perspective. In particular, we propose a geometric regularization technique to find the submanifold corresponding to an estimator of the class probability P(y|\vec x). The regularization term measures the volume of this submanifold, based on the intuition that overfitting produces rapid local oscillations and hence large volume of the estimator. This technique can be applied to regularize any classification function that satisfies two requirements: firstly, an estimator of the class probability can be obtained; secondly, first and second derivatives of the class probability estimator can be calculated. In experiments, we apply our regularization technique to standard loss functions for classification, our RBF-based implementation compares favorably to widely used regularization methods for both binary and multiclass classification.http://proceedings.mlr.press/v48/baia16.pdfPublished versio
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
A level set method with Sobolev Gradient and Haralick Edge Detection
Variational level set methods, which have been proposed with various energy functionals, mostly use the ordinary L type gradient in gradient descent algorithm to minimize the energy functional. The gradient flow is influenced by both the energy to be minimized and the norms, which are induced from inner products, used to measure the cost of perturbation of the curve. However, there are many undesired properties related to the gradient flows due to the 2 L type inner products. For example, there is not any regularity term in the definition of this inner product that causes non-smooth flows and inaccurate results. Therefore, in this work, Sobolev gradient has been used that is more efficient than the 2 L type gradient for image segmentation and has powerful properties such as regular gradient flows, independency to parameterization of curves, less sensitive to local features and noise in the image and also faster convergence rate than the standard gradient. In addition, Haralick edge detector has been used instead of the edge indicator function in this study. Because, the traditional edge indicator function, which is the absolute of the gradient of the convolved image with the aussian function, is sensitive to noise in level set methods. Experimental results on real images , which are abdominal magnetic resonance images, have been obtained for spleen and kidney segmentation. Quantitative analyses have been performed by using different measurements to evaluate the performance of the proposed approach, which can ignore topological noises and detect boundaries successfully
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