Elsevier

Image and Vision Computing

Volume 23, Issue 2, 1 February 2005, Pages 237-248
Image and Vision Computing

Towards a general sampling theory for shape preservation

https://doi.org/10.1016/j.imavis.2004.06.003Get rights and content

Abstract

Computerized image analysis makes statements about the continuous world by looking at a discrete representation. Therefore, it is important to know precisely which information is preserved during digitization. We analyze this question in the context of shape recognition. Existing results in this area are based on very restricted models and thus not applicable to real imaging situations. We present generalizations in several directions: first, we introduce a new shape similarity measure that approximates human perception better. Second, we prove a geometric sampling theorem for arbitrary dimensional spaces. Third, we extend our sampling theorem to two-dimensional images that are subjected to blurring by a disk point spread function. Our findings are steps towards a general sampling theory for shapes that shall ultimately describe the behavior of real optical systems.

Introduction

Computerized image analysis is increasingly used in applications where errors and mistakes can have critical consequences, like industrial inspection, surveillance, autonomous vehicle control, and medical imaging. Therefore, it becomes more and more important to understand and formalize the conditions of successful algorithm behavior. Errors may have many causes, some of them being external like viewing conditions and illumination, but others depending on the system itself. Since one has often no control over the external influences, it is all the more important to study how certain internal design choices influence performance limits of image analysis systems.

One key aspect is the role of sampling. What can be seen with a digital device looking at a continuous world? In this paper, we will be concerned with the relationship of sampling and shape recognition. How should an image analysis process be designed in order to preserve important shape characteristics of real objects, and when can this performance be guaranteed? Ultimately, this question should be answered by a general geometric sampling theorem analogous to Shannon's famous sampling theorem describing the preservation of waveforms.

Naturally, this paper is not the first attempt to answer this question, neither will it be the last. A detailed description of previous work is given in Section 2. This work is characterized by the fact that definite results could only be obtained by imposing very restrictive assumptions on the digitization process. The results are therefore quite far from reality and provide relatively little guidance for the design of practical applications. It is our goal to extend the existing work in ways that eventually bridge the gap between theory and praxis. In particular, we are going to present an improved measure of shape similarity (Section 3), we present the class of sets we are working on (Section 4), we define which digitization results have to be seen as correct under a certain digitization model (Section 5), we generalize existing sampling theorems for binary shapes to multiple dimensions and arbitrary grids (Section 6), we prove tight bounds for the two-dimensional (2D) case (Section 7), and we analyze how the sampling theorem has to be modified when the image is subjected to blurring prior to digitization, since blurring is unavoidable in any real optical system due to diffraction effects, defocus and finite sensor size (Section 8). These extensions represent significant steps towards a truly general geometric sampling theorem, although further generalizations are still needed to achieve the ultimate goal.

Section snippets

Prior work

Since image analysis tries to derive knowledge about the real world by analyzing discrete representations, it is important to understand which information is preserved during digitization. In particular, we are interested in the preservation of shape characteristics. There exist several approaches which describe how shapes—which can be understood as binary sets—behave under certain types of digitization. First results on this problem were independently presented by Pavlidis [5] and Serra [7] in

Shape similarity

Due to the information loss during digitization, a digital shape is almost never identical to its continuous original. In order to formally describe this information loss, a precise definition of shape similarity is needed. Ideally this definition should resemble human perception as good as possible. As we will see, previously used similarity criteria do not always fulfill this requirement. Therefore, we will define a new criterion called strong r-similarity that will be the basis of our

r-Regular sets

A shape can be defined as a set ARn. A geometric sampling theorem states which type of shapes retains characteristic properties during digitization. Surprisingly, in 1982 Pavlidis [5] and Serra [7] independently introduced totally different definitions for basically the same type of sets, which they used in their theorems. Also Latecki et al. [3], [4] and Tajine and Ronse [10] referred to this shape class. Since this shape class is fundamental, we introduce here a unified definition of r-

Sampling and reconstruction

In this section, we develop a precise definition of the digitization process. All definitions are given for arbitrary dimensional spaces, so that subsequent theorems can be formulated as general as possible. Since we identify shapes with subsets ARn and sets can be interpreted as binary functions, a set ARn can be transformed into an analog binary image by means of its characteristic function χA:Rn{0,1}, χA(x)=1 iff xA. Now the discretization is obtained by storing the values of this

Sampling in arbitrary dimensions

In the following we will show that in any dimension r-regular sets are all important class for shape preserving digitization. In 2D we will show that all r-regular set is weakly reconstructible by any r′-grid with r′<r, regardless of the grid structure and alignment. Nevertheless, it is not possible to prove a reconstruction to be r-similar to an original set of higher dimension in general. Thus, digitization in higher dimensions is not as easy as it looks like in 2D.

The following lemmas

Improved sampling theorems for 2D

The 2D sampling theorem we derived in Section 6 is a generalization of the known theorems of Pavlidis [5] and Serra [7], because the 2D square and hexagonal grids on the one hand and topology preservation and identity of homotopy trees on the other hand are special cases of our definitions. So the results of Serra and Pavlidis are direct corollaries of Theorem 2.

Corollary 1

Let S1=h1Z2 be the square grid with grid size (minimal sampling point distance) h1. Then every 2D r-regular set with r>h1/2 is weakly

Sampling of blurred 2D images

The subset digitization, which we used in Section 7, has the great disadvantage that it cannot be realized in practice. In real imaging devices the analog image is always subjected to blurring before being digitized. The blurring has various causes. First, the light is diffracted in the lens. Second, there is defocus blur for objects that are not exactly in focus. Third, real sensor elements are not zero-dimensional but rather have finite area. The net effect of these factors can be modeled by

Conclusions

In this paper we proved a powerful geometric sampling theorem. Put simply, it means the following. When an r-regular set in an arbitrary dimensional space is digitized, the number and the inclusion properties of connected components of the set and its complement are preserved, and the digital components are directly connected. Parts that were originally connected do not get separated, and the Hausdorff distance between the original and reconstructed boundaries is at most half the pixel

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