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23 September 1999 Errors in calculated moments of convex sets using digital images
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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.
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Reinhard Klette and Jovisa Zunic "Errors in calculated moments of convex sets using digital images", Proc. SPIE 3811, Vision Geometry VIII, (23 September 1999);

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