Mathematical morphology clarified geometrical properties of shape analysis algorithms for binary pictures. Results of labelling, distance transform, and adjacent numbering are, however, coded pictures. For full descriptions of shape analysis algorithms in the framework of mathematical morphology, it is necessary to extend morphological operations to code-labelled pictorial data. Nevertheless, extensions of morphology to code-labelled pictures have never discussed though the theory of gray morphology is well studied by several authors. Hence, this paper proposes a theory of the coded morphology which is based on the binary scaling of labels of pixels. The method uses n-layered binary sub-pictures for the processing of a picture with 2n labels. By introducing morphological operations for the coded point sets, we express some coding functions in the manner of the mathematical morphology. We also derive multidimensional array registers and gates which store and process coded pictures and morphological operations to them by proposing basic gates which compute parallelly logical operations for elements of Boolean layered arrays. These gates and registers are suitable for the implementation of the shape analysis processors on the three-dimensional VLSI and ULSI.

We introduce two probabilistic algorithms to determine the motion parameters of a planar shape without knowing a priori the point-to-point correspondences. If the target is limited to rigid objects, an Euclidean transformation can be expressed as a linear equation with six parameters, i.e. two translational parameters and four rotational parameters (the axis of rotation and the rotational speed about the axis). These parameters can be determined by applying the randomized Hough transform. One remarkable feature of our algorithms is that the calculations of the translation and rotation parameters are performed by using points randomly selected from two image frames that are acquired at different times. The estimation of rotation parameters is done using one of two approaches, which we call the triangle search and the polygon search algorithms respectively. Both methods focus on the intersection points of a boundary of the 2D shape and the circles whose centers are located at the shape's centroid and whose radii are generated randomly. The triangle search algorithm randomly selects three different intersection points in each image, such that they form congruent triangles, and then estimates the rotation parameter using these two triangles. However, the polygon search algorithm employs all the intersection points in each image, i.e. all the intersection points in the two image frames form two polygons, and then estimates the rotation parameter with aid of the vertices of these two polygons.