Local operations on labelled dot patterns

Transcription

1 Pattern Reconton Letters 9 (1989) May 1989 North-Holland Local operatons on labelled dot patterns Azrel RSENFEL and Jean-Mchel JLN Coputer Vson Laboratory, Center J~r Autoaton Research, Unversty ~! Mao'land, Collee Park, M , USA Receved 3 Auust 1988 Abstract: When a local operaton s perfored on the pxels n an array, the new value of the pxel s a functon of the old values of the pxel and ts nehbors. Ths paper ntroduces the ore eneral concept of local operatons on labelled dot patterns, where the new label of a dot s a functon of the old labels of the dot and a set of ts nehbors (e.., ts Vorono nehbors). Such operatons ay chane the postons of the dots, n addton to chann ther "values'. We llustrate these deas by vn exaples of operatons that perfor local feature detecton (e.., solated dot detecton, cluster ede detecton, dotted curve detecton) and 'enhanceent' (e.., 'soothn' the dot spacn or 'sharpenn" the edes of dffuse clusters), as well as 'orpholocal' operatons. Local operatons on labelled raphs are also brefly dscussed. 1. ntroducton The ost coon class of operatons used n dtal ae processn are the local operatons [1]. When such an operaton s perfored at a pxel P, t ves P a new value whch s a functon of the old values at P and at a set of ts nehbors. A strahtforward eneralzaton of a local operaton on an array of pxels s the concept of a local operaton on a set of labelled ponts ('dots') n the plane. Assue we are ven a defnton of the 'nehbors' of a dot - for exaple, the Vorono nehbors (see Secton 2). Then when we perfor a local operaton at dot P, P ets a new label whch s a functon of the old labels of P and of ts nehbors. n eneral, when we perfor a local operaton on a dot pattern, the new label of a dot should depend on the postons of the nehbors relatve to the dot. [Analoously, n ost local operatons on pxel arrays, the new value of a pxel depends not only on the set of nehbor values, but also on ther spatal The support of the Natonal Scence Foundaton and the U.S. Ar Force ffce of Scentfc Research under NSF Grant CR s ratefully acknowleded, as s the help of Sandy Geran n preparn ths paper. arraneent.] Ths suests that we should reard the poston of a dot as part of ts label, so we can use the relatve postons of the dot and ts nehbors n coputn the new label. Ths further ples that n eneral, a local operaton can odfy not only the 'value' of a dot (whch s one part of ts label) but also ts poston (whch s also part of ts label). We wll dscuss both value-odfyn and poston-odfyn local operatons n the follown sectons. 2. Local operatons Let P~... P, be dstnct dots n the plane, where P has coordnates (x, Yl) and "value' z, <_ < n. There are several standard ways of defnn the 'nehbors' of a ven dot P; soe of the are as follows: (a) The d-nehbors of P are the dots whose dstances fro P are at ost d. Note that for suffcently sall d, P s ts only d-nehbor, and for suffcently lare d, all the P's are d-nehbors. (b) The k nearest nehbors of P are the k dots that le closest to P. Note that, except for exact tes, P always has exactly k such nehbors. (c) The Vorono nehbors of P are defned as fol /89/$3.5 (~) 1989, Elsever Scence Publshers B.V. (North-Holland) 225

2 Volue 9, Nuber 4 PATTERN RECGNTN LETTERS May 1989 lows: Let R~ be the reon of the plane consstn of ponts that le closer to P~ than to any other Pj. Readly, the R's are open convex polyons. We call P and Pj Vorono nehbors f the closures of R~ and R~ ntersect. t can be shown that n a rando dot pattern, a dot has, on the averae, sx Vorono nehbors. (For a dscusson of the Vorono nehborhoods R~ and ther uses n analyzn dot patterns, see [2]; for further work on the subject see [3, 4].) The d-nehbor defnton has the dsadvantae that t requres us to choose d, and that the best choce of d ay vary fro one part of a dot pattern to another. The other two defntons, on the other hand, adjust theselves to the densty or sparseness of the pattern. The Vorono defnton has several advantaes over the k-nearest defnton; t s syetrc (f P s a Vorono nehbor of, the reverse s also true), and the Vorono nehbors of P 'surround t'. We shall use the Vorono defnton fro now on, but other defntons could be used f desred. Let P have coordnates (x,y) and value z; let the nehbors 1... k of P have coordnates (u, v~) and values w~, 1 _< _< k. A eneral local operaton at P ves P new coordnates and a new value that are functons of the old coordnates and value of P and the 's. n ths paper we wll consder only operatons that are shft-nvarant,.e. that do not depend on the absolute coordnates of P and the 's, but only on ther relatve coordnates (x-ul, y - vl)... (x - u~, y - v~). n the follown two sectons we llustrate two portant specal classes of local operatons: valueodfyn operatons, whch chane z but not (x,y), and poston-odfyn operatons, whch chane (x,y) but not z. Evdently, value-odfyn operatons are a eneralzaton of local operatons on dtal aes. 3. Value-odfyn operatons Several types of value-odfyn local operatons are coonly used n dtal ae processn. Feature detecton operatons take on hh values at pxels whose nehborhoods contan ven types of patterns of values e.., edelke, lnelke, or spot- lke. Enhanceent operatons sooth the ae, or 'sharpen' blurred edes n the ae. Morpholocal operatons 'shrnk' or 'expand' hh- valued reons n the ae. For an ntroducton to these classes of operatons see [1, Chapters 6 and 1]. Each of these classes of operatons can be eneralzed to value-odfyn operatons on dot patterns. n analoy wth feature detecton n a dtal ae, we can detect (cluster) edes, (dotted) curves, and spots or solated dots n a dot pattern. n analoy wth enhanceent, we can sooth or sharpen the dot values; and n analoy wth orpholoy, we can shrnk or expand the hh values Feature detecton (a) Value ede detecton The ost coon type of feature detecton used n dtal ae processn s ede detecton,.e. detectn pxels at whch the local rate of chane n value s hh. The 'edeness' at a pxel s coonly easured n several ways: (1) by coparn the pxel's nehborhood wth a set of teplates representn standardzed edes n varous orentatons; (2) by coputn frst dfferences of the pxel values n two perpendcular drectons, and cobnn the to fnd the axu rate of chane, or radent; (3) by coputn second dfferences of the pxel values n two perpendcular drectons, cobnn the nto a 'Laplacan' value, and fndn zerocrossns of that value. These ethods of ede detecton do not edately eneralze to dot patterns; snce the arraneent of the dots s rreular, t s not always obvous how to defne standard ede-lke patterns of values, or to estate drectonal dfferences. We can, however, easure the rate of chane n a ven drecton n the follown way: Let ~ be the drecton fro P to ~,.e., 1 = tan- ly- vl X -- U ' and let P be the dstance fro P to ~,.e., - u) 2 + (y - v) z. 226

3 - - Z Volue 9, Nuber 4 PATTERN RECGNTN LETTERS May 1989 / ~ o B ",.,-". zj. 1,/.../ _/! ",, _ / /~ /~(.. / /,,/ -- / /.,. / / M / / / / / (a) (b) / / /--N N N, / - (c) () Fure 1. Cluster ede detecton. (a) nput dot pattern consstn of three clusters (fro [3]). (b) Cluster ede values; the best (, 45", 9, or 135) at each dot s ndcated by the slope of the lne seent, and the postve edeness value for that s ndcated by the darkness of the seent. Note that snce the clusters are 'blurry', nearly every dot has a postve edeness. (c~t) Another exaple, usn a sharply bounded cluster on a sparse backround. Then the rate of chane of dot value n drecton can be easured by 1 k fw--- ""' Z)CS( -- 3 k=l P.e., the averae of the coponents, n drecton, of the rates of chane of value, per unt dstance, between P and ts nehbors. n ters of ths concept we can defne a 'radent' drecton n whch ths quantty s reatest, and we can defne the 'edeness' at P as the antude of ths reatest rate of chane. (b) Cluster ede detecton Another type of ede detecton, that can be defned for dot patterns but not for dtal aes, s the detecton of cluster edes, where there s a rapd chane n the local densty of the dots. Note that ths property depends only on the postons of the dots, not on ther values, so our forula for the rate 227

4 Volue 9, Nuber 4 PATTERN RECGNTN LETTERS May 1989 of chane (see below) wll not nvolve the values. We frst copute the local dot densty d(p) at each dot - e.., as easured by soe functon whch rows nversely wth the averae dstance fro P to ts nehbors, or wth the area of the Vorono polyon contann P. We can then copute 'edeness' usn d(p), rather than z, as the value at each dot, and then use the sae forula for the rate of chane as n (a) above. Two exaples of the edeness values coputed n ths way are shown n Fure 1, usn d(p)= 1/lo A(P), where A(P) s the area of the Vorono polyon contann P. (c) Value lne detecton n dtal aes, the 'lneness' at a pxel s usually easured by coparn the pxel's nehborhood wth a set of lnelke teplates n varous orentatons. Ths approach s not edately applcable to dot patterns, because of the rreular arraneent of the dots. However, we can easure the 'lneness' n a ven drecton by 1 k ]w-z 2(-,) where the notaton s as n (a) above. Ths quantty s hh f there are hh rates of chane n drectons and + n ('across' the lne) and low rates of chane n the perpendcular drectons ('alon' the lne); the 'lneness' at P s easured by fndn the for whch ths quantty s a axu. (d) otted lne detecton f there s a dotted lne throuh P n drecton, the nehbors of P n drectons near and + 7r are close to P, whle those n the perpendcular drectons are far fro P. (Note that s now the drecton alon the lne, not across the lne.) Ths stuaton can be detected by coputn 1 k ~Z= l 6() cos 2( - ), where 6() s soe functon of the closeness of to P, and fndn the for whch t s a axu. An exaple of 'dotted lneness' coputaton usn ths ethod, wth 6() = l/p~, s shown n Fure 2. (e) Spot detecton and solated dot detecton A 'spot' s a dot that has hh contrast wth all ts nehbors. Thus the hh-valued 'spotness' at P can be easured by ran (z - 1 <_Nk w)/p. :(a), t Fure 2. otted lne detecton. (a) nput dot pattern. (b) Lneness values and best 's (dsplayed as n Fure 1 b, d). b) 228

5 Volue 9, Nuber 4 PATTERN RECGNTN LETTERS May 1989 Slarly, an solated dot s one that s dstant fro all ts nehbors; thus the solatedness of P s easured by n P- 1 <Gk Note that because we are usn Vorono nehborhoods, we can copute edeness, lneness, etc. locally,.e., as functons of sall sets of nehbors, ndependently of the sparseness of the dot pattern. n a dtal ae, to detect features n sparse patterns we would need operators whose sze rew wth the sparseness of the pattern. Thus by operatn on Vorono nehborhoods we can handle sparse patterns wthout ncurrn hh coputatonal cost Enhanceent We can sooth the values n a dot pattern by replacn each dot's value by the averae of ts nehbors' values. Note that f the pattern contans value edes, ths process wll 'blur' the. We can sooth wthout blurrn value edes by usn the edan of the nehbors' values nstead of the ean. ther types of 'ede-preservn' soothn operatons used for dtal aes can also be defned for dot patterns; we ot the detals here. We can 'sharpen' the values n a dot pattern by usn a 'Laplacan' technque n whch soothed values are subtracted fro the ornal unsoothed values. As n the case of a dtal ae, ths has the effect of steepenn (and hence 'sharpenn') value edes Morpholoy We can 'expand' hh-valued reons n a dot pattern by replacn each dot's value (repeatedly) by the axu of ts nehbors' values. Slarly, we can 'shrnk' hh-valued reons (or expand lowvalued ones) by usn the nu. rectonal expanson or shrnkn can be perfored by vn the nehbors' values wehts that depend on ther drectons (e.., wehtn the by cos ). f desred, the values can also have wehts that depend on the nehbors' dstances. 4. Poston-odfyn operatons n ths secton we brefly dscuss local operatons that odfy the postons, rather then the values, of the dots. t s not clear how to defne 'feature detecton' operatons based on poston odfcaton; propertes such as the 'edeness' of a pxel are ost naturally expressed n ters of ts value, not ts poston. t ay be possble to defne 'orpholocal' operatons that odfy postons rather than values. For exaple, one ht want to ove hh-valued dots, or dots that have hh local dot denstes, farther apart fro one another, and low-valued or low-densty dots closer toether, or vce versa. We shall not attept to defne such operatons here. An portant class of poston-odfyn 'enhanceent' operatons nvolves ovn dots so as to sooth or 'sharpen' the dot denstes. We ve two sple exaples of ths class of operatons: (a) ot densty soothn. We can equalze the dot spacn by (repeatedly) ovn each dot n the drecton of the centrod of ts Vorono polyon. Ths oves t farther fro ts closer nehbors and closer to ts farther ones, thus tendn to equalze the nehbor dstances. An exaple of ths type of operaton s shown n Fure 3. (b) ot clustern. We can enhance dfferences n dot spacn (and ultately ~collapse') clusters of dots nto snle ponts) by (repeatedly) ovn each dot n the drecton of the centrod of ts set of Vorono nehbors. The effects of ths process are llustrated n Fure 4. Alternatvely, we can assocate asses wth the dots, and let the ove under the nfluence of ravty, whch wll also cause clusters to 'collapse'; ths concept has been used (n oderaton) as a ethod of enhancn clusters and akn the ore dstnushable fro ther backrounds [5]. 5. Concludn rearks We have shown that the concept of a local operaton on a dtal ae can be eneralzed to defne local operatons on labelled dots n a planar dot 229

6 Volue 9, Nuber 4 PATTERN RECGNTN LETTERS May t! B (a) (b) 6 )! o l (c) (d) Fure 3. ot densty soothn. (a) nput dot pattern. (b-h) Results of teratons 1, 3, 4, 7, 9, 11, 2 of poston adjustent. pattern. Such an operaton can odfy ether the value of a dot, ts poston, or both. [n a dtal ae, the pxel postons for a reular array, and cannot be odfed, except n specal ways, wthout destroyn the structure of the array. Thus postonodfyn local operatons are not approprate n dtal ae processn.] Evdently, the concepts ntroduced n ths paper can be strahtforwardly eneralzed to dot patterns n hher-densonal spaces. Value-odfyn local operatons can be further eneralzed to labelled raphs, n whch the values are the node la- bels, and the nehbors of a node P are ts nehbors n the raph (.e., the nodes joned to P by arcs). n such a raph, the nehbors could be dstnushed fro one another by arc labels, whch could specfy, for exaple, the relatve postons of the nodes, and the new value of a node can be a functon of ts old value, ts nehbors' values, and the relatve postons. [More precsely, an arc P should have two labels of opposte 'sn', one at each of ts ends, snce the poston of relatve to P s 18 away fro the poston of P relatve to.] Note that t s not easy to defne operatons that odfy the rela- 23

7 Volue 9, Nuber 4 PATTERN RECGNTN LETTERS May 1989 a B! B t(e) (f) R U q U q U U (91 " (h) " Fure 3 (contd.). tve postons; f we want to do poston odfcaton, t s better to specfy the absolute postons of the nodes n ther node labels, rather than ther relatve postons n ther arc labels, whch eans that n effect we are treatn the nodes as labelled dots. Labelled raphs can be used to represent aederved structures other than labelled dot patterns. For exaple, f we seent an ae nto reons, we can represent the reons as the nodes of a raph, labelled wth varous reon propertes (averae ray level, area, coordnates of centrod,...), and we can reard two nodes as joned by an arc f the correspondn reons are adjacent. [Here aan we can use arc labels, e.., representn the lenth of coon border between the two reons.] Soe value-odfyn local operatons on such a raph can be nterpreted as operatons on the ornal ae; for exaple, chann the averae ray level label corresponds to lhtenn or darkenn the reon. n the other hand, operatons that chane (node or arc) label values representn eoetrc propertes (e.., that chane reon area or poston) wll not usually be realzable as ae operatons. Thus local operatons on labelled raphs 231

8 Volue 9, Nuber 4 PATTERN RECGNTN LETTERS May 1989 l J e e :,,,. (a) (b) B o 2"-':'.:..'. " "" le l :.3." -.?.': : (c) (d) Fure 4. ot clustern. (a) nput dot pattern. (~d) Results of three teratons of poston adjustent. ay be of nterest n ther own rht, but they see to have only lted usefulness when we want to n- terpret the raphs as representn ae parts. References [1] Rosenfeld A., and A.C. Kak (1982). tal Pcture Processn. Acadec Press, New York. [2] Ahuja N. (1982). ot pattern processn usn Vorono nehborhoods, 1EEE Trans. Pattern Anal. Machne lntell., 4, [3] Tuceryan M., and N. Ahuja (1987). Extractn perceptual structure n dot patterns: an nterated approach. Techncal Report ULU-ENG , Unversty of llnos at Urbana-Chapan, January [4] Tuceryan M., A.K. Jan, and Y. Lee (1988). Texture seentaton usn Vorono polyons. Proc. EEE Conj. Coputer Vson Pattern Reconton, June 1988, 94~99. [5] Watanabe S., (1985). Pattern Reconton: Huan and Mechancal. Wley, New York, 16(~