[...]... reviewed 1 INTRODUCTION T h e "shrinking" of binary images to similarly connected representations t h a t have smaller foregrounds (i.e., fewer Ts) has found application as a fundamental preprocessing step in image processing Two forms of such shrinking have been of greatest interest: (1) T h e image is transformed to a "topologically equivalent" image t h a t has fewer I's; (2) connected components... 3-layer image If t h e local table method is used in each layer, its complexity for an n x n x n image is O(n^) + (n^ times t h e complexity of RESOLVE for each row) -f (n times t h e complexity of RESOLVE for each layer) 2.6 P A R A L L E L C O N N E C T E D C O M P O N E N T S A L G O R I T H M S T h e use of parallel architectures can speed up the execution of most image processing algorithms For example,... equivalence table for row L." EQTABLE := CREATE( ); "Process the row." for P := 1 to NCOLS do 10 if LABEL(L,P) < > 0 then begin LA := LABELS(NEIGHBORS(L,P)); for X in LA and X < > LABEL(L,P) ADD (X,LABEL(L,P), EQTABLE) end for end end for "Find equivalence classes." EQCLASSES := RESOLVE(EQTABLE); for E in EQCLASSES do EQLABEL(E) := MIN(LABELS(E)) end for "Relabel the pixels of row L one last time." for P :=... the computer vision community 26 REFERENCES 1 G Agin, "Computer Vision System for Industrial Inspection and Assembly," IEEE Transactions on Computers, Vol 14, 1980, pp 11-20 2 P.-E Danielsson and S.L Tanimoto, "Time Complexity for Serial and Parallel Propagation in Images", in Architecture and Algorithms for Digital Image Processing, A Oosterlinck and P.-E Danielsson, (eds.) Proceedings of the SPIE,... satisfied at the start of the pass: of a 2-d or performs a t h a t in the bottom-up • For 1 < z < n, whenever y G Ri is adjacent to x € Ri-i-, ^'s label is less t h a n or equal to ar's label A consequence of this is that in rule R , during pass 2, ?/'s label would always be the m i n i m u m of the labels of x and y Topological Algorithms for Digital Image Processing T.Y Kong and A Rosenfeld (Editors)... T H M T h e images handled by our algorithms so far are two-dimensional images Both the definition of connected components and the algorithms can be generahzed to threedimensional images, which are sequences of two-dimensional images called layers T h e generalization of the definition results straightforwardly from the generalization of the concept of a neighborhood Suppose that a 3D image consists... shows the initial binary image, and (b) the labeling after the first top down pass of the algorithm The equivalence classes found are 1: { 1,12,7,8,9,10,5 } and 2: { 2,3,4,6,11,13 } LABEL(L,P) := M; end for end for; "Find equivalence classes." EQCLASSES := RESOLVE(EQTABLE); for E in EQCLASSES do EQLABEL(E) := MIN(LABELS(E)) end for; "Top-down pass 2" for L := 1 to NROWS do for P := 1 to NCOLS do if... "Digital Topology: Introduction and Survey", Computer Vision, Graphics, and Image Processing, Vol 48, 1989, pp 357-393 5 R Lumia, L.G Shapiro, and 0 Zuniga, "A New Connected Components Algorithm for Virtual Memory Computers," Computer Vision, Graphics, and Image Processing, Vol 22, 1983, pp 287-300 6 R Lumia, "A New Three-Dimensional Connected Components Algorithm", Computer Vision, Graphics, and Image. .. for L:=l to NROWS do "Initialize local equivalence table for row L." EQTABLE := CREATE( ); "Initialize all labels on row L to zero." for P := 1 to NCOLS do LABEL(L,P) := 0 end for; "Process the row." for P := 1 to NCOLS do if I(L,P) = 1 then begin A := NEIGHBORS((L,P)); if ISEMPTY(A) t h e n M := NEWLABEL( ) else begin M := MIN(LABELS(A) ); for X in LABELS(A) and X < > M do ADD (X,M, EQTABLE) end for. .. component LABEL(L,P) := M; end end for; "Find equivalence classes detected on this row." EQCLASSES := RESOLVE(EQTABLE); for E in EQCLASSES do EQLABEL(E) := MIN(LABELS(E)) end for; "Relabel the parts of row L with their equivalence class labels." for P := 1 to NCOLS do if I(L,P) = 1 t h e n LABEL(L,P) := EQLABEL(CLASS(LABEL(L,P))) end for end for; "Bottom-up pass" for L := N R O W S - 1 to 1 by - 1 . sequential passes over the image. All the algorithms process a row of the image at a time. Modifications to process a rectangular window subimage at a time are straightforward. All the algorithms assign. (a) 0 0 1 0 0 0 0 0 0 0 a a a a a a a 2 a a a 1 b 4 a a a 3 a a a a a a a (b) a a a a a a a 1 a a a 1 a 1 a a a 1 a a a a a a a a. properties. Topological properties and algorithms play a fundamental role in the analysis of two- and three-dimensional digital images. This book deals with basic topological algorithms; it presents