What Can We Learn from Discrete Images about the Continuous World?
Image analysis attempts to perceive properties of the continuous real world by means of digital algorithms. Since discretization discards an infinite amount of information, it is difficult to predict if and when digital methods will produce reliable resul
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Abstract. Image analysis attempts to perceive properties of the continuous real world by means of digital algorithms. Since discretization discards an infinite amount of information, it is difficult to predict if and when digital methods will produce reliable results. This paper reviews theories which establish explicit connections between the continuous and digital domains (such as Shannon’s sampling theorem and a recent geometric sampling theorem) and describes some of their consequences for image analysis. Although many problems are still open, we can already conclude that adherence to these theories leads to significantly more stable and accurate algorithms.
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Introduction
As far as computer vision is concerned, the real world can be considered as a continuous domain.1 With few exceptions, discrete atoms and molecules are below the relevant scales of image analysis. In contrast, computer-based reasoning is always discrete – an infinite amount of information is inevitably lost before algorithms produce answers. Experience tells us that these answers are nevertheless useful, so enough information is apparently preserved in spite of the loss. But failures in automatic image analysis are not infrequent, and the fundamental question under which conditions discrete methods provide valid conclusions about the analog real world is still largely unsolved. This problem can be approached in different ways. In one approach (which is, for example, popular in physics), theories and models are formulated in the analog domain, e.g. by means of differential equations. Discretisation is then considered as an implementation issue that doesn’t affect the theory itself. The correctness of discrete solutions is ensured by asymptotic convergence theorems: Discretization errors can be made as small as desired by choosing sufficiently fine
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The author gratefully acknowledges essential contributions by Hans Meine and Peer Stelldinger and many fruitful discussions with Bernd Neumann, Hans-Siegfried Stiehl, and Fred Hamprecht. To avoid ambiguity, we shall use the term analog domain in the sequel. Otherwise, it might remain unclear whether a “continuous image” is an image function without discontinuities, or an image defined on a non-discrete domain. We will call the latter “analog image” instead.
D. Coeurjolly et al. (Eds.): DGCI 2008, LNCS 4992, pp. 4–19, 2008. c Springer-Verlag Berlin Heidelberg 2008
What Can We Learn from Discrete Images about the Continuous World?
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discretizations, and automatic or interactive refinement mechanisms adaptively adjust the resolution as required. Unfortunately, this approach doesn’t work well in image analysis: Usually, we don’t have the opportunity to take refined images if the original resolution turns out to be insufficient. Thus, we need absolute accuracy bounds (rather than asymptotic ones) to determine the correct resolution beforehand, or to decide what kind of additional information can be utilized if the desired resolution cannot be reached. Moreover, new discretisation schemes may be required be
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