A general nonparametric technique is proposed for the analysis of a complex multimodal feature space and to delineate arbitrarily shaped clusters in it. The basic computational module of the technique is an old pattern recognition procedure, the mean shift. We prove for discrete data the convergence of a recursive mean shift procedure to the nearest stationary point of the underlying density function and thus its utility in detecting the modes of the density. The equivalence of the mean shift procedure to the Nadaraya–Watson estimator from kernel regression and the robust M-estimators of location is also established. Algorithms for two low-level vision tasks, discontinuity preserving smoothing and image segmentation are described as applications. In these algorithms the only user set parameter is the resolution of the analysis, and either gray level or color images are accepted as input. Extensive experimental results illustrate their excellent performance.

This paper deals with establishing relations between a number of widely-used nonlinear filters for digital image processing. We cover robust statistical estimation with (local) M-estimators, local mode filtering in image or histogram space, bilateral filtering, nonlinear diffusion, and regularisation approaches. Although these methods originate in different mathematical theories, we show that their implementation reveals a highly similar structure. We demonstrate that all these methods can be cast into a unified framework of functional minimisation combining nonlocal data and nonlocal smoothness terms. This unification contributes to a better understanding of the individual methods, and it opens the way to new techniques combining the advantages of known filters. Keywords: image analysis, M-estimators, mode filtering, nonlinear diffusion, bilateral filter, regularisation

First we explain the interplay between robust loss functions, nonlinear lters and Bayes smoothers for edge-preserving image reconstruction. Then we prove the surprising fact that maximum posterior smoothers are nonlinear lters. A (generalized) Potts prior for segmentation and piecewise smoothing of noisy signals and images is adopted. For one-dimensional signals, an exact solution for the maximum posterior mode- based on dynamic programming- is derived. After, some results on the performance of nonlinear lters on jumps and ramps we nally introduce a cascade of nonlinear lters with varying scale parameters and discuss the choice of parameters for segmentation and piecewise smoothing.

We introduce some recent and very recent smoothing methods which focus on the preservation of boundaries, spikes and canyons in presence of noise. We try to point out basic principles they have in common; the most important one is the robustness aspect. It is reflected by the use of `cup functions' in the statistical loss functions instead of squares; such cup functions were introduced early in robust statistics to downweight outliers. Basically, they are variants of truncated squares. We discuss all the methods in the common framework of `energy functions', i.e we associate to (most of ) the algorithms a `loss function' in such a fashion that the output of the algorithm or the `estimate' is a global or local minimum of this loss function. The third aspect we pursue is the correspondence between loss functions and their local minima and nonlinear filters. We shall argue that the nonlinear filters can be interpreted as variants of gradient descent on the loss functions. This way we can ...

Models of spatial variation in images are central to a large number of low-level computer vision problems including segmentation, registration, and 3D structure detection. Often, images are represented using parametric models to characterize (noise-free) image variation, and, additive noise. However, the noise model may be unknown and parametric models may only be valid on individual segments of the image.

A local smoothing procedure is proposed for detecting jump location curves of regression surfaces. This procedure simpli es the computation of some existing jump detectors in the statistical literature. It also generalizes the Sobel edge detector in the image processing literature such that more observations can be used to smooth away random noise in the data. The problem to evaluate the performance of jump detectors is discussed and a new measurement of jump detection performance is suggested.

Abstract—In this paper, we present a new and efficient method to implement robust smoothing of low-level signal features: B-spline channel smoothing. This method consists of three steps: encoding of the signal features into channels, averaging of the channels, and decoding of the channels. We show that linear smoothing of channels is equivalent to robust smoothing of the signal features if we make use of quadratic B-splines to generate the channels. The linear decoding from B-spline channels allows the derivation of a robust error norm, which is very similar to Tukey’s biweight error norm. We compare channel smoothing with three other robust smoothing techniques: nonlinear diffusion, bilateral filtering, and mean-shift filtering, both theoretically and on a 2D orientation-data smoothing task. Channel smoothing is found to be superior in four respects: It has a lower computational complexity, it is easy to implement, it chooses the global minimum error instead of the nearest local minimum, and it can also be used on nonlinear spaces, such as orientation space.

It is known that a surface fitted by conventional local smoothing procedures is not statistically consistent at the jump locations of the true regression surface. In this paper, a procedure is suggested for modifying conventional local smoothing procedures such that the modified procedures can fit the surface with jumps preserved automatically. Taking the local linear kernel smoothing procedure as an example, in a neighborhood of a given point, we fit a bivariate piecewisely linear function with possible jumps along the boundaries of four quadrants. The fitted function provides four estimators of the surface at the given point, which are constructed from observations in the four quadrants, respectively. When the difference among the four estimators is smaller than a threshold value, the given point is most likely a continuous point and the surface at that point is then estimated by the average of the four estimators. When the difference is larger than the threshold value, the given point is likely a jump point and at least one of the four estimators estimates the surface well under some regularity conditions. By comparing the weighted residual sums of squares of the four estimators, the best one is selected to define the surface estimator at the given point. Like most conventional estimators, the current surface estimator has an explicit mathematical formula. Therefore it is easy to compute and convenient to use. It can be applied directly to image reconstruction problems and other jump surface estimation problems including mine surface estimation in geology and equi-temperature surface estimation in meteorology and oceanography.

Surface estimation is important in many applications. When conventional smoothing procedures, such as the running averages, local polynomial kernel smoothing procedures, and smoothing spline procedures, etc., are used for estimating jump surfaces from noisy data, jumps would be blurred at the same time when noise is removed. In recent years, new smoothing methodologies have been proposed in the statistical literature for detecting jumps in surfaces and for estimating jump surfaces with jumps preserved. We provide a review of these methodologies in this paper. Because a monochrome image can be regarded as a jump surface of the image intensity function, with jumps at the outlines of objects, edge detection and image restoration problems in image processing are closely related to the jump surface estimation problem in statistics. In this paper, we also review major methodologies on edge detection and image restoration, and discuss about connections and differences between these methods and the related methods in the statistical literature.

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