Geometric Heat Research Articles

Geometric Heat Comparison Criteria for Riemannian Manifolds

The main results of this paper are small-time heat comparison results for two points in two manifolds with characteristic functions as initial temperature distributions (Theorems 1 and 2). These results are based on the geometric concepts of (essential) distance from the complement and spherical area function. We also discuss some other geometric results about the heat development and illustrate them by examples.

A New Active Convex Hull Model for Image Regions

This paper presents a new active convex hull model with the following advantages: invariant with respect to the number of pixels to be enveloped; the number of time iterations is invariant, with respect to the image size; time-cheap for large image regions. The model is based on the geometric heat differential equations, derived from parabolic equation, and parameterized by arc length. To prevent the active contour from intruding into concavities and evolve it to the proper convex hull we use a vector field given as a difference between normal and tangent forces. The vector field is also used to segment an image to shells, such that a single region is present in each shell. A penalty function is developed to stop evolvement of those arc segments, whose vectors encountered boundary points of an image region. Based on the model a discrete algorithm is designed and coded by Mathematica 5.2. A condition is developed, with respect to the image size, to guarantee stable convergence of the active contour to the convex hull of the desired region. To validate the advantages and contributions a set of experiments is performed using synthetic, groundwater and medical images of different size and modalities. The paper concludes with a discussion and comparison of the active convex hull model with set of existing convex hull algorithms.

BEM Solution With Minimal Energy Technique for the Geometrical Inverse Heat Conduction Problem in a Doubly Connected Domain

This article addresses the use of boundary element method in conjunction with minimal energy technique for solving a geometrical inverse heat conduction problem. The problem considered in this study is to estimate the unknown inner boundary position in an irregular-shaped hollow body of which the inner boundary surface is subjected to a specified temperature condition. For solving the problem, first boundary element equations are converted into the quadratic programming problem by minimizing the energy functional with a constraint, next a hypothetical inner boundary is defined such that the actual inner boundary is located interior of the hypothetical solution domain, then temperatures at hypothetical inner boundary are determined to meet the constraints of measurement error in inner surface temperatures, and finally boundary element analysis is performed for the position of an unknown boundary. Based on these main solution procedures, an effective detection algorithm is provided. In addition, the solution method is numerically tested to investigate the effects of measurement errors on the accuracy of estimation.

Stochastic differential equations and geometric flows

In previous years, curve evolution, applied to a single contour or to the level sets of an image via partial differential equations, has emerged as an important tool in image processing and computer vision. Curve evolution techniques have been utilized in problems such as image smoothing, segmentation, and shape analysis. We give a local stochastic interpretation of the basic curve smoothing equation, the so called geometric heat equation, and show that this evolution amounts to a tangential diffusion movement of the particles along the contour. Moreover, assuming that a priori information about the shapes of objects in an image is known, we present modifications of the geometric heat equation designed to preserve certain features in these shapes while removing noise. We also show how these new flows may be applied to smooth noisy curves without destroying their larger scale features, in contrast to the original geometric heat flow which tends to circularize any closed curve.

Geometric heat trapping in niobium superconductor–insulator– superconductor mixers due to niobium titanium nitride leads

We analyze the current–voltage characteristics of a Nb superconductor–insulator–superconductor mixer with NbTiN leads to identify the heating processes in this device. We argue that the electron–electron interaction is much faster than the electron–phonon interaction, and show that the heat flow to the bath temperature is limited by the electron–phonon interaction time in the junction. A solution is suggested to considerably reduce the heating.

Enhancing CSS-based shape retrieval for objects with shallow concavities

Curvature scale space (CSS) image is a multi-scale organisation of the inflection points of a closed planar curve as it is smoothed. It consists of several arch shape contours, each related to a concavity or a convexity of the curve. The maxima of these contours have already been used as shape descriptors to find similar shapes in large image databases. In this article, we address the problem of shallow concavities. These may give rise to large contours in the CSS image. These contours may then match those corresponding to deep and wide concavities during the matching process. The phenomenon can be explained by recalling the fact that Gaussian smoothing leads to an approximation of geometric heat equation deformation. We have introduced a method to enrich the CSS image and create different contours for different types of concavities. We tested the proposed method on a database of 1100 images of marine creatures. A significant improvement was observed in the performance of the system on shapes with shallow segments.

Stress Intensity Factor Analysis of Nozzle Considering Pressure and Heat Transfer on Crack Face

In order to investigate the effect of nozzle on stress concentration in pressure vessels, three dimensional finite element analyses were performed. The results were compared with those for corresponding two dimensional axisymmetric finite element analyses. A three dimensional finite element model with a surface crack was also designed to evaluate the effect of internal pressure and heat transfer on crack face, and the resulting stress intensity factors from the finite element analyses were compared with those for ASME Sec. XI and Raju-Newman's stress intensity factor solution. As a result, the validity of currently available stress intensity factor solutions for a surface crack was reviewed in the presence of geometrical complexity, heat transfer and internal pressure.

The Green Function and Its Application to Heat Transfer in a Low Permeability Porous Channel

This paper describes an analytical technique for heat transfer in a low permeability porous channel with initially a point heat source and then multiple discrete sources. The solution of the temperature field for the point source is derived as a Green function, from which the temperature field for other geometrical heat sources can be obtained by using the Green integral. An application of the Green function and Green integral to the porous channel with discrete heat sources is presented. The results are compared with the existing numerical results and good agreements are achieved. [S1043-7398(00)01503-6]

Affine Invariant Detection: Edge Maps, Anisotropic Diffusion, and Active Contours

In this paper we undertake a systematic investigation of affine invariant object detection and image denoising. Edge detection is first presented from the point of view of the affine invariant scale-space obtained by curvature based motion of the image level-sets. In this case, affine invariant maps are derived as a weighted difference of images at different scales. We then introduce the affine gradient as an affine invariant differential function of lowest possible order with qualitative behavior similar to the Euclidean gradient magnitude. These edge detectors are the basis for the extension of the affine invariant scale-space to a complete affine flow for image denoising and simplification, and to define affine invariant active contours for object detection and edge integration. The active contours are obtained as a gradient flow in a conformally Euclidean space defined by the image on which the object is to be detected. That is, we show that objects can be segmented in an affine invariant manner by computing a path of minimal weighted affine distance, the weight being given by functions of the affine edge detectors. The gradient path is computed via an algorithm which allows to simultaneously detect any number of objects independently of the initial curve topology. Based on the same theory of affine invariant gradient flows we show that the affine geometric heat flow is minimizing, in an affine invariant form, the area enclosed by the curve.

CORNER ENHANCEMENT IN CURVATURE SCALE SPACE

Corner detection is very important for pattern recognition, computer vision and other works. However, successfully extracting corners from a planar curve is difficult because we do not know how to select the appropriate scale for the extraction. In this paper, the problem of corner enhancement via the curve scale space is addressed. The main idea of the approach is to deform the original curve (or construct the curve scale space) and the corners on the deformed curve become distinguished from other structures which eases the process of corner detection. Started from the general geometric heat flow (GGHF), we study under what conditions the GGHF satisfies the scale space causality criteria. This is very important because many different evolution approaches, which have regular scale space properties and other specific properties, can then be found from these conditions. The criteria of corner enhancement are also proposed. Having all these constraints, a new curve evolution scheme which can enhance strong corners suppress noise and satisfies the scale space criteria is presented.

Planar Shape Enhancement and Exaggeration

A local smoothing operator applied in the reverse direction is used to obtain planar shape enhancement and exaggeration. Inversion of a smoothing operator is an inherently unstable operation. Therefore, a stable numerical scheme simulating the inverse smoothing effect is introduced. Enhancement is obtained for short time spans of evolution. Carrying the evolution further yields shape exaggeration or caricaturization effect. Introducing attraction forces between the evolving shape and the initial one yields an enhancement process that converges to a steady state. These forces depend on the distance of the evolving curve from the original one and on local properties. Results of applying the unrestrained and restrained evolution on planar shapes, based on a stabilized inverse geometric heat equation, are presented showing enhancement and caricaturization effects.

On the affine heat equation for non-convex curves

In this paper, we extend to the non-convex case theaffine invariant geometric heat equationstudied by Sapiro and Tannenbaum for convex plane curves. We prove that a smooth embedded plane curve will converge to a point when evolving according to this flow. This result extends the analogy between the affine heat equation and the well-knownEuclidean geometric heat equation.

Geometric heat kernel coefficient for APS-type boundary conditions

Geometric heat kernel coefficient for APS-type boundary conditions

Geometric Heat Equation and Nonlinear Diffusion of Shapes and Images

Visual tasks often require a hierarchical representation of shapes and images in scales ranging from coarse to fine. A variety of linear and nonlinear smoothing techniques, such as Gaussian smoothing, anisotropic diffusion, regularization, etc., have been proposed, leading to scalespace representations. We propose ageometricsmoothing method based on local curvature for shapes and images. The deformation by curvature, or the geometric heat equation, is a special case of thereaction–diffusionframework proposed in [41]. For shapes, the approach is analogous to the classical heat equation smoothing, but with a renormalization by arc-length at each infinitesimal step. For images, the smoothing is similar to anisotropic diffusion in that, since the component of diffusion in the direction of the brightness gradient is nil, edge location is left intact. Curvature deformation smoothing for shape has a number of desirable properties: it preserves inclusion order, annihilates extrema and inflection points without creating new ones, decreases total curvature, satisfies the semigroup property allowing for local iterative computations, etc. Curvature deformation smoothing of an image is based on viewing it as a collection of iso-intensity level sets, each of which is smoothed by curvature. The reassembly of these smoothed level sets into a smoothed image follows a number of mathematical properties; it is shown that the extension from smoothing shapes to smoothing images is mathematically sound due to a number of recent results [21]. A generalization of these results [14] justifies the extension of the entireentropy scale spacefor shapes [42] to one for images, where each iso-intensity level curve is deformed by a combination of constant and curvature deformation. The scheme has been implemented and is illustrated for several medical, aerial, and range images.

Geometrical characteristic and heat transfer in building’s protruding structure

A non-equilibrium thermodynamical and geometrical analysis of the buildings’ protruding structures is developed. The geometrical conditions for the buildings’ protruding structures to obtain a decreasing in the buildings lost heat are obtained.

An inverse heat conduction algorithm for the thermal detection of arbitrarily shaped cavities

An inverse geometric heat conduction problem algorithm is presented for the detection of irregular shaped cavities located within irregular shaped bodies. The inverse problem is solved using the additional information provided by specifying a Cauchy condition at the exposed boundary. The exposed surface temperature distribution is determined experimentally using an infrared scanner. The inverse problem solution algorithm is based on solving a sequence of forward steady state heat conduction problems and defining a residual measuring the difference between actual and computed boundary conditions under the current system configuration. The boundary element method is used to solve the forward problems. The iterative procedure is automated through the use of a multivariable nonlinear solver along with the linear and cubic spline based Anchored Grid Pattern (AGP) method described herein. Numerical simulation is used to investigate the practical implementation of the AGP algorithm in detecting a small cavity. T...

A geometric heat flow for one-forms on three dimensional manifolds

A geometric heat flow for one-forms on three dimensional manifolds

Area and length preserving geometric invariant scale-spaces

In this paper, area preserving multi-scale representations of planar curves are described. This allows smoothing without shrinkage at the same time preserving all the scale-space properties. The representations are obtained deforming the curve via geometric heat flows while simultaneously magnifying the plane by a homethety which keeps the enclosed area constant. When the Euclidean geometric heat flow is used, the resulting representation is Euclidean invariant, and similarly it is affine invariant when the affine one is used. The flows are geometrically intrinsic to the curve, and exactly satisfy all the basic requirements of scale-space representations. In the case of the Euclidean heat flow, it is completely local as well. The same approach is used to define length preserving geometric flows. A similarity (scale) invariant geometric heat flow is studied as well in this work. >

CUBIC SPLINE ANCHORED GRID PATTERN ALGORITHM FOR HIGH-RESOLUTION DETECTION OF SUBSURFACE CAVITIES BY THE IR-CAT METHOD

Abstract An algorithm is presented for the high-resolution detection of irregular-shaped subsurface cavities within irregular-shaped bodies by the IR-CAT method. The theoretical basis of the algorithm is rooted in the solution of an inverse geometric steady-state heat conduction problem. A Cauchy boundary condition is prescribed at the exposed surface, and the inverse geometric heat conduction problem is formulated by specifying the thermal condition at the inner cavities walls, whose unknown geometries are to be detected. The location of the inner cavities is initially estimated, and the domain boundaries are discretized. Linear boundary elements are used in conjunction with cubic splines for high resolution of the cavity walls. An anchored grid pattern (AGP) is established to constrain the cubic spline knots that control the inner cavity geometry to evolve along the AGP at each iterative step. A residual is defined measuring the difference between imposed and computed boundary conditions. A Newton-Raphs...

Automated algorithm for the nondestructive detection of subsurface cavities by the IR-CAT method

An algorithm is presented to automate the detection of irregular-shaped subsurface cavities within irregular shaped bodies by the IR-CAT method. The algorithm is based on the solution of an inverse geometric steady state heat conduction problem. Cauchy boundary conditions are prescribed at the exposed surface. An inverse heat conduction problem is formulated by specifying the thermal boundary condition along the inner cavities whose unknown geometries are to be determined. An initial guess is made for the location of the inner cavities. The domain boundaries are discretized, and an Anchored Grid Pattern (AGP) is established. The nodes of the inner cavities are constrained to move along the AGP at each iterative step. The location of inner cavities is determined by using the Newton Raphson method with a Broyden update to drive the error between the imposed boundary conditions and computed boundary conditions to zero. During the iterative procedure, the movement of the inner cavity walls is restricted to physically realistic intermediate solutions. A dynamic relocation of the AGP is introduced in the Traveling Hole Method to adaptively refine the detection of inner cavities. The proposed algorithm is general and can be used to detect multiple cavities. Results are presented for the detection of single and multiple irregular shaped cavities. Convergence under grid refinement is demonstrated.