A General Algorithm for Computing Distance Transforms in Linear Time

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1 Ths chapter has been pblshed as: A. Mejster, J. B. T. M. Roerdnk and W. H. Hesselnk, A general algorthm for comptng dstance transforms n lnear tme. In: Mathematcal Morphology and ts Applcatons to Image and Sgnal Processng, Klwer Acad. Pbl., 2, pp Chapter 2 A General Algorthm for Comptng Dstance Transforms n Lnear Tme Abstract A general algorthm for comptng dstance transforms of dgtal mages s presented. The algorthm conssts of two phases. Both phases consst of two scans, a forward and a backward scan. The frst phase scans the mage colmn-wse, whle the second phase scans the mage row-wse. Snce the comptaton per row (colmn) s ndependent of the comptaton of other rows (colmns), the algorthm can be easly parallelzed on shared memory compters. The algorthm can be sed for the comptaton of the exact Ecldean, Manhattan (L 1 norm), and chessboard dstance (L norm) transforms. 2.1 Introdcton Dstance transforms play an mportant role n many morphologcal mage processng applcatons. They have been extensvely stded and sed n comptatonal geometry, mage processng, compter graphcs and pattern recognton, e.g., [17, 18, 25, 81]. The two-dmensonal dstance transform can be descrbed as follows. Let B be a set of grd ponts taken from a rectanglar grd of sze m n. The problem s to assgn to every grd pont (x,y) the dstance to the nearest pont n B. If we se the Ecldean metrc for comptng dstances, and represent B by a boolean array b[, ], we ths want to compte the two dmensonal array dt[x,y] = EDT(x,y), where EDT(x,y) = MIN(, j : < m j < n b[, j] : (x ) 2 + (y j) 2 ). Here we se the notaton MIN(k : P(k) : f (k)) for the mnmal vale of f (k) where k ranges over all vales that satsfy P(k). Snce the exact Ecldean dstance transform s often regarded as too comptatonally ntensve, several algorthms have been proposed that se some mask whch s swept over the mage n two scans, to compte approxmatons lke the Manhattan (cty-block) dstance, the chessboard dstance, or chamfer dstances (see [17, 18, 25, 81]). The tme complexty s lnear n the nmber of pxels of the mage (.e. O(m n)), bt t does not yeld the exact Ecldean dstance, whch

2 The frst phase s reqred for some applcatons. Another drawback of these algorthms s that they are hard to parallelze for execton on parallel compters snce prevosly compted reslts are propagated drng the comptaton, makng the process hghly seqental. A recrsve algorthm of order mnlogm for the exact EDT s gven n [45]. In [75] a recrsve algorthm of order mn for the exact EDT s gven by redcng the problem to a matrx search algorthm. In ths chapter, whch s based pon [54], we present an algorthm that also comptes dstance transforms n lnear tme, s smpler and more effcent than [75], and s easy to parallelze. It can compte the Ecldean (EDT), the Manhattan (MDT), and the chessboard dstance (CDT) transform, defned by EDT(x,y) = MIN(, j : < m j < n b[, j] : (x ) 2 + (y j) 2 ), MDT(x,y) = MIN(, j : < m j < n b[, j] : x + y j ), CDT(x,y) = MIN(, j : < m j < n b[, j] : x max y j ). If we defne the mnmm of the empty set to be, and se the rle z + = for all z, we fnd wth some calclaton EDT(x,y) = MIN( : < m : (x ) 2 + G(,y) 2 ), MDT(x,y) = MIN( : < m : x + G(,y)), CDT(x,y) = MIN( : < m : x max G(,y)), where G(,y) = MIN( j : j < n b[, j] : y j ). The algorthm can be smmarzed as follows. In a frst phase each colmn C x (defned by ponts (x,y) wth x fxed) s separately scanned. For each pont (x,y) on C x, the dstance G(x,y) of (x,y) to the nearest ponts of C x B s determned. In a second phase each row R y (defned by ponts (x,y) wth y fxed) s separately scanned, and for each pont (x,y) on R y the mnmm of (x x ) 2 + G(x,y) 2 for EDT, x x + G(x,y) for MDT, and x x max G(x,y) for CDT s determned, where (x,y) ranges over row R y. 2.2 The frst phase The object of the frst phase s to determne the fncton G. We frst observe that we can splt G nto two fnctons GT (top) and GB (bottom), sch that G(,y) = GT(,y) mn GB(,y), where GT(,y) = MIN( j : j y b[, j] : y j) GB(,y) = MIN( j : y j < n b[, j] : j y) We start wth the comptaton of GT by ntrodcng an array g to store ts vales. It s easy to see that GT(,y) = f b[,y] holds, and that, otherwse, GT(,y) = GT (,y 1) + 1 (or f y = ). We can therefore compte g[x,y] := GT(x,y) sng only g[x,y 1] n a smple colmn scan from top to bottom. Smlarly, we fnd GB(,y) = GB(,y + 1) + 1. The second scan rns from bottom to top, and comptes G(x,y) drectly, sng GT from the prevos scan, and GB from the crrent

3 A General Algorthm for Comptng Dstance Transforms n Lnear Tme 25 forall x [..m 1] do ( scan 1 ) f b[x,] then g[x,] := else g[x,] := ; endf for y := 1 to n 1 do f b[x,y] then g[x,y] := else g[x,y] := 1 + g[x,y 1]; endf ( scan 2 ) for y := n 2 downto do f g[x,y + 1] < g[x,y] then g[x,y] := (1 + g[x,y + 1]) endf end forall Fgre 2.1. Program fragment for the frst phase. one. After some smplfcaton, ths reslts n the code fragment gven n Fg Clearly, the tme complexty s lnear n the nmber of pxels (.e. O(m n)). In actal mplementatons t s convenent to replace by m + n, snce all dstances n the mages are less than m + n f the set B s non-empty. 2.3 The second phase In the second phase we want to compte EDT, MDT, or CDT row by row,.e. for all x wth fxed y. Therefore, n ths secton we regard y as a constant and omt t as a parameter n axlary fnctons, and ntrodce g() = G(, y). Instead of developng an algorthm for each metrc separately, we am at a more general algorthm for DT(x,y) = MIN( : < m : f (x,)). (2.1) The choce of the fncton f depends on the metrc we wsh to se,.e. (x ) 2 + g() 2 for EDT, f (x,) = x + g() for MDT, x max g() for CDT. It s helpfl to ntrodce a geometrcal nterpretaton of the mnmzaton problem of Eq For any wth < m, denote by F the fncton x f (x,) on the real nterval [,m 1].

4 The second phase (a) EDT (b) MDT (c) CDT Fgre 2.2. DT as the lower envelope (sold lne) of crves F, < m (dotted lnes). The dashed vertcal lnes ndcate the transtons between regons. We call the ndex of F. In the case of EDT, the graph of F s a parabola wth vertex at (, g()). In the case of MDT the parabolas are replaced by V-shaped approxmatons, whle n the case of CDT we deal wth topped off V-shaped approxmatons (see Fg. 2.2). We can nterpret DT geometrcally as the lower envelope of the collecton {F < m} evalated at nteger coordnates, cf. Fg The lower envelopes consst of a nmber of consectve crve segments, whose ndex we denote by s[],s[1],...,s[q] contng from left to rght. The projectons of the segments on the x-axs are called regons, and form a partton of the nterval [,m) by consectve segments. The comptaton of DT now conssts of two scans. In a forward (left-to-rght) scan the set of regons s determned sng an ncremental algorthm. In a backward (rght-to-left) scan the vales DT(x,y) are trvally compted for all x. We start by replacng the pper bond m n (2.1) by a varable and defne FL(x,) = MIN( : < : f (x,)). The geometrc nterpretaton s that we restrct the set B to the half plane to the left of. Clearly, DT(x,y) = FL(x,m). For gven pper bond >, we defne an ndex h to be a mnmzer at x f, n the expresson for FL(x,), the mnmal vale of f (x,) occrs at h. In general, x may have more than one mnmzer. The least mnmzer H(x,) of x w.r.t. s defned as the least ndex h wth h < sch that f (x,h) f (x,) for all n the same range,.e. H(x,) = MIN(h : h < ( : < : f (x,h) f (x,)) : h). (2.2) We clearly have FL(x,) = f (x,h(x,)), hence DT(x,y) = f (x,h(x,m)). Therefore, the problem redces to the comptaton of H(x,m). We consder the sets S() of the least mnmzers that occr drng the scan from left to rght, and the sets T (h,) of ponts wth the same least mnmzer h. We ths defne S() = {H(x,) x < m}, (2.3) T (h,) = {x x < m H(x,) = h } f h <.

5 A General Algorthm for Comptng Dstance Transforms n Lnear Tme (a) above (b) below (c) ntersecton Fgre 2.3. Locaton of F (dashed crve) w.r.t. the lower envelope (sold lne). Clearly, S() s a nonempty sbset of [,), and S() = {h T (h,) /}. We defne the regons for to be the sets T (h,) that are nonempty. It s easy to see that the regons for form a partton of [,m). The am s the case where = m. Indeed, for x T (h,m), we have H(x,m) = h and hence DT(x,y) = f (x,h). The second phase of the algorthm therefore conssts of two scans: scan 3 comptes the partton of [,m) that conssts of the regons for m and scan 4 ses these regons to compte DT. For gven, only the crves wth ndces from to 1 are taken nto accont. The mnmzer of x corresponds to the ndex of the crve segment whose projecton on the horzontal axs contans x. Let the crrent lower envelope consst of q + 1 segments,.e. S() = {s[],s[1],...,s[q]}, wth s[l] the ndex of the l-th segment. Consder what happens when F s added. Three statons may occr: (a) F s above the crrent lower envelope on [,m 1], cf. Fg. 2.3(a). Then S( + 1) = S(), snce the set T (, + 1) s empty. (b) F s below the crrent lower envelope on [,m 1], cf. Fg. 2.3(b). Then S( + 1) = {},.e., all old regons have dsappeared, and there s one new regon T (, + 1) = [,m). (c) F ntersects the crrent lower envelope on [,m 1], cf. Fg. 2.3(c). The crrent regons wll ether shrnk or dsappear, and there s one new regon T (, + 1). We start searchng from rght to left for the crrent regon whch s ntersected by F. Ths can be determned by comparng the vales of F and F l at the begn pont t[l] of each crrent regon l = q,q 1,..., ntl we fnd the frst l = l sch that F (t[l ]) F s[l ](t[l ]). Then F s not the least mnmzer at t[l ], and there mst be an ntersecton of F wth F l n regon l. Let x be the horzontal coordnate of the ntersecton. If l = q and x m we have case (a); f l < we have case (b); otherwse case (c) pertans. To fnd x, we ntrodce a fncton Sep, where Sep(,) s the frst nteger larger or eqal than the horzontal coordnate of the ntersecton pont of F and F wth <,.e. F (x) F (x) x Sep(,). (2.4)

6 The second phase forall y [..n 1] do q := ; s[] := ; t[] := ; for := 1 to m 1 do ( scan 3 ) whle q f (t[q],s[q]) > f (t[q],) do q := q 1; f q < then q := ; s[] := else w := 1 + Sep(s[q],); f w < m then q := q + 1; s[q] := ; t[q] := w end f end f end for for := m 1 downto do ( scan 4 ) dt[,y] := f (,s[q]); f = t[q] then q := q 1 end for end forall Fgre 2.4. Program fragments for the second phase. We ths have x = Sep(s[l ],). Clearly, the fncton Sep s dependent on whch dstance transform we want to compte. In the next secton we wll derve the expressons for the fncton Sep, bt n the remander of ths secton we smply assme that Sep s avalable. We ntrodce an nteger program varable. It s convenent to represent S() by an ncreasng seqence of elements. Snce the regons form a partton of [, m) by consectve segments, we can represent them by the seqence of ther least elements. Accordng to the case analyss above, the regons are to be adapted at ther end. We can therefore mplement these seqences n two nteger arrays, s and t, wth an nteger varable q as ndex of the end pont. We start wth the forward scan, see scan 3 n Fg We have S(1) = {}, and T (,1) = [,m), and ths start wth q =, s[] =, and t[] =. In a loop, varable s ncremented, and ths the representatons of S and T mst be pdated by means of the case analyss above. To nvestgate the complexty of the forward scan, we consder the expresson q + 2(m ), whch s ntally 2m. In every execton of the body of the oter loop (scan 3 n Fg. 2.4), and also n every execton of the body of ts nner loop, the vale of the expresson decreases. Ths mples that the tme complexty of the scan s lnear n m. Note that, the average nmber of teratons of the nner loop s at most two. The algorthm ses less than 2m comparsons of f vales, and fncton Sep s evalated less than m tmes. When the forward scan s fnshed, we have completely determned the partton of [,m) n regons. Gven these regons, we can trvally compte dt-vales n a smple backward scan (see scan 4 n Fg. 2.4).

7 A General Algorthm for Comptng Dstance Transforms n Lnear Tme 29 g() g() g() g() (a) g() g() + (b) g() > g() + (c) otherwse g() g() x * Fgre 2.5. Cases for fndng Sep for MDT. 2.4 Dervaton of the fncton Sep The dervaton n the prevos secton was ndependent of the actal metrc sed. The fnctons dependent on the metrc are f and Sep. In ths secton we compte expressons for Sep for EDT, MDT, and CDT. The easest s EDT. We fnd for < F (x) F (x) {defnton of F, F } (x ) 2 + g() 2 (x ) 2 + g() 2 {calcls; < ; x s an nteger} x ( g() 2 g() 2 ) dv (2( )). Here, we denote nteger dvson wth rondng off towards zero by dv. Ths, we fnd for EDT that Sep(,) = ( g() 2 g() 2 ) dv (2( )). If we se the Manhattan metrc, the analyss s slghtly more complcated. Snce we have to deal wth absolte vales n the expressons, awkward case analyss s necessary f we want to compte Sep analytcally. Therefore we prefer a geometrc argment. We have to consder three cases (see Fg. 2.5). If g() g() +, the graph of F les entrely above the graph of F for all x, ths we choose Sep(,) =. If g() > g()+, the graph of F les entrely above the graph of F, so F (x) F (x) for no x at all. Ths, we mst choose Sep(,) = to satsfy (2.4). In all other cases, F ntersects F at x = (g() g() + + )/2. So, f we want to compte MDT we se f g() g() +, Sep(,) = f g() > g() +, (g() g() + + ) dv 2 otherwse. For the case of CDT we have x max g() x max g(). We consder two man cases, whch each can be splt p n three sb-cases. Frst we consder the case g() g(). From Fg. 2.6(a)-(c), we see that the ncreasng segment of F (y = x ) ntersects the decreasng part of F (y = x), or the constant part (y = g()). Let γ be the vertcal coordnate correspondng

8 3 2.4 Dervaton of the fncton Sep ( )/2 g() g() g() ( )/2 g() (+)/2 (+)/2 (a) g() γ g() γ (b) g() γ g() > γ g() g() ( )/2 ( )/2 g() g() (+)/2 (+)/2 g() ( )/2 (c) g() > γ g() g() ( )/2 (d) g() γ g() (+)/2 (+)/2 (e) g() > γ g() γ (f) g() > γ g() > γ Fgre 2.6. Cases for fndng Sep for CDT, where γ = ( )/2. Cases (a)-(c): g() g(). Cases (d)-(f): g() > g(). wth the mddle of and (x = ( + )/2),.e. γ = ( )/2. From Fg. 2.6(a), we see that f g() γ g() γ, we have F (x) F (x) f x ( + )/2. From Fg. 2.6(b)-(c), we see that the ncreasng part of F ntersects the constant segment of F at + g(), and ths we have F (x) F (x) f x + g(). Pttng the three cases together, we can conclde g() g() ( F (x) F (x) x + 2 max ( + g()) ). The other man case s g() > g(). Agan, n Fg. 2.6(d), we see that f g() γ, the ntersecton at (+)/2 s the separator. If g() > γ (see Fg. 2.6(e)-(f)), the horzontal segment of F ntersects the decreasng part of F at x = g(). Jst lke n the prevos case, we can pt these cases together. Ths reslts n the followng expresson for Sep: Sep(,) = { ( + g()) max (( + ) dv 2) f g() g(), ( g()) mn (( + ) dv 2) otherwse.

9 A General Algorthm for Comptng Dstance Transforms n Lnear Tme 31 Table 2.1. Tmng reslts n ms. From left to rght: EDT, MDT, and CDT. sze p=1 p=2 p=3 p=4 p=1 p=2 p=3 p=4 p=1 p=2 p=3 p= Parallelzaton, tmng reslts and conclsons Snce the comptaton per row (colmn) s ndependent of the comptaton on any other row (colmn), the algorthm s well sted for parallelzaton on a shared memory machne. In the frst (second) phase, the colmns (rows) are dstrbted over the processors. The two phases mst be separated by a barrer, whch assres that all processors have completed the frst phase before any of them starts wth the second phase. The theoretcal tme complexty of the parallel algorthm for p processors (where p m mn n) s O(mn/p). We ran experments on an Intel Pentm III based shared memory parallel compter wth 4 cp s, rnnng at a 55MHz clock freqency. We performed tme measrements sng several bnary mages, and fond that the execton tme s almost ndependent of mage content, and scales well w.r.t. the nmber of processors. Ths s as expected, snce the amont of work per row and colmn s almost the same. In table 2.1 the tmngs for sqare mages are gven for p = 1 to p = 4 processors. Note that the comptaton of MDT and CDT s only slghtly faster than the exact EDT. We also mplemented the seqental algorthm of [81] for CDT, and fond that or algorthm s less than a factor of 2 slower, whch can easly be overcome by parallel processng. The algorthm can be easly extended to d-dmensonal dstance transforms by separatng the problem nto d phases, each solvng a one-dmensonal problem, as carred ot above for the case d = 2.

10 4 1.1 Introdcton to Mathematcal Morphology X Y Dlaton of X by Y Eroson of X by Y Fgre 1.1. The geometrcal nterpretaton of dlaton and eroson (n the contnos case). In order to avod havng to wrte many parentheses that make expressons hard to read, we ntrodce the notaton Y h = (Y ) h = {h y y Y }. Dlaton and eroson have a smple geometrcal nterpretaton, whch helps to nderstand ther behavor (see fgre 1.1): δ Y (X) = {h X Y h /} ε Y (X) = {h Y h X} In words, the dlaton of X by Y s the set of ponts h sch that the translaton of the reflected set Y over the vector h hts the set X. Smlarly, the eroson of X by Y s the set of ponts h sch that after translatng Y over the vector h t fts n X. Several algebrac propertes of dlaton and eroson exst. The most mportant propertes are: Dlaton and eroson are ncreasng operators, whch means that f X Y we have δ B (X) δ B (Y ), and ε B (X) ε B (Y ) for any strctrng element B. Eroson s decreasng n ts second argment (the strctrng element). If B 1 B 2, then ε B2 (X) ε B1 (X), for every set X. Eroson and dlaton form an adjncton,.e. δ B (A) C A ε B (C), for any sets A,B and C. Dlaton s commtatve: δ B (A) = δ A (B), or eqvalently A B = B A, for any sets A and B.

11 Parallelzaton, tmng reslts and conclsons

Modeling Local Uncertainty accounting for Uncertainty in the Data

Modeling Local Uncertainty accounting for Uncertainty in the Data Modelng Local Uncertanty accontng for Uncertanty n the Data Olena Babak and Clayton V Detsch Consder the problem of estmaton at an nsampled locaton sng srrondng samples The standard approach to ths problem