In this paper, we propose a method to solve PDEs on surfaces with arbitrary topologies by using the global conformal parametrization. The main idea of this method is to map the surface conformally to 2D rectangular areas and then transform the PDE on the 3D surface into a modified PDE on the 2D parameter domain. Consequently, we can solve the PDE on the parameter domain by using some well-known numerical schemes on ℝ2. To do this, we have to define a new set of differential operators on the manifold such that they are coordinates invariant. Since the Jacobian of the conformal mapping is simply a multiplication of the conformal factor, the modified PDE on the parameter domain will be very simple and easy to solve. In our experiments, we demonstrated our idea by solving the Navier-Stoke’s equation on the surface. We also applied our method to some image processing problems such as segmentation, image denoising and image inpainting on the surfaces.
The need to regularize tensor fields arise recently in various applications. We treat in this paper tensors that belong to matrix Lie groups. We formulate the problem of these SO(N) flows in terms of the principal chiral model (PCM) action. This action is defined over a Lie group manifold, By minimizing the PCM action with respect to the group element, we obtain the equations of motion for the group element (or the corresponding connection). Then, by writing the gradient descent equations we obtain the PDE for the Lie group flows. We use these flows to regularize in particular the group of N-dimensional orthogonal matrices with determinant one i.e. SO(N). This type of regularization preserves their properties (i.e., the orthogonality and the determinant). A special numerical scheme that preserves the Lie group structure is used. However, these flows regularize the tensor field isotropically and therefore discontinuities are not preserved. We modify the functional and thereby the gradient descent PDEs in order to obtain an anisotropic tensor field regularization. We demonstrate our formalism with various examples.
We propose a variational approach for deblurring and impulsive noise removal in multi-channel images. A robust data fidelity measure and edge preserving regularization are employed. We consider several regularization approaches, such as Beltrami flow, Mumford-Shah and Total-Variation Mumford-Shah. The latter two methods are extended to multi-channel images and reformulated using the Γ-convergence approximation. Our main contribution is in the unification of image deblurring and impulse noise removal in a multi-channel variational framework. Theoretical and experimental results show that the Mumford-Shah and Total Variation Mumford Shah regularization methods are superior to other color image restoration regularizers. In addition, these two methods yield a denoised edge map of the image.
PDE-based image inpainting efficiently recovers structured features. We expand this to textures. We adjust the coordinates to proper directions, and embed in anisotropy terms the brightness correlation between pixels adjoining on the new grid. A simple elliptic equation then repairs both oriented textures and edges by one uniform, automated algorithm. Extensive experimental results on a variety of standard natural images show the technique's generality and stability.
In this paper we propose a surface reconstruction method for highly noisy and non-uniform data based on minimal surface model and tensor voting method. To deal with ill-posedness, noise and/or other uncertainties in the data we processes the raw data first using tensor voting before we do surface reconstruction. The tensor voting procedure allows more global and robust communications among the data to extract coherent geometric features and saliency independent of the surface reconstruction. These extracted information will be used to preprocess the data and to guide the final surface reconstruction. Numerically the level set method is used for surface reconstruction. Our method can handle complicated topology as well as highly noisy and/or non-uniform data set, Moreover, improvements of efficiency in implementing the tensor voting method are also proposed. We demonstrate the ability of our method using synthetic and real data.
In this paper we describe a new framework for the tracking of closed curves described through implicit surface modeling. The approach proposed here enables a continuous tracking along an image sequence of deformable object contours. Such an approach is formalized through the minimization of a global spatio-temporal continuous cost functional stemming from a Bayesian Maximum a posteriori estimation of a Gaussian probability distribution. The resulting minimization sequence consists in a forward integration of an evolution law followed by a backward integration of an adjoint evolution model, This latter pde include also a term related to the discrepancy between the curve evolution law and a noisy observation of the curve. The efficiency of the approach is demonstrated on image sequences showing deformable objects of different natures.
Converting point samples and/or triangular meshes to a more compact spline representation for arbitrarily topology is both desirable and necessary for computer vision and computer graphics. This paper presents a C1 manifold interpolatory spline that can exactly pass through all the vertices and interpolate their normals for data input of complicated topological type. Starting from the Powell-Sabin spline as a building block, we integrate the concepts of global parametrization, affine atlas, and splines defined over local, open domains to arrive at an elegant, easy-to-use spline solution for complicated datasets. The proposed global spline scheme enables the rapid surface reconstruction and facilitates the shape editing and analysis functionality.
This paper addresses the problem of calibrating camera lens distortion, which can be significant in medium to wide-angle lenses. Our approach is based on the analysis of distorted images of straight lines. We use a PDE-based level set method to find the lens distortion parameters that straighten these lines. One advantage of this method is that it integrates the extraction of image distorted lines and the computation of distortion parameters within one energy functional which is minimized during level set evolution. Some experiments to evaluate the performance of this approach are reported.
We present a computationally efficient segmentation–restoration method, based on a probabilistic formulation, for the joint estimation of the label map (segmentation) and the parameters of the feature generator models (restoration). Our algorithm computes an estimation of the posterior marginal probability distributions of the label field based on a Gauss Markov Random Measure Field model. Our proposal introduces an explicit entropy control for the estimated posterior marginals, therefore it improves the parameter estimation step. If the model parameters are given, our algorithm computes the posterior marginals as the global minimizers of a quadratic, linearly constrained energy function; therefore, one can compute very efficiently the optimal (Maximizer of the Posterior Marginals or MPM) estimator for multi–class segmentation problems. Moreover, a good estimation of the posterior marginals allows one to compute estimators different from the MPM for restoration problems, denoising and optical flow computation. Experiments demonstrate better performance over other state of the art segmentation approaches.
This paper explores new aspects of the image decomposition problem using modern variational techniques. We aim at splitting an original image f into two components u and v, where u holds the geometrical information and v holds the textural information. Our aim is to provide the necessary variational tools and suggest the suitable functional spaces to extract specific types of textures. Our modeling uses the total-variation semi-norm for extracting the structural part and a new tunable norm, presented here for the first time, based on Gabor functions, for the textural part. A way to select the splitting parameter based on the orthogonality of structure and texture is also suggested.
We study in this paper the problem of regularization of mappings between manifolds of arbitrary dimension and codimension using variational methods. This is of interest in various applications such as diffusion tensor imaging and EEG processing on the cortex. We consider the cases where the source and target manifold are represented implicitly, using multiple level set functions, or explicitly, as functions of the spatial coordinates. We derive the general implicit differential operators, and show how they can be used to generalize previous results concerning the Beltrami flow and other similar flows. As examples, We show how these results can be used to regularize gray level and color images on manifolds, and to regularize tangent vector fields and direction fields on manifolds.
While methods based on partial differential equations (PDEs) and variational techniques are powerful tools for denoising and inpainting digital images, their use for image compression was mainly focussing on pre- or postprocessing so far. In our paper we investigate their potential within the decoding step. We start with the observation that edge-enhancing diffusion (EED), an anisotropic nonlinear diffusion filter with a diffusion tensor, is well-suited for scattered data interpolation: Even when the interpolation data are very sparse, good results are obtained that respect discontinuities and satisfy a maximum–minimum principle. This property is exploited in our studies on PDE-based image compression. We use an adaptive triangulation method based on B-tree coding for removing less significant pixels from the image. The remaining points serve as scattered interpolation data for the EED process. They can be coded in a compact and elegant way that reflects the B-tree structure. Our experiments illustrate that for high compression rates and non-textured images, this PDE-based approach gives visually better results than the widely-used JPEG coding.
Current state-of-the-art methods in variational image segmentation using level set methods are able to robustly segment complex textured images in an unsupervised manner. In recent work, [18,19] we have explored the potential of AM-FM features for driving the unsupervised segmentation of a wide variety of textured images. Our first contribution in this work is at the feature extraction level, where we introduce a regularized approach to the demodulation of the AM-FM-modelled signals. By replacing the cascade of multiband filtering and subsequent differentiation with analytically derived equivalent filtering operations, increased noise-robustness can be achieved, while discretization problems in the implementation of the demodulation algorithm are alleviated. Our second contribution is based on a generative model we have recently proposed [18,20] that offers a measure related to the local prominence of a specific class of features, like edges and textures. The introduction of these measures as weighting terms in the evolution equations facilitates the fusion of different cues in a simple and efficient manner. Our systematic evaluation on the Berkeley segmentation benchmark demonstrates that this fusion method offers improved results when compared to our previous work as well as current state-of-the-art methods.
In this paper we propose a novel variational technique for the knowledge based segmentation of two dimensional objects. One of the elements of our approach is the use of higher order implicit polynomials to represent shapes. The most important contribution is the estimation of uncertainties on the registered shapes, which can be used with a variable bandwidth kernel-based non-parametric density estimation process to model prior knowledge about the object of interest. Such a non-linear model with uncertainty measures is integrated with an adaptive visual-driven data term that aims to separate the object of interest from the background. Promising results obtained for the segmentation of the corpus callosum in MR mid-sagittal brain slices demonstrate the potential of such a framework.
Many problems in image analysis and computer vision involving boundaries and regions can be cast in a variational formulation. This means that m-surfaces, e.g. curves and surfaces, are determined as minimizers of functionals using e.g. the variational level set method. In this paper we consider such variational problems with constraints given by functionals. We use the geometric interpretation of gradients for functionals to construct gradient descent evolutions for these constrained problems. The result is a generalization of the standard gradient projection method to an infinite-dimensional level set framework. The method is illustrated with examples and the results are valid for surfaces of any dimension.
For non-rigid registration, the objects in medical images are usually treated as a single deformable body with homogeneous stiffness distribution. However, this assumption is invalid for certain parts of the human body, where bony structures move rigidly, while the others may deform. In this paper, we introduce a novel registration technique that models local rigidity of pre-identified rigid structures as well as global non-rigidity in the transformation field using triangular B-splines. In contrast to the conventional registration method based on tensor-product B-splines, our approach recovers local rigid transformation with fewer degrees of freedom (DOFs), and accurately simulates sharp features (C 0 continuity) along the interface between deformable regions and rigid structures, because of the unique advantages offered by triangular B-splines, such as flexible triangular domain, local control and space-varying smoothness modeling. The accurate matching of the source image with the target one is accomplished through the use of a variational framework, in which a composite energy, measuring the image dissimilarity and enforcing local rigidity and global smoothness, is minimized subject to pre-defined point-based constraints. The algorithm is tested on both synthetic and real 2D images for its applicability. The experimental results show that, by accurately modeling sharp features using triangular B-splines, the deformable regions in the vicinity of rigid structures are less constrained by the global smoothness regularization and therefore contribute extra flexibility to the optimization process. Consequently, the registration quality is improved considerably.
We develop a fast numerical method to approximate the solutions of a wide class of equations associated to the Shape From Shading problem. Our method, which is based on the control theory and the interfaces propagation, is an extension of the "Fast Marching Method" (FMM) [30,25]. In particular our method extends the FMM to some equations for which the solution is not systematically decreasing along the optimal trajectories. We apply with success our one-pass method to the Shape From Shading equations which are involved by the most relevant and recent modelings [22,21] and which cannot be handled by the most recent extensions of the FMM [26,8].
In this paper we generalize the iterated refinement method, introduced by the authors in [8],to a time-continuous inverse scale-space formulation. The iterated refinement procedure yields a sequence of convex variational problems, evolving toward the noisy image. The inverse scale space method arises as a limit for a penalization parameter tending to zero, while the number of iteration steps tends to infinity. For the limiting flow, similar properties as for the iterated refinement procedure hold. Specifically, when a discrepancy principle is used as the stopping criterion, the error between the reconstruction and the noise-free image decreases until termination, even if only the noisy image is available and a bound on the variance of the noise is known. The inverse flow is computed directly for one-dimensional signals, yielding high quality restorations. In higher spatial dimensions, we introduce a relaxation technique using two evolution equations. These equations allow accurate, efficient and straightforward implementation.
We present a novel variational model to find shape-based correspondences between two sets of level curves. While the usual correspondence techniques work with parametrized curves, we use a level-set formulation that enables us to handle curves with arbitrary topology. Given the functions $\Phi_{1}: (\Omega_{1} \subseteq IR^{2}) \longrightarrow IR$ and $\Phi_{2}: (\Omega_{2} \subseteq IR^{2}) \longrightarrow IR$ whose 0-level curves we want to match, we search for a diffeomorphism that minimizes the rate of change of the difference in tangential orientation of the zero-level sets. To make the formulation symmetric and to simplify computations, we map the domains of the level-set functions Φi to a common domain Ω by initial diffeomorphisms that are chosen arbitrarily. We then search for diffeomorphisms from Ω to itself, generating them by flows of certain vector fields on Ω. The resulting correspondences are scale- and rotation-invariant with respect to the curves. We show how this model can be used as a basis to compare curves of different topology. The model was tested on synthetic and MRI cardiac data,with good results.
