Deep Structure, Singularities, and Computer Vision DSSCV

The deep structure of an image is investigated, and a Multi-Scale Singularity Tree (MSST) is constructed based on the pair-wise annihilations of critical points. This report contains two main contributions. Firstly, we describe a fast, simple, and robust method of extracting feature lines from data sets of up to four dimensions, which we apply in order to extract critical paths from scale-spaces. Secondly, we investigate the extracting of MSSTs using either support regions or extrema partitions. Given an image, both methods produce a binary tree that mathematically represents the topological structures of the image. The software described in this report is available through the EU-project, Deep Structure, Singularities, and Computer Vision. 1 Scales, Deep Structures, and Trees The concept of scales is applied everywhere, explicitly otherwise implicitly. Images of the same scene might look completely different when measured at different scales. Without relevant prior knowledge, we do not know what scale is better than another and all scales are equally important. We have to take all of them into account resulting in multi-scale schemes. In this report, we use a simple mathematical frame-work for working with multi scales called the Gaussian scale-space[8, 21, 10]. In practice, the finest and the coarsest scales are limited by the inner scalethe resolution of the measurement device and the outer scalethe size of the image, respectively. Between the inner and the outer scale, critical points move, annihilate and possibly are created. The movements and interactions of these critical points at all scales are called the deep structure of the image. It incorporates the structure of an image at all possible scales. The deep structure depends on the dimensionality of the image[10, 3]. Only annihilations of critical points occur in one-dimensional signal, while creations of critical points are also possible for images of higher dimensionality. For generic images1, only pair-wise creations and annihilations of critical points of opposite Hessian signatures2 can occur. These annihilations and creations are called catastrophe events and the points that they occur in scale-spaces are called catastrophe points. The merging of image structures, suggests a hierarchy which can be represented mathematically using trees [11]. We call such a tree a MultiScale Singularity Tree(MSST). The long-term goal of our work is to use the extracted MSSTs as images descriptors. The approach is relatively new in computer-vision literatures. Using trees as image descriptors transforms the computervision problems into tree manipulation problems, which are well-studied mathematical problems. We expect that this approach will be useful in applications, such as image matching and recognition, image database indexing, image compression, etc. 2 Introduction to Gaussian Scale-Space The dimensional Gaussian scale-space of an dimensional image is an ordered stack of images, where each image is a blurred version of the former. The blurring is performed according to the diffusion equation, (1) where is the first partial-derivative of the image in the scale direction , and is the Laplace operator, which in 3 dimensions reads ! #" $#" % . Images whose structure is not perturbed by noise. The sign of the determinant of the matrix of all second-order derivatives. Version: October 7, 2003 2 3 COMPUTING THE DEEP STRUCTURE IN GENERIC SCALE-SPACE IMAGES The Gaussian kernel is the Green’s function of the heat diffusion equation, i.e. (2) ! "! (3) where # is the image at scale , is the original image, is the convolution operator, is the Gaussian kernel at scale , is the dimensionality of the problem, and %$ &(' , given $ being the standard deviation of the Gaussian kernel. The Gaussian scale-space is the only scale-space we will consider in this report, it will henceforth be called the scale-space. Convolution in spatial domain is equivalent to multiplication in Fourier domain, ) +* ,*. */ 0 (4) with * as the Fourier transformation operator. Further, since differentiation commutes with convolution and the Gaussian kernel is infinitely differentiable, differentiation of scale-spaces is conveniently computed, ! 1 # ! 1 ) 2 ! 1 3 (5) Alternative implementations of the scale-space are spatial convolutions, finite differencing schemes for the heat diffusion equation, additive operator splitting[20], and recursive implementation[4, 19]. The Fourier implementation is the most convenient for specifying derivative operators and it is precise and fast for coarse scales. 3 Computing the Deep Structure in Generic Scale-Space Images Constructing a scale-space of an image adds a scale dimension, e.g. the scale-space of a two-dimensional image has dimensionality of three. We call such scale-space a ' scale-space. Similarly, constructing the scale-space of a three-dimensional image gives a scale-space. Although the dimensionality of the constructed scale-space depends on the dimensionality of the original image, critical points, namely maxima, minima and saddles, in the image at each scale are always points, and critical paths, which are the paths of critical points when the scale is varied, are therefore also always curves embedded in the scalespace. The following sections describe a method used to extract critical paths, the scale-space saddles and the catastrophe points on them. At each scale in a scale-space, points with vanishing spatial gradients are called critical points. The type of a critical point is derived from the configuration of the eigenvalues of the Hessian matrix computed at that position. Critical points with all positive eigenvalues are minima, critical points with all negative eigenvalues are maxima, and critical points with a mixture of both negative and positive eigenvalues are saddles. As we increase the scale parameter, the critical points move smoothly forming critical paths. At some scales, a pair of critical points with opposite Hessian signatures might meet and annihilate or be created. Such events are called catastrophe events, and the points where they occur are called catastrophe points. Generically3 , there are only two types of generic catastrophe events in scale-space namely creation events and annihilation events. It is further known that these events only occur between pairs of critical points differing in the sign of one and only one eigenvalue of the Hessian matrix. The notion of genericity is useful for regarding only the likely catastrophes, i.e. non-generic events disappear with small perturbations such as added random noise, while generic events do not. Version: October 7, 2003 3 3 COMPUTING THE DEEP STRUCTURE IN GENERIC SCALE-SPACE IMAGES