Vector-space solution for a morphological shape-decomposition problem

Abstract

We define a restricted domain as the discrete set of points representing any convex, four-connected, filled polygon whose (i) vertices lie on the lattice points, (ii) interior angles are multiples of 45°, and (iii) number of sides are at most eight. We describe the boundary code and discrete half-plane representation and use them for representing restricted domains. Morphological operations of dilation and n-fold dilation on the restricted domains with structuring elements that are also restricted domains are expressed in terms of the above representations. We give algorithms for these operations and prove that they are of O(1) complexity and hence are independent of the size of the objects.

We prove that there is a set of 13 restricted domains {K ₁, K ₂, ..., K ₁₃} such that any given restricted domain K is expressible as % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2% da9iaadUeadaWgaaWcbaGaaGimaaqabaGccqGHvksXdaqadeqaamaa% xababaGaeyyLIumaleaacaWGRbWaaSbaaWqaaiaaigdaaeqaaaWcbe% aakiaadUeadaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaacqGH% vksXdaqadeqaamaaxababaGaeyyLIumaleaacaWGRbWaaSbaaWqaai% aaikdaaeqaaaWcbeaakiaadUeadaWgaaWcbaGaaGymaaqabaaakiaa% wIcacaGLPaaacqGHvksXcaGGUaGaaiOlaiaac6cacqGHvksXdaqade% qaamaaxababaGaeyyLIumaleaacaWGRbWaaSbaaWqaaiaaigdacaaI% ZaaabeaaaSqabaGccaWGlbWaaSbaaSqaaiaaigdacaaIZaaabeaaaO% GaayjkaiaawMcaaaaa!5B9E!\[K = K_0 \oplus \left( {\mathop \oplus \limits_{k_1 } K_1 } \right) \oplus \left( {\mathop \oplus \limits_{k_2 } K_1 } \right) \oplus ... \oplus \left( {\mathop \oplus \limits_{k_{13} } K_{13} } \right)\] where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWabeaada% WfqaqaaiabgwPifdWcbaGaam4AamaaBaaameaacaWGPbaabeaaaSqa% baGccaWGlbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaaaaa!3DCB!\[\left( {\mathop \oplus \limits_{k_i } K_i } \right)\] represents the k _i -fold dilation of K _i and K ₀ is a translation. We show that this entails a linear transformation from a 13-dimensional space in which restricted domains are represented in terms of n-fold dilations of the 13 basis structuring elements, to an eight-dimensional space in which restricted domains are represented in terms of their eight side lengths. Furthermore, we show that any particular decomposition forms a particular solution of this transformation and that finding all possible dilation decompositions of a restricted domain is equivalent to finding the general solution of this transformation. Finally, we derive a finite-step algorithm for finding a particular decomposition and then give an algorithm for finding all possible decompositions.