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This paper presents a method combining shape and shading models in order to obtain estimations
of 3D shape parameters directly from image grey values. The problem is considered as an
application of optimal parameter estimation theory, according to Liebelt 8 This theory has been
applied previously, where the emphasis was laid on time-delay 2, and motion estimation 3, 5, 9. It
is applied here to provide an environment in which somewhat more complicated models can be
designed with relative ease and to indicate how the behaviour of the parameters can be
investigated. A shading model is added, offering explicit prediction of image grey values. We
consider the problem for a single image and for an image pair, showing the shade of the object at
two consecutive points of time. The last problem requiresalso a model for the motion of the body.
The resulting non-linear estimation problem is linearized about a last parameter guess 8,so that a
linear estimator can be applied to compute a new estimate. The various stages of the modelling
process are separated by introducing several coordinate systems. Coordinate transformations will
show the object from other points of view, and perform an orthographic projection of the 3D scene
into the 2D image plane. The explicit grey value prediction yields a template, having a definite
extent in the image. Because of the shading model this method requires no gradient images, as in
the case of motion estimation 6 or stereo 5. The gradients can be computed analytically. To
demonstrate the usefulness and the flexibility of our method, we consider a solid cylinder,
irradiated with X-rays. The image is a shadow image originating from the absorption of radiation
by the cylinder.
In section 2 some background is given about the theory of parameter estimation from digital
images. In section 3 the various models for the shape and motion of the body and the imaging
process are given. In section 4 and 5 we investigate the properties of the estimator. In section 4
identifiability and uniqueness of the parameters are considered, yielding the parameters, that can
be estimated uniquely from the image data. In section 5 some examples are given, elucidating the
stabilty properties of the algorithm.
To conclude we mention the possibility to replace the motion model with a model connecting
images taken from two different positions. Thus this method is also suited to handle a stereo
configuration.
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