This paper presents a database containing ‘ground truth ’ segmentations produced by humans for images of a wide variety of natural scenes. We define an error measure which quantifies the consistency between segmentations of differing granularities and find that different human segmentations of the same image are highly consistent. Use of this dataset is demonstrated in two applications: (1) evaluating the performance of segmentation algorithms and (2) measuring probability distributions associated with Gestalt grouping factors as well as statistics of image region properties. 1.

We develop a framework for learning generic, expressive image priors that capture the statistics of natural scenes and can be used for a variety of machine vision tasks. The approach extends traditional Markov Random Field (MRF) models by learning potential functions over extended pixel neighborhoods. Field potentials are modeled using a Products-of-Experts framework that exploits nonlinear functions of many linear filter responses. In contrast to previous MRF approaches all parameters, including the linear filters themselves, are learned from training data. We demonstrate the capabilities of this Field of Experts model with two example applications, image denoising and image inpainting, which are implemented using a simple, approximate inference scheme. While the model is trained on a generic image database and is not tuned toward a specific application, we obtain results that compete with and even outperform specialized techniques. 1.

Most simple nonlinear thresholding rules for wavelet-based denoising assume that the wavelet coefficients are independent. However, wavelet coefficients of natural images have significant dependencies. In this paper, we will only consider the dependencies between the coefficients and their parents in detail. For this purpose, new non-Gaussian bivariate distributions are proposed, and corresponding nonlinear threshold functions (shrinkage functions) are derived from the models using Bayesian estimation theory. The new shrinkage functions do not assume the independence of wavelet coefficients. We will show three image denoising examples in order to show the performance of these new bivariate shrinkage rules. In the second example, a simple subband-dependent data-driven image denoising system is described and compared with effective data-driven techniques in the literature, namely VisuShrink, SureShrink, BayesShrink, and hidden Markov models. In the third example, the same idea is applied to the dual-tree complex wavelet coefficients.

The statistics of photographic images, when represented using multi-scale (wavelet) bases, exhibit two striking types of non-Gaussian behavior. First, the marginal densities of the coefficients have extended heavy tails. Second, the joint densities exhibit variance dependencies not captured by second-order models. We examine properties of the class of Gaussian scale mixtures, and show that these densities can accurately characterize both the marginal and joint distributions of natural image wavelet coefficients. This class of model suggests a Markov structure, in which wavelet coefficients are linked by hidden scaling variables corresponding to local image structure. We derive an estimator for these hidden variables, and show that a nonlinear ``normalization'' procedure can be used to Gaussianize the coefficients.

This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coherent picture of this framework. A second goal is to describe how this topic fits into the even larger field of MR methods and concepts–in particular making ties to topics such as wavelets and multigrid methods. A third is to provide several alternate viewpoints for this body of work, as the methods and concepts we describe intersect with a number of other fields. The principle focus of our presentation is the class of MR Markov processes defined on pyramidally organized trees. The attractiveness of these models stems from both the very efficient algorithms they admit and their expressive power and broad applicability. We show how a variety of methods and models relate to this framework including models for self-similar and 1/f processes. We also illustrate how these methods have been used in practice. We discuss the construction of MR models on trees and show how questions that arise in this context make contact with wavelets, state space modeling of time series, system and parameter identification, and hidden

Statistical analysis of images reveals two interesting properties: (i) invariance of image statistics to scaling of images, and (ii) non-Gaussian behavior of image statistics, i.e. high kurtosis, heavy tails, and sharp central cusps. In this paper we review some recent results in statistical modeling of natural images that attempt to explain these patterns. Two categories of results are considered: (i) studies of probability models of images or image decompositions (such as Fourier or wavelet decompositions), and (ii) discoveries of underlying image manifolds while restricting to natural images. Applications of these models in areas such as texture analysis, image classification, compression, and denoising are also considered.

We attempt to recover a high-dimensional vector observed in white noise, where the vector is known to be sparse, but the degree of sparsity is unknown. We consider three different ways of defining sparsity of a vector: using the fraction of nonzero terms; imposing power-law decay bounds on the ordered entries; and controlling the ℓp norm for p small. We obtain a procedure which is asymptotically minimax for ℓr loss, simultaneously throughout a range of such sparsity classes. The optimal procedure is a data-adaptive thresholding scheme, driven by control of the False Discovery Rate (FDR). FDR control is a recent innovation in simultaneous testing, in which one seeks to ensure that at most a certain fraction of the rejected null hypotheses will correspond to false rejections. In our treatment, the FDR control parameter q also plays a controlling role in asymptotic minimaxity. Our results say that letting q = qn → 0 with problem size n is sufficient for asymptotic minimaxity, while keeping fixed q>1/2prevents asymptotic minimaxity. To our knowledge, this relation between ideas in simultaneous inference and asymptotic decision theory is new. Our work provides a new perspective on a class of model selection rules which has been introduced recently by several authors. These new rules impose complexity penalization of the form 2·log ( potential model size / actual model size). We exhibit a close connection with FDR-controlling procedures having q tending to 0; this connection strongly supports a conjecture of simultaneous asymptotic minimaxity for such model selection rules.

in signal and image processing, including image denoising, coding, and super-resolution. # 2001 Academic Press 1. INTRODUCTION Stochastic models of natural images underlie a variety of applications in image processing and low-level computer vision, including image coding, denoising and 1 MW supported by NSERC 1967 fellowship; AW and MW by AFOSR Grant F49620-98-1-0349 and ONR Grant N00014-91-J-1004. Address correspondence to MW. 2 ES supported by NSF Career Grant MIP-9796040 and an Alfred P. Sloan fellowship. 89 1063-5203/01 $35.00 Copyright # 2001 by Academic Press All rights of reproduction in any form reserved. 90 WAINWRIGHT, SIMONCELLI, AND WILLSKY restoration, interpolation and synthesis. Accordingly, the past decade has witnessed an increasing amount of research devoted to developing stochastic models of images (e.g., [19, 38, 45, 48, 55]). Simultaneously, wavel

We introduce structured importance sampling, a new technique for efficiently rendering scenes illuminated by distant natural illumination given in an environment map. Our method handles occlusion, high-frequency lighting, and is significantly faster than alternative methods based on Monte Carlo sampling. We achieve this speedup as a result of several ideas. First, we present a new metric for stratifying and sampling an environment map taking into account both the illumination intensity as well as the expected variance due to occlusion within the scene. We then present a novel hierarchical stratification algorithm that uses our metric to automatically stratify the environment map into regular strata. This approach enables a number of rendering optimizations, such as pre-integrating the illumination within each stratum to eliminate noise at the cost of adding bias, and sorting the strata to reduce the number of sample rays. We have rendered several scenes illuminated by natural lighting, and our results indicate that structured importance sampling is better than the best previous Monte Carlo techniques, requiring one to two orders of magnitude fewer samples for the same image quality.

Abstract. We develop a scale-invariant version of Matheron’s “dead leaves model ” for the statistics of natural images. The model takes occlusions into account and resembles the image formation process by randomly adding independent elementary shapes, such as disks, in layers. We compare the empirical statistics of two large databases of natural images with the statistics of the occlusion model, and find an excellent qualitative, and good quantitative agreement. At this point, this is the only image model which comes close to duplicating the simplest, elementary statistics of natural images—such as, the scale invariance property of marginal distributions of filter responses, the full co-occurrence statistics of two pixels, and the joint statistics of pairs of Haar wavelet responses. natural images, stochastic image model, non-Gaussian statistics, scaling, dead leaves model, occlu-Keywords: sions, clutter 1.