We formulate a general energy and method for segmentation that is designed to have preference for segmenting the coarse structure over the fine structure of the data, without smoothing across boundaries of regions. The energy is formulated by considering data terms at a continuum of scales from the scale space computed from the Heat Equation within regions, and integrating these terms over all time. We show that the energy may be approximately optimized without solving for the entire scale space, but rather solving time-independent linear equations at the native scale of the image, making the method computationally feasible. We provide a multi-region scheme, and apply our method to motion segmentation. Experiments on a benchmark dataset shows that our method is less sensitive to clutter or other undesirable fine-scale structure, and leads to better performance in motion segmentation.
Scale space theory @cite_25 @cite_13 @cite_22 @cite_47 has a long and rich history as a theory for analyzing images, and we only provide brief highlights. The basic idea is that an image consists of structures at various different scales (e.g., a leaf of a tree exists at a different scale than a forest), and thus to analyze an image without a-priori knowledge, it is necessary to consider the image at scales. This is accomplished by blurring the image at a continuum of kernel sizes. The most common kernel is a Gaussian, which is known to be the only scale space satisfying certain axioms such as not introducing any new features as the image is blurred @cite_9 . Scale space has been used to analyze structures in images (e.g., @cite_14 @cite_15 @cite_9 @cite_39 ). This has had wide ranging applications in stereo and optical flow @cite_28 , reconstruction @cite_37 @cite_16 , key -point detection in wide-baseline matching @cite_11 , design of descriptors for matching @cite_23 , shape matching @cite_21 , and curve evolution @cite_2 , among others.
@cite_37 @cite_14 @cite_22 @cite_15 @cite_28 @cite_9 @cite_21 @cite_39 @cite_23 @cite_2 @cite_47 @cite_16 @cite_13 @cite_25 @cite_11
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This paper describes a generalized axiomatic scale-space theory that makes it possible to derive the notions of linear scale-space, affine Gaussian scale-space and linear spatio-temporal scale-space using a similar set of assumptions (scale-space axioms). The notion of non-enhancement of local extrema is generalized from previous application over discrete and rotationally symmetric kernels to continuous and more general non-isotropic kernels over both spatial and spatio-temporal image domains. It is shown how a complete classification can be given of the linear (Gaussian) scale-space concepts that satisfy these conditions on isotropic spatial, non-isotropic spatial and spatio-temporal domains, which results in a general taxonomy of Gaussian scale-spaces for continuous image data. The resulting theory allows filter shapes to be tuned from specific context information and provides a theoretical foundation for the recently exploited mechanisms of shape adaptation and velocity adaptation, with highly useful applications in computer vision. It is also shown how time-causal spatio-temporal scale-spaces can be derived from similar assumptions. The mathematical structure of these scale-spaces is analyzed in detail concerning transformation properties over space and time, the temporal cascade structure they satisfy over time as well as properties of the resulting multi-scale spatio-temporal derivative operators. It is also shown how temporal derivatives with respect to transformed time can be defined, leading to the formulation of a novel analogue of scale normalized derivatives for time-causal scale-spaces. The kernels generated from these two types of theories have interesting relations to biological vision. We show how filter kernels generated from the Gaussian spatio-temporal scale-space as well as the time-causal spatio-temporal scale-space relate to spatio-temporal receptive field profiles registered from mammalian vision. Specifically, we show that there are close analogies to space-time separable cells in the LGN as well as to both space-time separable and non-separable cells in the striate cortex. We do also present a set of plausible models for complex cells using extended quasi-quadrature measures expressed in terms of scale normalized spatio-temporal derivatives. The theories presented as well as their relations to biological vision show that it is possible to describe a general set of Gaussian and or time-causal scale-spaces using a unified framework, which generalizes and complements previously presented scale-space formulations in this area. The fact that objects in the world appear in different ways depending on the scale of observation has important implications if one aims at describing them. It shows that the notion of scale is of utmost importance when processing unknown measurement data by automatic methods. In their seminal works, Witkin (1983) and Koenderink (1984) proposed to approach this problem by representing image structures at different scales in a so-called scale-space representation. Traditional scale-space theory building on this work, however, does not address the problem of how to select local appropriate scales for further analysis. This article proposes a systematic methodology for dealing with this problem. A framework is presented for generating hypotheses about interesting scale levels in image data, based on a general principle stating that local extrema over scales of different combinations of γ-normalized derivatives are likely candidates to correspond to interesting structures. Specifically, it is shown how this idea can be used as a major mechanism in algorithms for automatic scale selection, which adapt the local scales of processing to the local image structure. Support for the proposed approach is given in terms of a general theoretical investigation of the behaviour of the scale selection method under rescalings of the input pattern and by integration with different types of early visual modules, including experiments on real-world and synthetic data. Support is also given by a detailed analysis of how different types of feature detectors perform when integrated with a scale selection mechanism and then applied to characteristic model patterns. Specifically, it is described in detail how the proposed methodology applies to the problems of blob detection, junction detection, edge detection, ridge detection and local frequency estimation. In many computer vision applications, the poor performance of the low-level vision modules constitutes a major bottleneck. It is argued that the inclusion of mechanisms for automatic scale selection is essential if we are to construct vision systems to automatically analyse complex unknown environments. Scale space theory from computer vision leads to an interesting and novel approach to nonparametric curve estimation. The family of smooth curve estimates indexed by the smoothing parameter can be represented as a surface called the scale space surface. The smoothing parameter here plays the same role as that played by the scale of resolution in a visual system. In this paper, we study in detail various features of that surface from a statistical viewpoint. Weak convergence of the empirical scale space surface to its theoretical counterpart and some related asymptotic results have been established under appropriate regularity conditions. Our theoretical analysis provides new insights into nonparametric smoothing procedures and yields useful techniques for statistical exploration of features in the data. In particular, we have used the scale space approach for the development of an effective exploratory data analytic tool called SiZer. We seek a practical method for establishing dense correspondences between two images with similar content, but possibly different 3D scenes. One of the challenges in designing such a system is the local scale differences of objects appearing in the two images. Previous methods often considered only few image pixels; matching only pixels for which stable scales may be reliably estimated. Recently, others have considered dense correspondences, but with substantial costs associated with generating, storing and matching scale invariant descriptors. Our work is motivated by the observation that pixels in the image have contexts—the pixels around them—which may be exploited in order to reliably estimate local scales. We make the following contributions. (i) We show that scales estimated in sparse interest points may be propagated to neighboring pixels where this information cannot be reliably determined. Doing so allows scale invariant descriptors to be extracted anywhere in the image. (ii) We explore three means for propagating this information: using the scales at detected interest points, using the underlying image information to guide scale propagation in each image separately, and using both images together. Finally, (iii), we provide extensive qualitative and quantitative results, demonstrating that scale propagation allows for accurate dense correspondences to be obtained even between very different images, with little computational costs beyond those required by existing methods.