Purpose and Limits of Models

The term model refl ects the fact that any natural phenomenon can only be described to a certain degree of accuracy and correctness. It is one of the most powerful principles throughout all natural sciences to seek the simplest and most general description that still describes the obser- vations with minimum deviations. A handful of basic laws of physics describe an enormously wide range of phenomena in a quantitative way. Along the same lines, models are a useful and valid approach for im- age processing tasks. However, models must be used with caution. Even if the data seem to be in perfect agreement with the model assumptions,

there is no guarantee that the model assumptions are correct.

Figure 17.1 shows an illustrative example. The model assumptions include a fl at black object lying on a white background that is illuminated homogeneously (Fig. 17.1a). The object can be identifi ed clearly by low gray values in the image, and the discontinuities between the high and low values mark the edges of the object.

If the black object has a non-negligible thickness, however, and the scene is illuminated by an oblique parallel light beam (Fig. 17.1c), we receive exactly the same type of profi le as for Fig. 17.1a. Thus, we do

17.2 Purpose and Limits of Models                                               443

g

a

c                       illumination                        g

 

thick black object

 

 

white

shaded area

 

wrong edge detection

 

Figure 17.1: Demonstration of a systematic error which cannot be inferred from the perceived image. a , c sketch of the object and illumination conditions; b and d resulting gray value profi les for a and c , respectively.

 

A                                                 b                                            c

Histogram

p(g)

 

 

?

 

 

g

 

Figure 17.2: Demonstration of a systematic deviation from a model assump- tion (object is black, background white) that cannot be inferred from the image histogram.

 

not detect any deviation from the model assumption. Still only the right edge is detected correctly. The left edge is shifted to the left because of the shadowed region resulting in an image too large for the object.

Figure 17.2 shows another case. A black fl at object fi lls half of the im- age on a white background. The histogram (the distribution of the gray values) clearly shows a bimodal shape with two peaks of equal height. This tells us that basically only two gray values occur in the image, the lower being identifi ed as the black object and the higher as the white background, each fi lling half of the image.

444                                                                     17 Regularization and Modeling

 

This does not mean, however, that any bimodal histogram stems from an image where a black object fi lls half of the image against a white background. Many other interpretations are possible. For instance, also a white object could be encountered on a black background. The same bimodal histogram is also gained from an image in which both the ob- ject and the background are striped black and white. In the latter case, a segmentation procedure that allocates all pixels below a certain thresh- old to the object and the others to the background would not extract the desired object but the black stripes. This simple procedure only works if the model assumption is met that the objects and the background are of uniform brightness.

The two examples discussed above clearly demonstrate that even in simple cases we can run into situations where the model assumptions appear to be met — as judged by the image or quantities derived from the image such as histograms — but actually are not. While it is quite easy to see the failure of the model assumption in these simple cases, this may be more diffi cult if not impossible in more complex cases.

 

17.3 Variational Image Modeling†

As discussed in the introduction (Section 17.1), a mathematically well- founded approach to image modeling requires the setup of a model func- tion and an error functional that measures the residual deviations of the measured data from the computed model data.

For image segmentation, a suitable modeling function could be a piecewise fl at target function f ( x ). Regions with constant values cor- respond to segmented objects and discontinuities to object boundaries. The free parameters of this model function would be gray values in the diff erent regions and the boundaries between the regions. The bound- aries between the objects and the gray values of the regions should be varied in such a way that the deviation between the model function f ( x ) and the image data g( x ) are minimal.

The global constraints of this segmentation example are rather rigid. Smoothness constraints are more general. They tend to minimize the spatial variations of the feature. This concept is much more general than using a kind of fi xed model saying that the feature should be constant as in the above segmentation example or vary only linearly.

Such global constraints can be handled in a general way using vari- ation calculus. Before we turn to the application of variation calculus in image modeling, it is helpful to start with a simpler example from physics.

17.3 Variational Image Modeling†                                                                    445

 

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