Theoretical Computer Science. Reconstruction of convex lattice sets from tomographic projections in quartic time

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1 Theoretcal Computer Scence 406 (2008) Contents lsts avalable at ScenceDrect Theoretcal Computer Scence journal homepage: Reconstructon of convex lattce sets from tomographc projectons n quartc tme Sara Brunett a, Alan Daurat b, a Dpartmento d Scenze Matematche e Informatche, Unverstà d Sena, Pan de Mantelln 44, 53100, Sena, Italy b LSIIT UMR 7005 CNRS-ULP, Pôle API, Boulevard Sébasten Brant, Illkrch-Graffenstaden, France a r t c l e n f o a b s t r a c t A large part of ths paper s extracted from the conference artcle [7] whch was a jont work wth Attla Kuba. To our deep sorrow, Attla dd not see the end of the story. Ths paper s dedcated to hs memory Keywords: Dscrete tomography Convexty Fllng operatons Fllng operatons are procedures whch are used n Dscrete Tomography for the reconstructon of lattce sets havng some convexty constrants. Many algorthms have been publshed gvng fast mplementatons of these operatons, and the best runnng tme [S. Brunett, A. Daurat, A. Kuba, Fast fllng operatons used n the reconstructon of convex lattce sets, n: Proc. of DGCI 2006, n: Lecture Notes n Comp. Sc., vol. 4245, 2006, pp ] s O(N 2 log N) tme, where N s the sze of projectons. In ths paper we mprove ths result by provdng an mplementaton of the fllng operatons n O(N 2 ). As a consequence, we reduce the tme-complexty of the reconstructon algorthms for many classes of lattce sets havng some convexty propertes. In partcular, the reconstructon of convex lattce sets satsfyng the condtons of Gardner Grtzmann [R.J. Gardner, P. Grtzmann, Dscrete tomography: Determnaton of fnte sets by X-rays, Trans. Amer. Math. Soc. 349 (1997) ] can be performed n O(N 4 )-tme Elsever B.V. All rghts reserved. 1. Introducton One of the most ntensvely studed problems of dscrete tomography s the reconstructon of lattce sets. Several algorthms have been publshed for reconstructng such sets. One of the most studed questons s whch sub-class of lattce sets can be reconstructed n polynomal tme. In most cases some knd of (dscrete) convexty s assumed for the sets. For example, Attla Kuba publshed an algorthm [14] to reconstruct so-called hv-convex lattce sets from two projectons. As t turned out later the reconstructon problem n ths class s NP-complete [15]. Barcucc et al. showed [2] that a sub-class of hv-convex lattce sets, namely, the class of hv-convex polyomnoes can be reconstructed n polynomal tme. Ths result was extended also to a bgger class, that of hv-convex 8-connected lattce sets [8]. Fnally, the reconstructon has been proved to be polynomal for the class of convex lattce sets (ntersecton of a convex polygon wth Z 2 ) f the projectons are taken w.r.t. to certan sets of four drectons [5]. Most of the algorthms reconstructng sets presentng some convexty propertes use specal procedures called fllng operatons. These operatons can be appled n teratve procedures to approach the fnal solutons wth two sequences of sets. The frst sequence s a sequence of decreasng upper bounds, and the second s a sequence of ncreasng lower bounds of the solutons. The classcal (four) fllng operatons were defned n [2]. If N = max(m, n), the whole teratve process runs n O(N 4 )- tme n [2]. In [13] an effcent algorthm was gven to apply the fllng operatons n O(N 2 log N)-tme. In [8] a ffth fllng operaton was ntroduced to decrease the overall complexty of the reconstructon algorthm. Unfortunately, the algorthm for the fllng operatons of [13] cannot be generalzed wth ths ffth operaton (n [8] ths pont was not treated). In [7], Correspondng author. Tel.: ; fax: E-mal addresses: sara.brunett@uns.t (S. Brunett), daurat@dpt-nfo.u-strasbg.fr (A. Daurat) /$ see front matter 2008 Elsever B.V. All rghts reserved. do: /j.tcs

2 56 S. Brunett, A. Daurat / Theoretcal Computer Scence 406 (2008) the authors provde an mplementaton of all the fve fllng operatons havng the same complexty as the algorthm of [13]. In ths paper we modfy one of the fllng operatons such that they can be performed n O(N 2 ). As a result, we get an mprovement n the tme-complexty of the reconstructon algorthms for many classes of lattce sets presentng some convexty propertes. The structure of ths paper s as follows. Secton 2 contans the necessary defntons and notatons. The fllng operatons, the new reconstructon algorthm, ts analyss are descrbed n Secton 3. Secton 4 shows the applcaton of these operatons for reconstructng hv-convex polyomnoes and convex sets. Fnally, we conclude n Secton Notaton and defntons A lattce set s a fnte subset of Z 2. A lattce drecton s gven by an nteger vector p = (p x, p y ) (0, 0), and t can also be represented by a lnear form p(x, y) = p y x p x y. The horzontal drecton (resp. vertcal drecton) denoted by h (resp. v) s determned by the vector (1, 0) (resp. (0, 1)). The -th horzontal lne s denoted h =, and the j-th vertcal lne s denoted v = j. A lattce set s lne-convex wth respect to a drecton p f ts ntersecton wth each lne n the drecton p s made up of consecutve ponts. A set whch s lne-convex w.r.t. to the horzontal and vertcal drectons s called hv-convex. The (tomographc) projecton of a lattce set E along a drecton p, denoted by X p E, s the functon whch gves the number of ponts on any lne of drecton p, more precsely: X p E(k) = {M E : p(m) = k} for any k Z where p s the lnear form assocated to p. In ths artcle we are nterested n the reconstructon from projectons of set E whch satsfes some convexty constrants. More precsely f M s a class of lattce sets, and D s a fnte set of lattce drectons, the reconstructon problem for the class M and the drectons D s the followng. Reconstructon(M, D) Input: A functon f : D Z Z + whch gves a non-negatve nteger f (p, k) for any lne p = k wth p D, and such that {(p, k) : f (p, k) > 0} s fnte. Task: Reconstructng a lattce set E M such that X p E(k) = f (p, k) for any (p, k) D Z Throughout ths paper, [a, b] denotes the dscrete nterval {k Z : a k b}. 3. Fast fllng operatons 3.1. Prelmnares A fllng operaton s a procedure whch has been used n many reconstructon algorthms [2,4,5,8,14]. Formally, a fllng operaton takes a functon f of Reconstructon(M, D), and a par of sets (α, β) such that α β and returns a new par of sets (α, β ) wth α α β β. We now present classcal fllng operatons whch can be used for any subclass of that of lne-convex sets w.r.t. D. To smplfy the descrpton of these operatons, we frst descrbe them for the set D = {h, v} consstng of the horzontal and vertcal drectons. Let h = f (h, ) and v j = f (v, j), and suppose wthout loss of generalty that there exst m, n Z + such that h = 0 for / [1, m] and v j = 0 for j / [1, n], h 1, h m, v 1, v n > 0. Let E be a soluton of Reconstructon(M, D), then the horzontal feet of E are the ponts of E whch are on the lnes h = 1 and h = m. Smlarly the vertcal feet are the ponts of E whch are on the lnes v = 1 and v = n. For any [1, m] let α h = {(, j) : j Z, (, j) α} and β h = {(, j) : j Z, (, j) β}. Analogously, α v j = {(, j) : Z, (, j) α}, and β v j = {(, j) : Z, (, j) β} for j [1, n]. In the followng we use the conventons mn( ) = +, max( ) =. The extremtes of α h and β h are denoted by l(α h ) = mn({j : (, j) αh }), r(αh ) = max({j : (, j) αh }) l(β h ) = mn({j : (, j) βh }), r(βh ) = max({j : (, j) βh }), respectvely. Wth ths notaton, the four fllng operatons of [2] on horzontal lnes can be defned as: If α h then α h = {(, j) β h : l(α h ) j r(αh )}. α h = {(, j) β h : r(β h ) h < j < l(β h ) + h }. If α h, j = max({j : (, j) / β h and j < l(α h )}, j = mn({j : (, j) / β h and j > r(α h )} then βh = {(, j) β h : j < j < j }. If α h, then β h = {(, j) β h : r(α h ) h < j < l(α h ) + h }.

3 S. Brunett, A. Daurat / Theoretcal Computer Scence 406 (2008) Fg. 1. The fllng operatons. It s easy to see that operatons and ncrease α h and operatons and reduce β h : α h α h, αh α h β h β h, βh β h. Addtonally, the followng propertes hold: the fllng operatons appled on h = make α h connected, and f α h, then β h s connected and β h = 2h α h. A ffth fllng operaton has been ntroduced n [4,5,8]: t removes the components of β h (maxmum sequences of consecutve elements of β h ) whch are smaller than the correspondng projecton. To defne t formally we need a notaton for the extremtes of each component. So the sequence (c k ) 1 k 2r = c(β h ) s defned by: r c k < c k+1 and {j : (, j) β h } = [c 2k 1, c 2k 1]. (1) k=1 Then the operaton s defned by: β h = [c 2k 1, c 2k 1]. 1 k r c 2k c 2k 1 h The fllng operatons are llustrated n Fg. 1. We can analogously defne the fve fllng operatons on the vertcal lnes. The reconstructon algorthms descrbed n [2,5,8] teratvely apply the fllng operatons n a fxed order (,,, ) on all the lnes of the grd [1, m] [1, n] (teratve step). The k-th teraton gves rse to a new par (α k, β k ) from (α k 1, β k 1 ), and the teratve process ends when an nvarant par s obtaned, that s, (α k, β k ) = (α k 1, β k 1). There are several methods to construct the ntal par of sets (α 0, β 0 ). For example n [2], β 0 concdes wth the grd and α 0 conssts of a set of the vertcal and horzontal feet. More generally, we assume that α 0 β 0 [1, m] [1, n]. The frst step of the reconstructon algorthms usng fllng operatons conssts n fxng arbtrarly the feet. If the four feet are chosen, one can fnd an ntal α such that α h and α v j for all and j [2]. The operaton permts to reduce the number of fxed feet from four to two (opposte) so reducng the overall complexty of the reconstructon algorthm. The man pont s that t s suffcent for the step followng the fllng operatons (the 2-SAT reducton) to obtan a par of sets (α, β) whch s nvarant w.r.t.,,, operatons and whch satsfes β h = 2h α h, for all the lnes h = and βv = 2v j j α v j, for all the lnes v = j. Wth fxed horzontal feet, t s easy to obtan an ntal (α, β) whch satsfes α h or β h = 2h, for all. It mples that, after the applcaton of,,, operatons, α and β satsfy β h = 2h α h, for all (see [6, Proposton 2] or [5, Proposton 3.7]). The operaton has been ntroduced because after ts applcaton on v = j, we have: α v j or β v 2v j j. Hence after the applcaton of the other four fllng operatons, we get: β v 2v j j α v j. Snce β v = β = j β h, α v = α = j α h, vj = β = h, the propertes β h = 2h α h and βv 2v j j α v j, for all and j, mply that βv = 2v j j α v j,

4 58 S. Brunett, A. Daurat / Theoretcal Computer Scence 406 (2008) for all j. (See also [5, p ] or [6, p ] for more detals.) Ths dscusson shows that the operaton can be appled only to the lnes whch satsfy β v j < 2v j α v j. Therefore, n the paper the operaton s restrcted to the lnes p = whch satsfy β p < 2f (p, ) αp. Ths restrcton of the operaton s the man new trck of ths paper comparng to the algorthm descrbed n [7]: t permts to decrease the complexty of the algorthm that mplements the fllng operatons, and moreover t reflects n a smplfcaton and reducton of the necessary data structures. Let us recall that the best tme-complexty wth the fve fllng operatons s O(N 2 log N) [7]. Now we descrbe a procedure whch performs these operatons, wth the restrcton on the operaton descrbed above, n O(N 2 )-tme The new algorthm for the fllng operatons In ths secton we descrbe an algorthm whch performs the fllng operatons untl nvarance occurs n O(N 2 ) tme. For sake of smplcty, n a frst step we descrbe these operatons for the set of horzontal and vertcal drectons. Hence we consder two projecton-vectors (h ) [1,m] and (v j ) j [1,n] and an ntal par of sets (α 0, β 0 ) (obtaned n some way). Notce that the algorthm works for any choce of (α 0, β 0 ) The used data-structures The algorthm uses a smple two-dmensonal array of booleans to encode α and β. These structures allows us to check the membershp of a pont n O(1) tme. For each horzontal lne h = the followng structures are used: 8 varables l 1 (α h ), l 2(α h ), r 1(α h ), r 2(α h ), l 1(β h ), l 2(β h ), r 1(β h ), r 2(β h ) whch are updated n such a way that the followng propertes are always satsfed:. All the ponts (, j) such that j [l 1 (α h), r 1(α h )] are n α. After the applcaton of the frst four fllng operatons on h = (procedure treat_lne1), α h s connected and l 1 (α h ), r 1(α h ) hold the smallest and the largest ndex of the ponts n α h, respectvely. They are ntalzed as: l 1(α h) = + and r 1(α h) =.. l 2 (α h) = mn({j : (, j) αh }), r 2(α h) = max({j : (, j) αh }). They ndcate the smallest and the bggest ponts n α h. In general, αh s not connected. They are updated every tme that any fllng operaton recognzes a pont of β h as a pont of α h (procedure (, j) and remove_from_beta h (, j)).. l 1 (β h) = mn({j : (, j) βh }), r 1(β h) = max({j : (, j) βh }). They ndcate the smallest and the bggest ponts n β h. They are updated every tme that any fllng operaton removes a pont (, j) wth j = l 1(β h) or j = r 1(β h) (procedure remove_from_beta h (, j)).. f α h =, then l 2 (β h) = and r 2(β h) = + ; otherwse l 2(β h) = max({j : (, j) / βh and j < l 2 (α h )}) + 1, r 2 (β h) = mn({j : (, j) / βh and j > r 2 (α h)}) 1. Snce changes n βh due to the applcaton of the fllng operatons on any lne can remove ponts from β h dsconnectng t, l 2 (β h) and r 2(β h ) ndcate the connected subset of β h contanng α h. Consequently, they are updated every tme that any fllng operaton removes any pont (, j) dsconnectng β h (procedure remove_from_beta h (, j)). the varables cardα h and cardβ h, equal to the cardnaltes of the sets αh and β h, respectvely. the nteger array next_n_beta h, defned by: next_n_betah[j] = mn({k > j : (, k) βh }). the nteger array prev_n_beta h defned by: prev_n_beta h[j] = max({k < j : (, k) βh }). We use smlar structures for the vertcal lnes. Moreover we need the structure lnes_to_treat defned n [7] whch memorzes the lnes whch have to be treated by the fllng operatons. We need three operatons on ths structure: sempty(lnes_to_treat) whch ndcates f there are some lnes to be treated. extract(lnes_to_treat) whch returns one of the lnes to be treated, and removes t from the structure lnes_to_treat. add_lne(lnes_to_treat, l) whch adds a lne to be treated agan. At the begnnng all the lnes are added to the structure. Then, a lne s nserted f any change to the state (membershp) of one of ts ponts occurs durng the applcatons of the fllng operatons to another lne. These operatons can be executed n constant tme f the set s mplemented as an array of booleans coupled wth an array (mplementng a stack) of the elements both ndexed n [1, m + n] and an nteger varable for the cardnalty (see [7, pages ] for more detals) Updatng the structures The two followng procedures ndcate how the data structures are updated when a pont changes ts state,.e. t s added to α, or removed from β. We dstngush the case where the modfcaton to the state of a pont orgnates from a fllng operaton appled to a horzontal lne or a vertcal lne because n the frst case any vertcal lne, whereas n the second case any horzontal lne s added to the set lnes_to_treat, respectvely.

5 S. Brunett, A. Daurat / Theoretcal Computer Scence 406 (2008) (, j) f (, j) / β then EXIT(no soluton) f (, j) α then return α α {(, j)} for (p,, j ) {(h,, j), (v, j, )} do f cardα p = 0 then l 2 (β p ) max({j : (, j ) / β p and j < j }) + 1 r 2 (β p ) mn({j : (, j ) / β p and j > j }) 1 l 2 (α p ) mn(l 2 (α p ), j ) r 2 (α p ) max(r 2 (α p ), j ) cardα p cardα p + 1 add_lne(lnes_to_treat, v = j) remove_from_beta h (, j) f (, j) α then EXIT(no soluton) f (, j) / β then return β β \ {(, j)} for (p,, j, x) {(h,, j, h ), (v, j,, v j )} do f j = l 1 (β p ) then l 1 (β p ) next_n_beta p [j ] f j = r 1 (β p ) then r 1 (β p ) prev_n_beta p [j ] f cardα p 0 then f j < l 2 (α p ) then l 2 (β p ) max(l 2 (β p ), j + 1) f j > r 2 (α p ) then r 2 (β p ) mn(l 2 (β p ), j 1) cardβ p cardβ p 1 next_n_beta p [prev_n_beta p [j ]] next_n_beta p [j ] prev_n_beta p [next_n_beta p [j ]] prev_n_beta p [j ] add_lne(lnes_to_treat, v = j) The procedures put_n_alpha v (, j), remove_from_beta v (, j) are smlar. // Frst pont of α; the two followng nstructons // take O(N) tme but are executed O(N) tmes Fllng operatons on a lne The followng operaton apples the,,, operatons on the horzontal lne h = : treat_lne1(h = ) f cardα h 0 then f l 1 (α h ) = + then for all j [l 2 (α h) + 1, r 2(α h ) 1] do (, j) else for all j [l 2 (α h) + 1, l 1(α h) 1] [r 1(α h) + 1, r 2(α h ) 1] do (, j) // Operaton

6 60 S. Brunett, A. Daurat / Theoretcal Computer Scence 406 (2008) l 1 (α h) l 2(α h ); r 1(α h) r 2(α h) for all j [l 1 (β h), l 2(β h) 1] [r 2(β h) + 1, r 1(β h )] do // Operaton remove_from_beta h (, j) f r 1 (β h) h + 1 l 1 (β h) + h 1 then // Operaton f cardα p = 0 then for all j [r 1 (β h) h + 1, l 1 (β h) + h 1] do (, j) else for all j [r 1 (β h) h + 1, l 1 (α h) 1] [r 1(α h) + 1, l 1(β h) + h 1] do (, j) l 1 (α h) l 2(α h ); r 1(α h) r 2(α h) f cardα p 0 then // Operaton for all j [l 1 (β h), r 1(α h) h ] [l 1 (α h) + h, r 1 (β h )] do remove_from_beta h (, j) The followng procedure apples the,,,, operatons on the horzontal lne h = : treat_lne(h = ) treat_lne1(h = ) f cardβ h < 2h cardα h then // when true mples cardα h = 0 whle r 1 (β h) l 1(β h) + 1 > cardβh do // β h has more than one component // The number of teratons of ths loop s at most equal to the number // removed components j 1 l 1 (β h ); j 2 r 1 (β h) whle (, j 1 ) β and (, j 2 ) β do j 1 j 1 + 1; j 2 j 2 1 end whle f (, j 1 ) / β then for all j [l 1 (β h ), j 1 1] do remove_from_beta h (, j) else for all j [j 2 + 1, r 1 (β h )] do remove_from_beta h (, j) end whle treat_lne1(h = ) The procedures treat_lne1(v = j), treat_lne(v = j) are smlar. // Remove the component wth smallest // length // If the length of the remanng component s less than h // treat_lne1(h = ) gves an EXIT(no soluton) error The fllng_operatons-procedure Gven two ntal sets α 0, β 0 satsfyng α 0 β 0 [1, m] [1, n], the followng algorthm apples the fllng operatons untl nvarance. fllng_operatons(α 0, β 0 ) α α 0 ; β β 0 β β \ {(, j) : h = 0 or v j = 0} for all l {h = : 1 m and h > 0} {v = j : 1 j n and v j > 0} do add_lne(lnes_to_treat, l) for all the lnes p = of lnes_to_treat do l 1 (α p ) +, r 1(α p ) ntalze l 2 (α p ), r 2(α p ), l 1(β p ), l 2(β p ), r 1(β p ), r 2(β p ), next_n_betap, prev_n_betap, cardαp, cardβp accordng to the propertes of Secton

7 S. Brunett, A. Daurat / Theoretcal Computer Scence 406 (2008) whle not(sempty(lnes_to_treat)) do l extract(lnes_to_treat) treat_lne(l) end whle return(α, β) 3.3. Correctness and complexty At the end of the executons of the procedure put_n_alpha and remove_from_beta, the varables l 1 (β h ), r 1(β h ), l 2 (α h ), r 2(α h ), l 2(β h ), r 2(β h ), next_n_beta, prev_n_beta are updated accordng to the current α and β. If an nstructon EXIT(no soluton) s executed, the fllng operatons lead to a stuaton where α β. Procedure treat_lne1(h = ) mplements the,,, operatons n ths order and at the end of an executon, lne h = s nvarant w.r.t. these four fllng operatons. Procedure treat_lne(h = ) mplements the,,,, and addtonally operatons. Last operaton s performed only f β h < 2h α h. Snce h = s nvarant wth respect to the frst four fllng operatons we have αh = 0 and β h s made up of at least two connected component, at most one of them havng less than h ponts. Ths proves the correctness of the procedure treat_lne because the operaton removes ponts of ether the component wth smallest or largest ndex teratvely untl there remans only one component wth length greater than the projecton. So at the end of an executon of treat_lne(h = ), lne h = s nvarant. Consequently, all the lnes whch are not n lnes_to_treat are nvarant w.r.t. to the fllng operatons, and fnally, when lnes_to_treat s empty, (α, β) s nvarant w.r.t. to the fllng operatons. The algorthm halts after a fnte number of steps because ( β \ α, lnes_to_treat ) decreases lexcographcally at each teraton of fllng_operatons. Let N = max({m, n}). Structure sempty(lnes_to_treat) s ntalzed wth less than 2N lnes, and, snce add_lne(lnes_to_treat, l) takes O(1), the ntalzaton s done n lnear tme. Procedures sempty(lnes_to_treat) and extract(lnes_to_treat) take constant tme, and so the tme complexty of fllng_operatons depends on the number of tmes treat_lne s executed and on ts cost. lnes_to_treat s flled frst wth less than 2N lnes, and then a lne s added to t only by put_n_alpha and remove_from_beta. Moreover the tme complexty of treat_lne s proportonal to the number of calls to put_n_alpha or remove_from_beta. Therefore, the tme complexty of fllng_operatons s proportonal to the global cost of the executon of both put_n_alpha and remove_from_beta. The procedures put_n_alpha and remove_from_beta are executed at most once for each pont, snce the state of a pont can change no more than once. Hence these two procedures are executed less than N 2 tmes, globally. Besdes remove_from_beta runs n O(1) tme. If put_n_alpha s called for a pont (, j) such that α h, t runs n O(1) tmes, otherwse n O(N), but globally the latter case can happen no more than N tmes. (The same holds for the procedures put_n_alpha v and remove_from_beta v ). We conclude that the tmecomplexty of fllng_operatons s O(N 2 ). We deduce: Theorem 1. For any ntal par of sets (α 0, β 0 ) such that α 0 β 0 [1, m] [1, n], fllng_operatons runs n O(N 2 ) tme, where N = max(m, n) Extenson to any fnte set of lattce drectons Now we consder the general case: D s a fnte set of lattce drectons, and M s a class of lattce sets contanng the lne-convex sets w.r.t. D. We suppose that f s a functon as n Reconstructon(M, D). The sze of f s gven by N = max p D (max({k : f (p, k) > 0}) mn({k : f (p, k) > 0}) + 1). The fllng operatons descrbed above can be easly generalzed n ths stuaton: The consdered data s exactly the same, except that we consder all the lnes parallel to one drecton of D, so each lne p = wth p D s assocated to the followng data: l 1 (α p ), l 2(α p ), r 1(α p ), r 2(α p ), l 1(β p ), l 2(β p ), cardαp, cardβp, next_n_beta, prev_n_beta. The procedures put_n_alpha and remove_from_beta update the data assocated to all the lnes p = through the pont nto consderaton except l 1 (α p ) and r 1(α p ). Moreover, all the lnes p = through the pont nto consderaton are added to the set lnes_to_treat except the lne whch has caused the procedure to be called. The procedures treat_lne1 and treat_lne are unchanged. The ntal β 0 s always ncluded n G = {M Z 2 : p D mn({k : f (p, k) > 0}) p(m) max({k : f (p, k) > 0})} whch contans less than N 2 ponts. The tme-complexty of the whole algorthm s stll O(N 2 ) as the procedures put_n_alpha and remove_from_beta are done at most two tmes for each pont and each drecton. We deduce: Theorem 2. For any ntal par of sets (α 0, β 0 ) such that α 0 β 0 G, the applcaton of,,, operatons, and operaton restrcted to the lnes p = whch satsfy β p < 2f (p, ) αp, untl (α, β) s nvarant or α β, can be performed n O(N 2 ) tme, where N = max p D (max({k : f (p, k) > 0}) mn({k : f (p, k) > 0}) + 1).

8 62 S. Brunett, A. Daurat / Theoretcal Computer Scence 406 (2008) Fast reconstructon of convex sets By usng the reconstructon method descrbed n [8] and the fllng operatons procedure fllng_operatons of Theorem 1, we can prove: Theorem 3. If D = {h, v} and P s the class of hv-convex polyomnoes, then Reconstructon(P, D) can be solved n O(N 4 ) tme, where N s the maxmum of the sze of the horzontal and vertcal projecton. Ths result was already known for an algorthm whch dd not use any fllng operaton [9]. The fast fllng operatons have also some consequences on the reconstructon of convex lattce sets. We recall that all the convex lattce sets are unquely determned by ther projectons w.r.t. to a set D of drectons f and only f D contans four drectons whch cross-rato s not n {4/3, 3/2, 2, 3, 4} [11,12]. It s also known, that n ths case, the reconstructon can be acheved n polynomal tme [4,5]. In [6], the authors desgn an algorthm whch permts to reconstruct convex lattce sets w.r.t. the drectons satsfyng the above property n O(N 2 (N 2 + C(N))) tme, where N = max p D (max({k : f (p, k) > 0}) mn({k : f (p, k) > 0}) + 1) and C(N) s the tme complexty for applyng the fllng operatons untl nvarance occurs. Hence, by Theorem 2 we deduce: Theorem 4. If D s a set of drectons whch contans four drectons whose cross-rato s not n {4/3, 3/2, 2, 3, 4} and C s the class of convex lattce sets, then Reconstructon(C, D) can be solved n O(N 4 ) tme, where N = max p D (max({k : f (p, k) > 0}) mn({k : f (p, k) > 0}) + 1). Ths s an mprovement compared to the complexty O(N 5 ) of [5] and the complexty O(N 4 log(n)) of [7]. 5. Concluson Ths paper s the last of a long seres [1 8,13] descrbng algorthms for reconstructng sets whch have convexty propertes from projectons. These algorthms have the same structure: Choosng arbtrarly some ponts of the set Applyng the fllng operatons Reducng the problem to 2-SAT The contrbuton of ths paper s to provde an algorthm for fllng operatons that runs n O(N 2 ) tme. Ths result reflects n an mprovement n the speed of the reconstructon algorthms that follow the scheme above. Indeed the tme complexty of these algorthms can speed up to be quadratc n the sze of the mage. (Recall that the oldest paper of the seres [2] runs n O(N 8 ) tme.) Ths can be acheved because t s suffcent to choose only two feet and the fllng operatons can be done n lnear tme n the sze of the mage. Hence, n order to decrease further the complexty of the reconstructon wth ths approach, one should prove that the choce of the feet s unnecessary. Experments seem to suggest the conjecture that n the reconstructon from more than two drectons, many cases do not requre ths choce (see for example [10, annex B.1]) but there s no theoretcal proof that confrms ths. Acknowledgements The authors wsh to thank the anonymous referees for ther careful readng and suggestons whch mproved the qualty of the present paper. References [1] E. Barcucc, S. Brunett, A. Del Lungo, M. Nvat, Reconstructon of lattce sets from ther horzontal, vertcal and dagonal X-rays, Dscrete Math. 241 (1 3) (2001) [2] E. Barcucc, A. Del Lungo, M. Nvat, R. Pnzan, Reconstructng convex polyomnoes from horzontal and vertcal projectons, Theoret. Comput. Sc. 155 (1996) [3] E. Barcucc, A. Del Lungo, M. Nvat, R. Pnzan, Medans of polyomnoes: A property for the reconstructon, Int. J. Imagng Syst. Technol. 9 (2 3) (1998) [4] S. Brunett, A. Daurat, Reconstructon of dscrete sets from two or more X-rays n any drecton, n: Proc. of IWCIA 2000, Unversté de Caen, 2000, pp [5] S. 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Grtzmann, Dscrete tomography: Determnaton of fnte sets by X-rays, Trans. Amer. Math. Soc. 349 (1997) [13] M. Gębala, The reconstructon of convex polyomnoes from horzontal and vertcal projectons, n: Proc. of SOFSEM 98, n: Lecture Notes n Comp. Sc., vol. 1521, 1998, pp [14] A. Kuba, Reconstructon of two-drectonally connected bnary patterns from ther two orthogonal projectons, Comp. Vs. Graph. Image Process. 27 (1984) [15] G.H. Woegnger, The reconstructon of polyomnoes from ther horzontal and vertcal projectons, Inform. Process. Lett. 77 (5 6) (2001)