An ideal image texture descriptor for describing, classifying and recognizing textures should be able to capture essential perceptual properties of the texture structure, and it also should be invariant to environmental changes, such as changes in view-point, luminance and geometry of the underlying surface. In the Computer Vision literature, the search for invariances started in the nineties [17] , when researcher dug up the mathematical literature on algebraic and geometric invariants. Great importance has been given to the study of quantities which are invariant to the viewpoint from which the image is taken. A number of projective invariants [13] have been found, which are defined on feature sets of points and lines and planar curves [17] , and they have been used for object recognition and calibration. However, none of these descriptors provides a high-level characterization of textures. Numerous texture descriptors have been proposed. Most of them are either statistics-based or filter-based ([1, 4, 5, 15]), which makes them sensitive to changes in viewpoint. Lazebnik, Schmid and Ponce ([6]) proposed a texture representation which is invariant to view-point changes in a weak form (it is locally invariant to affine transforms). Here we introduce a novel texture descriptor called the MFS (multifractal spectrum vector), which is based on fractal geometry theory. The MFS is globally invariant under the bi-Lipschitz transform, a very general transform which includes perspective transforms (viewpoint changes) and smooth texture surface deformations. Furthermore, the MFS has extraordinary low dimension (26 using one feature and three times that much using three features as in our implementation here for a typical 640 × 480 image texture). We demonstrate with a number of experiments on synthesized and real image textures that, first the MFS in practice is very robust to 3D transforms and non-rigid smooth transforms. Second, its performance is similar to the top methods in traditional texture retrieval and classification on standard texture datasets, but by using much lower dimensional feature vectors. Fractal geometry has been used before in the description of textures ([2, 14]) and texture segmentation ([8, 9, 10]). However, the invariance of the fractal dimension to biLipschitz maps has not been utilized in the vision community. Furthermore, existing approaches either simply compute a scalar number, which does not provide enough information for a good characterization of the texture; or they utilize the multi-fractal-spectrum for local feature description, in which there is no global invariance to bi-Lipschitz maps. Moreover, the fractal dimension was defined only on the image intensity, which makes it sensitive to changes in luminance. Intuitively, the MFS (multi-fractal-spectrum) is the extension of the fractal dimension. The MFS encodes the fractal dimension of every level set under a level set categorization of image points. Different categorizations of image points lead to different MFS vectors. The categorizations in our MFS approach are based on the density function defined on (functions of) the image intensity, and this makes it robust to luminance variations. The paper is organized as follows. Section 2 gives a brief introduction to fractal geometry and multi-fractal spectrum theory. Section 3 presents our MFS texture descriptor and its invariance to spatial bi-Lipschitz transforms and local affine illumination changes. Section 4 provides experiments on texture retrieval and classification and a comparison to
Recently, REF proposed the use of multifractal spectrum vectors to describe textures while achieving global invariance under bi-Lipschitz transforms, a general class of transforms which includes perspective transforms and smooth texture surface deformations.
A Projective Invariant for Textures
2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'06)
2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'06)
Computer Science