Multi-scale anisotropic heat diffusion based on normal-driven shape representation

Multi-scale local feature detection enables downstream registration and recognition tasks in medical image analysis. This paper articulates a novel robust method for multi-scale local feature extraction on volumetric data. The central idea is the elegant unification of local/global eigen-structures within the powerful framework of anisotropic heat diffusion. First, the local vector field is constructed by way of Hessian matrix and its eigenvectors/eigenvalues. Second, anisotropic heat kernels are computed using the vector field’s global graph Laplacian. Robust local features are manifested as extrema across multiple time scales, serving as volumetric heat kernel signature. To tackle the computational challenge for massive volumetric data, we propose a multiresolution strategy for hierarchical feature extraction based on our feature-preserving down-sampling approach. As a result, heat kernels and local feature identification can be approximated at a coarser level first, and then are pinpointed in a localized region at a finer resolution. Another novelty of this work lies at the initial heat design directly using local eigenvalue for anisotropic heat diffusion across the volumetric domain. We conduct experiments on various medical datasets, and draw comparisons with 3D SIFT method. The diffusion property of our local features, which can be interpreted as random walks in statistics, makes our method robust to noise, and gives rise to intrinsic multi-scale characteristics.

There are problems needed to be solved when extracting centerline of structured light strip on multi-pass seam. The problems include complex shape, non-uniform intensity distribution, disconnected caused by welding beans or occlusion and environmental noise etc. To solve these problems, a novel centerline extraction algorithm is proposed based on anisotropic heat diffusion. Putting pixels gray values as the temperature, the algorithm could get centerline composed by highest energy pixels, through heat diffusion along same temperature curve normal directions. The higher accuracy and efficiency of our algorithm is proved by experiments.

Road detection is a basic functionality for driver assistance system such as vehicle and pedestrian detection. The main challenge is to deal with the presence of different objects (pedestrians and vehicles), the continuously varying background, different imaging conditions (different viewpoints, varying weather conditions and changing illumination), different road types (color, shape) and different environments (off-road, urban, highways). Accordingly, we propose a novel road detection approach based on superpixels and anisotropic heat diffusion. The idea is to previously denoise and deblur road images using directional filters. And then to oversegment the enhanced image into small homogeneous regions which are called superpixels with the superpixel segmentation method. According to the thermodynamic energy diffusion theory, we cluster superpixels into several region blocks. Finally, the algorithm extracts the road region combining the prior knowledge with the largest and most coherent principle of cosegmentation. Quantitative and qualitative experiments show that the approach is robust to lighting variations, heavy traffic and shadows. Moreover, the proposed method provides the better performance when compared with several state-of-the-art methods.

Visual tasks often require a hierarchical representation of images in scales ranging from coarse to fine. A variety of linear and nonlinear smoothing techniques, such as Gaussian smoothing, anisotropic diffusion, regularization, wavelet thresholding etc... have been proposed. In this work, we propose a geometrical multiscale anisotropic diffusion based on the geometrical flow for denoising multivalued images. The geometrical flow is determined by the Bandelet transform of the image being processed. Consequently, the image is segmented into a quadtree where each square regroups pixels sharing the same geometrical flow direction. The motivation of this work is to introduce a new multiscale multistructure bandelet-based diffusion tensor to adjust the anisotropic diffusion toward the direction of the optimal geometrical flow. Therefore, multiple dyadic squares in the quadtree have multiple structure tensors. Hence, a more accurate geometrically driven noise suppression is obtained where the homogeneity of different image regions is well maintained.

One of the major goals of computer vision is the development of flexible and efficient methods for shape representation. This is true, especially for non-rigid 3D shapes where a great variety of shapes are produced as a result of deformations of a non-rigid object. Modeling these non-rigid shapes is a very challenging problem. Being able to analyze the properties of such shapes and describe their behavior is the key issue in research. Also, considering photometric features can play an important role in many shape analysis applications, such as shape matching and correspondence because it contains rich information about the visual appearance of real objects. This new information (contained in photometric features) and its important applications add another, new dimension to the problem's difficulty. Two main approaches have been adopted in the literature for shape modeling for the matching and retrieval problem, local and global approaches. Local matching is performed between sparse points or regions of the shape, while the global shape approaches similarity is measured among entire models. These methods have an underlying assumption that shapes are rigidly transformed. And Most descriptors proposed so far are confined to shape, that is, they analyze only geometric and/or topological properties of 3D models. A shape descriptor or model should be isometry invariant, scale invariant, be able to capture the fine details of the shape, computationally efficient, and have many other good properties. A shape descriptor or model is needed. This shape descriptor should be: able to deal with the non-rigid shape deformation, able to handle the scale variation problem with less sensitivity to noise, able to match shapes related to the same class even if these shapes have missing parts, and able to encode both the photometric, and geometric information in one descriptor. This dissertation will address the problem of 3D non-rigid shape representation and textured 3D non-rigid shapes based on local features. Two approaches will be proposed for non-rigid shape matching and retrieval based on Heat Kernel (HK), and Scale-Invariant Heat Kernel (SI-HK) and one approach for modeling textured 3D non-rigid shapes based on scale-invariant Weighted Heat Kernel Signature (WHKS). For the first approach, the Laplace-Beltrami eigenfunctions is used to detect a small number of critical points on the shape surface. Then a shape descriptor is formed based on the heat kernels at the detected critical points for different scales. Sparse representation is used to reduce the dimensionality of the calculated

In this paper, an intelligent and automatic moving object edge detection algorithm is proposed, based on heat flow analogy. This algorithm starts with anisotropic heat diffusion in the spatial domain to remove noise and sharpen region boundaries for the purpose of obtaining high quality edge data. Then, isotropic heat diffusion is applied in the temporal domain to calculate the total amount of heat flow. The moving edges are represented as the total amount of heat flow out from the reference frame. The overall process is completed by non-maxima suppression and hysteresis thresholding to obtain binary moving edges. Evaluation results indicate that this approach has advantages in handling noise in the temporal domain because of the averaging inherent of isotropic heat flow. Results also show that this technique can detect moving edges in image sequences.

Pattern forming instabilities of the nematic smectic-B interface

Image transform, which transforms one image to another image by co-ordinate transformation, is often conducted in remote sensing applications. Image transform methods include forward mapping and reverse mapping. In this paper we propose a new forward image mapping method. Since a grid pixel in the source image may be no longer mapped to a grid pixel in the target image, gap phenomenon, i.e. some grids in the target may not receive information from any source pixel, occurs via forward mapping. Our method fills the gaps with information surrounding them using the image inpainting technique. Imagine that the image is a temperature field, i.e. the pixel value is the temperature value. The heat will transfer to the gaps, thus the gaps are filled after some time. This can be achieved by solving the classical diffusion equation. Furthermore, if the features in the image are rather complicated, we can use anisotropic heat diffusion (total variation diffusion) to fill the gaps in order to well preserve the edge features. The experimental result demonstrates the effectiveness of the proposed inpainting method.

Graph spectral image smoothing using the heat kernel

Morse theory is a powerful mathematical tool that uses the local differential properties of a manifold to make conclusions about global topological aspects of the manifold. Morse theory has been proven to be a very useful tool in computer graphics, geometric data processing and understanding. This work is divided into two parts. The first part is concerned with constructing geometry and symmetry aware scalar functions on a triangulated $2$-manifold. To effectively apply Morse theory to discrete manifolds, one needs to design scalar functions on them with certain properties such as respecting the symmetry and the geometry of the surface and having the critical points of the scalar function coincide with feature or symmetry points on the surface. In this work, we study multiple methods that were suggested in the literature to construct such functions such as isometry invariant scalar functions, Poisson fields and discrete conformal factors. We suggest multiple refinements to each family of these functions and we propose new methods to construct geometry and symmetry-aware scalar functions using mainly the theory of the Laplace-Beltrami operator. Our proposed methods are general and can be applied in many areas such as parametrization, shape analysis, symmetry detection and segmentation. In the second part of the thesis we utilize Morse theory to give topologically consistent segmentation algorithms.

We construct default-free interest rate models in the spirit of the well-known Markov functional models: our focus is analytic tractability of the models and generality of the approach. We work in the setting of state price densities and construct models by means of the so called propagation property. The propagation property can be found implicitly in all of the popular state price density approaches, in particular heat kernels share the propagation property (wherefrom we deduced the name of the approach). As a related matter, an interesting property of heat kernels is presented, too.

A mathematical model consisting of a complete set of the linearized Boussinesq equations governing atmospheric convection has been solved. The significant new features of the model include (1) anisotropic eddy heat and momentum diffusion coefficients with the eddy viscosity for momentum varying with height through the convecting layer; (2) convection cells of general horizontal geometry as determined by parameter values appearing in a modified form of the generalized Christopherson shape function; (3) a non-zero complex stability factor allowing for steady and time-dependent convective modes; (4) a diabatic heating term in the conservation of energy equation allowing for the development of a moist atmospheric Rayleigh number (Ra); and (5) an atmospheric Prandtl number (Pγ) where molecular viscosity and conductivity have been replaced by their eddy counterparts. The important physical parameters in specifying the convective state are Ra, Pγ, and the eddy anisotropy coefficients for momentum and heat diffusion.Results obtained show that: (1) aspect ratio, cell geometry, and stability of the convective mode are very sensitive to variations in Ra, Pγ, and the anisotropy coefficients; (2) Regime-Stability (R-S) diagram reveals a convective order which closely corresponds to stability diagrams obtained in laboratory studies using different species of fluids; (3) cell flatness, as determined by values of the aspect ratio, is found to be inversely proportional to the square of the convective depth, with the proportionality constant being an increasing function of the momentum diffusion anisotropy coefficient; (4) a family of curves is found to exist where the diameter to depth ratio increases with increasing eddy anisotropy, supporting the observational evidence that such atmospheric convective phenomena as open and closed mesoscale cellular convection (MCC), radar cells in clear air, isolated tropical rings and thunderstorm cells are ordered according to their degree of anisotropy on a cell flatness versus depth diagram.Using an assumed periodic form of the perturbation solutions, the eigenvalue problem is reduced to one of solving a sixth-order differential equation for the vertical wind field with variable coefficients and accompanying boundary conditions. By applying the two-variable small perturbation technique, a Regime-Stability diagram was constructed yielding information concerning: (1) the geometry of the convective mode; (2) the stability of the convective mode; and (3) the anisotropic character of the eddy diffusion coefficients.

Ringing artifact degradations always appear in the deconvolution of geophysical data. To address this problem, we propose a postprocessing approach to suppress ringing artifacts that uses a novel anisotropic diffusion based on a stationary wavelet transform (SWT) algorithm. In this paper, we discuss the ringing artifact suppression problem and analyze the characteristics of the deconvolution ringing artifact. The deconvolution data containing ringing artifacts are decomposed into different SWT subbands for analysis, and a new multiscale adaptive anisotropic filter is developed to suppress these degradations. Finally, we demonstrate the performance of the proposed method and describe the experiments in detail.

Kernel representation and properties of discrete-time input-output systems

This paper proposes a novel hypergraph skeletal representation for 3D shape based on a formal derivation of the generic structure of its medial axis. By classifying each skeletal point by its order of contact, we shout that generically the medial axis consists of five types of points which are then organized into sheets, curves, and points: (i) sheets (manifolds with boundary) which are the locus of bitangent spheres with regular tangency/sup 1/ A/sub 1//sup 2/. Two types of curves (ii) the intersection curve of three sheets and the locus of centers of tritangent spheres, A/sub 1//sup 3/, and (iii) the boundary of sheets which are the locus of centers of spheres whose radius equals the larger principle curvature, i.e., higher order contact A/sub 3/ points; and two types of points (iv) centers of quad-tangent spheres, A/sub 1//sup 4/, and, (v) centers of spheres with one regular tangency and one higher order tangency, A/sub 1/A/sub 3/ The geometry of the 3D medial axis thus consists of sheets (A/sub 1//sup 2/) bounded by one type of curve (A/sub 3/) on their free end, which corresponds to ridges on the surface, and attached to two other sheets at another type of curves (A/sub 1//sup 3/), which support a generalized cylinder description. The A/sub 3/ curves can only end in A/sub 1/A/sub 3/ points where they must meet an A/sub 1//sup 3/ curve. The A/sub 1//sup 3/ curves can either meet one A/sub 3/ curve or meet three other A/sub 1//sup 3/ curve at an A/sub 1//sup 4/ point. This formal result leads to a compact representation for 3D shape, referred to as the medial axis hypergraph representation consisting of nodes (A/sub 1//sup 4/ and A/sub 1/A/sub 3/ points), links between pairs of nodes (A/sub 1//sup 3/ and A/sub 3/ curves) and hyperlinks between groups of links (A/sub 1//sup 2/ sheets). The description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes, in analogy to 2D. We derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph. Thus, the hypergraph completely characterizes 3D shape and lays the theoretical foundation for its use in recognition, morphing, design and manipulation of shapes.