ON DIGITAL IMAGE INVERSION 
Yong-Jian Zheng 
Institute for Photogrammetry and Remote Sensing, University Karlsruhe 
Englerstrasse 7, D-7500 Karlsruhe 1, Germany 
Email: zheng@ipf.bau-verm.uni-karlsruhe.de 
Abstract 
Image understanding is the enterprise of automating 
and integrating a wide range of processes and repre- 
sentations used for vision perception. It includes tech- 
niques not only for geometric modeling but also for 
inference and reasoning. In this paper, we look at the 
issue of inductive inference in digital image under- 
standing. Many goals in both low-level and high-level 
image analysis can be formulated generally as a pro- 
blem of inferring object-properties from image data, 
having assistance of a priori knowledge. This process 
of information processing would be called image in- 
version, as the desired information about the objects 
is derived from image data. Based on the inverse pro- 
blem theory, we provide a sound theoretical basis for 
determination of generalizations, descriptions, rules, 
and laws, from a set of raw data, observations, featu- 
res or facts. To demonstrate this approach, we present 
its application in the limited domain of surface recon- 
struction from multiple images. 
1 Introduction 
Computer Vision includes techniques not only for geo- 
metric modeling but also for inference and reasoning. 
Many of its tasks require the ability to create explicit 
representations of knowledge from implicit ones and 
they can be therefore formulated as problems of infe- 
rence drawing. Drawing inference from image data is 
only plausible as the available information is incom- 
plete or inexact and it is inadequate to support the 
desired sorts of logical inferences. 
Plausible inference is a basic issue, of which we are 
all aware through our own experience in research on 
many vision problems, including feature extraction, 
image and boundary segmentation, object reconstruc- 
tion and image interpretation. In these cases, problem 
solvers have to reason with inconsistent and incom- 
plete information on the basis of beliefs, not only true 
(or false) facts. 
In this paper, we think of inference drawing from di- 
gital images as an inverse process which we call di- 
488 
gital image inversion (Zheng, 1990). Drawing plau- 
sible inference is therefore solving ill-posed inverse 
problems. Although this kind of problems have been 
considered for a long time almost exclusively as ma- 
thematical curiosities, it is now clear that many in- 
verse problems have ill-posed nature and their solu- 
tions are of great practical interest (Herman, 1980; 
Fawcett, 1985; Poggio et al., 1985). To deal with ill- 
posed inverse problems, one has to deal with several 
questions including: What is the nature of inverse pro- 
blems? How about their solvability? How to integrate 
a priori knowledge to deal with the ill-posedness of 
inverse problems? And how to evaluate the quality of 
solutions? 
We begin by introducing a paradigm for digital image 
inversion, which has three steps of representation, for- 
ward modeling and inversion. Then, we discuss the 
theory of inductive inference and inverse problems. 
Based on the Maximum A Posteriori (MAP) we des- 
cribe a framework for integrating a priori knowledge 
to solve the decision problem in the ill-posed inverse 
process. Next, we formulate the problem of surface 
reconstruction from digital images within this fra- 
mework. We then demonstrate shortly the result of 
our implementation and experiment results using real 
image data. 
2 Digital Image Inversion 
Generally, vision can be regarded as an inference pro- 
cess in which a description of the outside world is 
inferred from images of the world, having the aid of 
a priori knowledge about the world and about ima- 
ging process. Here, three kinds of information have to 
be dealt with, i.e. the desired information about the 
outside world, the available information contained in 
images, and the a priori information of image inter- 
preters. 
Now, let S represent a physical system (for instance 
the earth's surface, or an object in an image). Assume 
that we are able to define a set of model parameters 
which completely describes S, to some extent. These 
parameters may not all be directly measurable. We 
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