Jovisa D. Zunic

Interactions between number theory and image analysis

The conceptual design of many procedures used in image analysis starts with models which assume as an input sets in Euclidean space which we regard as real objects. However, the application finally requires that the Euclidean (real) objects have to be modelled by digital sets, i.e. they are approximated by their corresponding digitizations. Also continuous operations (for example integrations or differentiations) are replaced by discrete counterparts (for example summations or differences) by assuming that such a replacement has only a minor impact on the accuracy or efficiency of the implemented procedure. This paper discusses applications of results in number theory with respect to error estimations, accuracy evaluations, correctness proofs etc. for image analysis procedures. Knowledge about digitization errors or approximation errors may help to suggest ways how they can be kept under required limits. Until now have been only minor impacts of image analysis on developments in number theory, by defining new problems, or by specifying ways how existing results may be discussed in the context of image analysis. There might be a more fruitful exchange between both disciplines in the future.

A Characterization of Digital Disks by Discrete Moments

In this paper our studies are focused on the digital disks and problems of their characterization (coding) with an appropriate number of bits, and reconstruction of the original disk from the code that is used. Even though the digital disks appear very often in practice of the computer vision and image processing, only the problem of their recognition has been solved till now. In this paper a representation by constant number of integers, requireing optimal number of bits, is presented. One-to-one correspondence between the digital disks and their proposed codes, consisting of: n n n- the number of points of the digital disk n n n- the sum of x-coordinates of the points of digital disk n n n- the sum of y-coordinates of the points of digital disk, is proved.

Errors in calculated moments of convex sets using digital images

Moments have been widely used in shape recognition and identification. In general, the (k,1)-moment, denoted by mk,l(S), of a planar measurable set S is defined by mk,l(S) equals (integral) S(integral) xkyl dx dy. We assume situations in image analysis and pattern recognition where real objects are acquired (by thresholding, segmentation, etc.) as binary images D(S), i.e. as digital sets or digital regions. For a set S, in this paper its digitization is defined to be the set of all grid points with integer coordinates which belong to the region occupied by the given set S. Since in image processing applications, the exact values of the moments mk,l(S) remain unknown, they are usually approximated by discrete moments (mu) k,l(S) where (mu) k,l(S) equals (summation)/(i,j)(epsilon) D(S) ik (DOT) jl equals (summation)/i,j are integers (i,j)(epsilon) S ik (DOT) jl. Moments of order up to two (i.e. k + l less than or equal to 2) are frequently used and our attention is focused on them, i.e. on the limitation in their estimation from the corresponding digital picture. In this paper is it proved that mk,l(S) - 1/rk+l+2 (DOT) (mu) k,l(r (DOT) S) equals (Omicron) (1/r15/11+(epsilon )) approximately equals (Omicron) (1/r1.363636...) for k + l less than or equal to 2, where S is a convex set in the plane with a boundary having continuous third derivative and positive curvature at every point, r is the number of pixels per unit (i.e. 1/r is the size of the pixel), while r (DOT) S denotes the dilation of S by factor r.

Toward experimental studies of digital moment convergence

Digital moments approximate real moments where the accuracy depends upon grid resolution. There are theoretical results about the speed of convergence. However, there is a lack of more detailed studies with respect to selected shapes of regions, or with respect to experimental data about convergence. This paper discusses moments for specific shapes of regions, and provides some initial experimental data about measured convergence of digital moments.

Ellipses Estimation from their Digitization

Ellipses in general position, and problems related to their reconstruction from digital data resulting from their digitization, are considered. If the ellipse n n

The Discrete Moments of the Circles

E:tilde Aleft( {x - p} right)^2 + 2tilde Bleft( {x - p} right)left( {y - q} right) + tilde Cleft( {y - q} right)^2 leqslant 1, tilde Atilde C - tilde B^2 > 0,

Statistical characterization of digital lines

n nis presented on digital picture of a given resolution, then the corresponding digital ellipse is: n n