Robert A. Hummel

Scene Labeling by Relaxation Operations

Given a set of objects in a scene whose identifications are ambiguous, it is often possible to use relationships among the objects to reduce or eliminate the ambiguity. A striking example of this approach was given by Waltz [13]. This paper formulates the ambiguity-reduction process in terms of iterated parallel operations (i.e., relaxation operations) performed on an array of (object, identification) data. Several different models of the process are developed, convergence properties of these models are established, and simple examples are given.

On the Foundations of Relaxation Labeling Processes

A large class of problems can be formulated in terms of the assignment of labels to objects. Frequently, processes are needed which reduce ambiguity and noise, and select the best label among several possible choices. Relaxation labeling processes are just such a class of algorithms. They are based on the parallel use of local constraints between labels. This paper develops a theory to characterize the goal of relaxation labeling. The theory is founded on a definition of con-sistency in labelings, extending the notion of constraint satisfaction. In certain restricted circumstances, an explicit functional exists that can be maximized to guide the search for consistent labelings. This functional is used to derive a new relaxation labeling operator. When the restrictions are not satisfied, the theory relies on variational cal-culus. It is shown that the problem of finding consistent labelings is equivalent to solving a variational inequality. A procedure nearly identical to the relaxation operator derived under restricted circum-stances serves in the more general setting. Further, a local convergence result is established for this operator. The standard relaxation labeling formulas are shown to approximate our new operator, which leads us to conjecture that successful applications of the standard methods are explainable by the theory developed here. Observations about con-vergence and generalizations to higher order compatibility relations are described.

A Three-Dimensional Edge Operator

Modern scanning techniques, such as computed tomography, have begun to produce true three-dimensional imagery of internal structures. The first stage in finding structure in these images, like that for standard two-dimensional images, is to evaluate a local edge operator over the image. If an edge segment in two dimensions is modeled as an oriented unit line segment that separates unit squares (i.e., pixels) of different intensities, then a three-dimensional edge segment is an oriented unit plane that separates unit volumes (i.e., voxels) of different intensities. In this correspondence we derive an operator that finds the best oriented plane at each point in the image. This operator, which is based directly on the 3-D problem, complements other approaches that are either interactive or heuristic extensions of 2-D techniques.

Reconstructions from zero crossings in scale space

In computer vision, the one-parameter family of images obtained from the Laplacian-of-a-Gaussian-filtered version of the image, parameterized by the width of the Gaussian, has proved to be a useful data structure for the extraction of feature data. In particular, the zero crossings of this so-called scale-space data are associated with edges and have been proposed by D. Marr (1982) and others as the basis of a representation of the image data. The question arises as to whether the representation is complete and stable. The authors survey some of the studies and results related to these questions as well as several studies that attempt reconstructions based on this or related representations. They formulate a novel method for reconstruction from zero crossings in scale space that is based on minimizing equation error, and they present results showing that the reconstruction is possible but can be unstable. They further show that the method applies when gradient data along the zero crossings are included in the representation, and they demonstrate empirically that the reconstruction is then stable. >

Experiments with the intensity ratio depth sensor

Abstract This paper discusses a modification of the “plane-of-light” approach to range data acquisition which utilizes the ratio of two intensity images. The modification allows depth determination at each image pixel, avoiding the need to scan the plane of light, requiring only the acquisition and processing of two or three intensity images. The depth equation together with three experimental methods for its calculation are presented. Results of the three sensor implementations are given for test scenes.

Junctions: detection, classification, and reconstruction

Junctions are important features for image analysis and form a critical aspect of image understanding tasks such as object recognition. We present a unified approach to detecting, classifying, and reconstructing junctions in images. Our main contribution is a modeling of the junction which is complex enough to handle all these issues and yet simple enough to admit an effective dynamic programming solution. We use a template deformation framework along with a gradient criterium to detect radial partitions of the template. We use the minimum description length principle to obtain the optimal number of partitions that best describes the junction. The Kona detector presented by Parida et al. (1997) is an implementation of this model. We demonstrate the stability and robustness of the detector by analyzing its behavior in the presence of noise, using synthetic/controlled apparatus. We also present a qualitative study of its behavior on real images.

Deblurring Gaussian blur

Gaussian blur, or convolution against a Gaussian kernel, is a common model for image and signal degradation. In general, the process of reversing Gaussian blur is unstable, and cannot be represented as a convolution filter in the spatial domain. If we restrict the space of allowable functions to polynomials of fixed finite degree, then a convolution inverse does exist. We give constructive formulas for the deblurring kernels in terms of Hermite polynomials, and observe that their use yields optimal approximate deblurring solutions among the space of bounded degree polynomials. The more common methods of achieving stable approximate deblurring include restrictions to band-limited functions or functions of bounded norm.

Feature detection using basis functions

A class of feature-detection operations using orthonormal basis functions is introduced. The basis functions are derived from the Karhunen-Loe`ve expansion of the local image data. This theory is applied to define a new edge detector which can be adjusted to have increased sensitivity in any desired orientation.

A statistical viewpoint on the theory of evidence

The authors provide a perspective and interpretation regarding the Dempster-Shafer theory of evidence that regards the combination formulas as statistics of the opinions of experts. This is done by introducing spaces with binary operations that are simpler to interpret or simpler to implement than the standard combination formula, and showing that these spaces can be mapped homomorphically onto the Dempster-Shafer theory-of-evidence space. The experts in the space of opinions-of-experts combine information in a Bayesian fashion. Alternative spaces for the combination of evidence suggested by this viewpoint are presented. >

Motion parameter estimation from global flow field data

Presented are two methods for the determination of the parameters of motion of a sensor, given the vector flow field induced by an imaging system governed by a perspective transformation of a rigid scene. Both algorithms integrate global data to determine motion parameters. The first (the flow circulation algorithm) determines the rotational parameters. The second (the FOE search algorithm) determines the translational parameters of the motion independently of the first algorithm. Several methods for determining when the function has the appropriate form are suggested. One method involves filtering the function by a collection of circular-surround zero-mean receptive fields. The other methods project the function onto a linear space of quadratic polynomials and measures the distance between the two functions. The error function for the first two methods is a quadratic polynomial of the candidate position, yielding a very rapid search strategy. >