Bounds on the quality of reconstructed images in binary tomography

Bounds on he quali y o econs uc ed images in bina y omog aphy K. Joos Ba enbu ga,c, Wagne Fo esa,b, Lajos Hajdud,e, Robe Tijdemanb aCen um Wiskunde &In o ma ica, Science Pa k 123, 1098 XG Ams e dam, The Ne he lands bMa hema ical Ins i u e, Leiden Uni e si y, The Ne he lands cVision Lab, Uni e si y o An we p, Belgium dIns i u e o Ma hema ics, Uni e si y o Deb ecen, Hunga y eNumbe Theo y Resea ch G oup o he Hunga ian Academy o Sciences, Deb ecen, Hunga y Abs ac Bina y omog aphy deals wi h he p oblem o econs uc ing a bina y image om i s p ojec ions. In pa icula , he e is a ocus on highly unde de e mined econs uc ion p oblems o which many solu ions may exis . In such cases, i is impo an o ha e a quali y measu e o he econs uc ion wi h espec o he unknown o iginal image. In his a icle, we de i e a se ies o uppe bounds ha can be used o gua an ee he quali y o a econs uc ed bina y image. The bounds limi he numbe o pixels ha can be inco ec in he econs uc ed image wi h espec o he o iginal image. We p o ide se e al e sions o hese bounds, anging om bounds on he di e ence be ween any wo bina y solu ions o a omog aphy p oblem o bounds on he di e ence be ween app oxima e solu ions and he o iginal objec . The bounds a e e alua ed expe imen ally o a ange o es images, based on simula ed p ojec ion da a. Keywo ds: bina y omog aphy, disc e e omog aphy, e o bounds, s abili y 1. In oduc ion Tomog aphy is a echnique o econs uc ing an image o an objec om a se ies o p o- jec ions o his objec , acqui ed om a ange o iewing angles. The p ojec ion images a e ypically eco ded using a scanning de ice, which can employ a ious ypes o beams (e.g., X- ays, neu ons, elec ons) ha a e se he objec , a e which a de ec o measu es he esul o he beam-objec in e ac ion. P o ided ha a la ge numbe o high-quali y p ojec ions a e a ailable, sampled om a ull ange o angles, an accu a e econs uc ion o he objec can be compu ed using a omog aphic econs uc ion algo i hm [15, 19]. In p ac ice, he se o angles o which p ojec ions a e acqui ed is o en limi ed. Due o dose cons ain s, i can be desi able o eco d as ew p ojec ions images as possible, while s ill a aining su icien image quali y. Also, he angula ange can be es ic ed by he pa icula scanning se up, such as in elec on omog aphy, whe e he shape o he sample holde limi s he Email add esses: [email p o ec ed] (K. Joos Ba enbu g), [email p o ec ed] (Wagne Fo es), [email p o ec ed] (Lajos Hajdu), [email p o ec ed] (Robe Tijdeman) P ep in submi ed o Disc e e Applied Ma hema ics Sep embe 10, 2012 angula ange o he p ojec ions [20]. The esul ing image econs uc ion p oblems, based on jus a small numbe o p ojec ions, a e known as limi ed da a p oblems. Fo omog aphic econs uc ion om se e ely limi ed da a, classical algo i hms based on an- aly ical in e sion o he Radon ans o m, such as he Fil e ed Backp ojec ion algo i hm, o en yield low quali y econs uc ions ha a e pollu ed by s ong a e ac s. In such cases, i makes sense o exploi a ailable p io knowledge o he unknown objec . Inco po a ion o his knowl- edge in he econs uc ion algo i hm can po en ially esul in a educ ion o he equi ed numbe o p ojec ions, inc eased accu acy o he econs uc ion, o an imp o ed abili y o deal wi h noisy p ojec ion da a. A p io ha has ecei ed much a en ion ecen ly conce ns he spa si y o he image, o o i s g adien , which is exploi ed in he ield o Comp essed Sensing [12, 13, 23, 24]. A ela ed, bu mo e s ic ype o p io knowledge is exploi ed in Disc e e Tomog aphy, which ocuses on he econs uc ion o images ha consis o a small, disc e e se o g ey alues [16, 17]. The ac ual se o g ey le els is ypically assumed o be known in ad ance. He e, we ocus speci ically on he econs uc ion o bina y images, which consis o jus wo g ey le els, 0 and 1. Se e al econs uc ion algo i hms ha e been p oposed in li e a u e o econs uc ing bina y images om a small numbe o p ojec ions, o en e en less han 6 [5, 8, 17, 22]. These algo i hms do no wo k well o all ypes o bina y images, ye hey ha e demons a ed he abili y o accu a ely econs uc a ious phan om images. Despi e he s ong cons ain imposed on he g ey alues in disc e e omog aphy, many alid solu ions o he econs uc ion p oblem can exis , all co esponding o he same se o p ojec- ions. I he p ojec ions a e ob ained by pe o ming measu emen s on some unknown g ound u h objec , he econs uc ion can hen de ia e subs an ially om he ue objec . As a con- sequence, he e is a need o an uppe bound on he di e ence be ween bina y solu ions o he econs uc ion p oblem. As he g ound u h is a solu ion by i sel , his would also yield a bound on he econs uc ion e o wi h espec o he g ound u h. A ela ed p oblem in disc e e omog aphy is he so-called s abili y p oblem, which deals wi h he ques ion o how he econs uc ion changes i he p ojec ions a e sligh ly pe u bed. Fo images ep esen ed on a disc e e g id, bo h lowe and uppe bounds ha e been ob ained o he magni ude o such changes [1, 2, 3, 11, 28]. Fo he case o bina y image econs uc ion om jus wo p ojec ions, ho izon al and e ical, sha p bounds on he di e ence be ween bina y images ha ing he same p ojec ions ha e been ob ained by Alpe s and B une i [2] and ex ended by Van Dalen [26, 27]. In his a icle, we p esen a se ies o bounds which a e highly gene al. Ou bounds can be compu ed o any se o p ojec ions and in di e en geome ical se ings, o la ice images as well as disc e ized con inuous images. A key idea in de i ing hese bounds is an obse a ion i s made by Hajdu and Tijdeman in [14], conce ning he ac ha all bina y solu ions o he econs uc ion p oblem mus lie on a ce ain hype sphe e, o which bo h he cen e and adius can be compu ed. The cen e o his hype sphe e, which we call he cen al econs uc ion, is he sho es eal- alued solu ion o he omog aphy p oblem. This hype sphe e cons uc ion leads di ec ly o a simple bound on he dis ance be ween any wo bina y solu ions, based on he iangle inequali y. S onge bounds can be de i ed by ocusing on he dis ance be ween bina y solu ions o he omog aphy p oblem and he bina y image ha is ob ained by ounding he cen al econs uc ion. We de i e se e al bounds ha combine p ope ies o he eal- alued solu ion wi h combina o ial p ope ies ha a e sa is ied by he bina y solu ions. In pa icula , he ac ha he sum o he pixel alues in he unknown image is ully de e mined by he p ojec ion da a can be used o imp o e he e o bounds o bina y images. The s uc u e o his a icle is as ollows. Ou no a ion is in oduced in Sec ion 2, a e which 2 a gene al class o econs uc ion p oblems is in oduced. We hen discuss wo speci ic examples o such p oblems, based on he g id model and s ip model o compu ing he p ojec ions. In Sec ion 3, he cen al econs uc ion is in oduced, which plays an impo an ole in all esul s ha ollow. We discuss wo di e en s a egies o i s compu a ion. Sec ion 4 con ains he main esul s o his pape : a se ies o quali y bounds o bina y solu ions o he omog aphy p oblem. I is di ided in h ee pa s: in Sec ion 4.1, a gene al bound is de i ed on he di e ence be ween wo bina y images ha ing a gi en se o p ojec ions. Sec ion 4.2 deals wi h bounds ha a e based on p ope ies o he bina y images ha a e ob ained by ounding he cen al econs uc ion. These bounds a e subsequen ly e ined in Sec ion 4.3 by including he knowledge o he o al numbe o 1’s in any bina y solu ion, which can be de e mined om he p ojec ion da a. Sec ion 5 p esen s a se ies o simula ion expe imen s and hei esul s. F om hese esul s, he p ac ical alue o he p oposed bounds can be e alua ed o di e en ypes o images. The esul s a e u he discussed in Sec ion 6. Sec ion 7 concludes he pape . This a icle is an ex ended e sion o he con e ence pape [7]. Compa ed o he con e ence pape , se e al new bounds ha e been included in he p esen a icle, in pa icula (i) bounds on he di e ence be ween any bina y image, no necessa ily being a solu ion o he econs uc ion p oblem, and he unknown o iginal image, and (ii) bounds ha inco po a e knowledge o he o al numbe o 1’s in he unknown o iginal image. Va ious o he bounds ha e also been e ined. The expe imen al sec ion has been ex ended wi h an example, depic ing he di e en images and concep s in ol ed in he a ious bounds. A se ies o expe imen s on bounding he e o o bina y econs uc ions compu ed by he DART algo i hm, a heu is ic me hod ha is no gua an eed o yield an exac solu ion [8], bu yields high quali y econs uc ions in p ac ice, is also included. 2. No a ion and model Th oughou he disc e e omog aphy li e a u e, se e al imaging models ha e been consid- e ed. In he g id model, an image is o med by assigning a alue o each poin in a egula g id. In he case o bina y images, each poin is assigned a alue o ei he 0 o 1. He e, we conside squa e g ids o he o m A={(i,j)∈Z2: 1 ≤i,j≤s} o s∈N,s≥1; see Fig. 1(a). Fo he g id model, we e e o he poin s in Aas pixels. A bina y image de ined on Acan be ep esen ed by a map A→ {0,1}. A p ojec ion o an image is o med by conside ing he se o pa allel lines h ough one o mo e g id poin s in a ce ain di ec ion (a,b)∈Z2, wi h a≥0 and (a,b) cop ime, and summing he alues o he poin s on each line. Fo a line gi en by he equa ion ax −by = ( ∈Z), he line p ojec ion p is de ined as p=X (x,y)∈A:ax−by= (x,y). The g id model can be used o model nanoc ys als, ha consis o disc e e a oms posi ioned in a egula g id [18, 25]. In many omog aphy applica ions, a con inuous ep esen a ion o he objec is mo e ealis ic, as he e is no in insic g id s uc u e. In such cases, he unknown image is ypically app oxima ed by an image de ined on a disc e e pixel g id, using squa e pixels. A common model o compu - ing he p ojec ions o such a pixelized image is he s ip model [19, sec ion 7.4.1], [30]. In he s ip model, a p ojec ion is compu ed by conside ing a se o pa allel s ips in a gi en di ec ion and and o each s ip compu ing he weigh ed sum o all he pixels which in e sec ha s ip wi h a weigh equal o he in e sec ion a ea o he s ip and he pixel. 3 (a) G id model (b) S ip model Figu e 1: Two di e en p ojec ion models. We now de ine some gene al no a ion. An image is ep esen ed by a ec o x=(xi)∈Rn. We e e o he en ies o xas pixels, which co espond o uni squa es in he s ip model and o poin s in he g id model. The de i a ion o ou main esul s does no depend on he pa icula p ojec ion model. Th oughou his pape we assume ha all images a e squa e, consis ing o c ows and ccolumns, whe e n=c2. A bina y image co esponds wi h a ec o ¯x∈ {0,1}n. Fo a gi en se o kp ojec ion di ec ions, he p ojec ion map maps an image x o a ec o p∈ Rmo p ojec ion da a, whe e mdeno es he o al numbe o line measu emen s. As he p ojec ion map is a linea ans o ma ion, i can be ep esen ed by a ma ix W=(wi j)∈Rm×n, called he p ojec ion ma ix. En y wi j ep esen s he weigh o he con ibu ion o xj o p ojec ed line i. No e ha o he g id model he p ojec ion ma ix is a bina y ma ix, while o he s ip model i s en ies a e eal alues in [0,1]. The p ojec ion ma ix Wand ec o pcan be decomposed in o k blocks as W=            W1 . . . Wk             ,p=            p1 . . . pk             ,(1) whe e each block Wd(d=1,...,k) ep esen s he p ojec ion map o a single di ec ion and each block pd ep esen s he co esponding p ojec ion da a. F om his poin on, we assume ha he p ojec ion ma ix has he p ope y ha Pm i=1wi j =k o all j=1,...,n. This p ope y is ce ainly sa is ied o he g id model, as e e y xjis coun ed wi h weigh 1 o exac ly one line in each p ojec ion di ec ion. The p ope y is also sa is ied o he s ip p ojec ion model, as he o al pixel weigh o each p ojec ion angle is equal o he a ea o a pixel, which is 1. Fo mos o he p ojec ion models commonly used in omog aphy, such as he line model, whe e he weigh o a pixel is de e mined by he leng h o i s in e sec ion wi h a line, his p ope y is app oxima ely sa is ied, bu no always exac ly. The gene al econs uc ion p oblem consis s o inding a solu ion o he sys em Wx =p o gi en p ojec ion da a p, i.e., o ind an image ha has he gi en p ojec ions. In bina y o- mog aphy, one seeks a bina y solu ion o he sys em. Fo a gi en p ojec ion ma ix Wand gi en p ojec ion da a p, le SW(p)={x∈Rn:Wx =p}, he se o all eal- alued solu ions co espond- ing wi h he p ojec ion da a, and le ¯ SW(p)=SW(p)∩ {0,1}n, he se o bina y solu ions o he sys em. As he main goal o inco po a ing p io knowledge o he bina y g ey le els in he econ- s uc ion is o educe he numbe o equi ed p ojec ions, we ocus on he case whe e mis small wi h espec o n, such ha he eal- alued econs uc ion p oblem is se e ely unde de e mined. Despi e he s ong cons ain ha each pixel alue ximus belong o he se {0,1}, he bina y econs uc ion p oblem o en does no ha e a unique solu ion. Ins ead o posing he uniqueness 4 p oblem as a yes/no ques ion, we ocus in his pape on he numbe o di e en pixels be ween di e en bina y solu ions. Fo example, i he econs uc ion p oblem has no unique solu ion, bu all pai s o solu ions ha e a mos 4 di e en pixels, hen one can say ha any such solu ion mus be close o he o iginal objec om which he p ojec ions ha e been ob ained, e en i he exac se o di e ences canno be de e mined. Fo any wo ec o s ¯ u,¯ ∈ {0,1}n, de ine he di e ence se D(¯ u,¯ )={i: ¯ui,¯ i}and he numbe o di e ences d(¯ u,¯ )=#D(¯ u,¯ ), whe e he symbol # deno es he ca dinali y ope a o o a ini e se . No e ha d(¯ u,¯ )=k¯ u−¯ k1. 3. The cen al econs uc ion As he p ojec ion ma ix is ypically no a squa e ma ix, and also does no ha e ull ank, i does no ha e an in e se. Recall ha he Moo e-Pen ose pseudo in e se o an m×nma ix Ais an n×mma ix A†, which can be uniquely cha ac e ized by he wo geome ic condi ions A†b⊥ N(A) and (I−AA†)b⊥ R(A) o all b∈Rm, whe e N(A) is he nullspace o Aand R(A) is he ange o A, [10, page 15]. Le x∗=W†p. Then x∗has he p ope y (see Chap e 3 o [9]) ha i is he eal- alued solu ion o minimal Euclidean no m o he sys em Wx =p, p o ided ha he la e sys em is sol able. We call x∗ he cen al econs uc ion o p. The cen al econs uc ion plays an impo an ole in he bounds we de i e o he bina y econs uc ion p oblem. We will show in he nex sec ion ha all bina y solu ions o he sys em ha e equal dis ance o x∗, so ha one can conside he cen al econs uc ion as lying “in he middle” o all bina y solu ions. As all bounds p esen ed in his pape depend on x∗, accu a e compu a ion o x∗is necessa y o compu e he co esponding di e ence bounds. One app oach o compu ing he cen al econ- s uc ion o a consis en sys em Wx =pis o use he QR decomposi ion o WT. We will only ske ch he compu a ion he e and e e o [4] o de ails. Fo cla i y o p esen a ion, we assume ha Whas ull ow ank. In ac , his assump ion is no sa is ied o omog aphy, and he ex- ended QR decomposi ion should be used. The QR decomposi ion ac o izes he ma ix WTin o an o hogonal ma ix Qand an uppe iangula ma ix Ro ull column ank, as WT=Q R 0!. The cen al econs uc ion is hen gi en by x∗=Q(RT)−1p, which can be compu ed e icien ly by i s sol ing he sys em RTy=p o yby back subs i u ion, and hen compu ing x∗=Qy. Howe e , due o he size o he ma ix W, calcula ion o he QR decomposi ion is usually un- p ac ical o la ge images. Fo he case m<n, he QR decomposi ion equi es O(n3) ope a ions. Mo eo e , he n×nma ix Qis ypically dense, equi ing a as amoun o compu e memo y. As an al e na i e, an i e a i e me hod o sol ing he sys em Wx =p, called CGLS (Conjuga e G a- dien Leas Squa es), can be used [21]. The CGLS algo i hm can e ec i ely exploi he spa se s uc u e o he p ojec ion ma ix o educe he equi ed compu a ion ime, and does no equi e s o age o la ge, dense ma ices. Apa om nume ical e o s, applying CGLS o he sys em Wx =p esul s, a e con e gence, in he compu a ion o W†p, while no compu ing he ma ix W†explici ly (see also [29]). Fo all expe imen s in Sec ion 5, he ime equi ed o compu e he cen al econs uc ion a ied om a ew seconds up o wo minu es on a s anda d PC, depending on he numbe o p ojec ions and image size. 5 4. Quali y bounds o bina y solu ions In all he esul s in he ollowing subsec ions, we conside a ixed sys em Wx =pco e- sponding o a bina y omog aphy p oblem, and e e o he cen al econs uc ion o his sys em as x∗. As a subs an ial numbe o bounds will be gi en h oughou his pape , we in oduce he ollowing no a ion ha will be u he de ined in he emainde o he pape : •The exp essions a(i) (i=1,2,3,4) will ep esen bounds on he numbe o pixel di e ences be ween any wo bina y solu ions o he econs uc ion p oblem. •The exp essions b(i) (i=1,2,3,4) will ep esen bounds on he numbe o pixel di e ence be ween a ce ain gi en bina y image (no necessa ily a solu ion) and any bina y solu ion o he econs uc ion p oblem. •The exp essions c(i) (i=1,2) will ep esen bounds on he numbe o pixel di e ences be ween he ounded cen al econs uc ion ¯ and any bina y solu ion. The bounds wi hin each class a,b, and c ep esen uppe bounds o he same dis ance measu e and can he e o e be compa ed. 4.1. Elemen a y bounds based on he cen al econs uc ion In his subsec ion, a i s se o bounds a e de i ed. They ollow om he ac ha he Eu- clidean dis ance be ween he cen al econs uc ion and any bina y solu ion o he econs uc ion p oblem can be de e mined om he p ojec ions. We s a by no icing ha he Euclidean no m o any bina y solu ion o he omog aphy p oblem is de e mined by he p ojec ion da a: Lemma 1. Le ¯ x∈¯ SW(p). Then, k¯ xk2 2=kpk1 k. P oo . By he de ini ion o he `1-no m, kpk1=Pm i=1|pi|=Pm i=1pi, since pi≥0 (i=1,...,n). Also, m X i=1 pi= m X i=1        n X j=1 wi j ¯xj        = n X j=1       m X i=1 wi j      ¯xj= n X j=1 k¯xj,(2) and he e o e kpk1=kPn j=1¯xj. As ¯ x∈ {0,1}n, we ha e k¯ xk2 2=k¯ xk1=Pn j=1¯xj=kpk1 k. The ollowing lemma illus a es he impo ance o he cen al econs uc ion, he sho es eal- alued solu ion in SW(p), by showing ha he bina y solu ions a e he sho es among all in ege solu ions o he sys em. Lemma 2. Le ¯ x∈¯ SW(p)and y∈SW(p)∩Zn. Then k¯ xk2≤ kyk2, wi h equali y i and only i y∈¯ SW(p). P oo . No e ha he s a emen is p o ed in [14], see P oblem 2 and he subsequen pa ag aph. Howe e , o he con enience o he eade we gi e he p oo he e. We ha e || ¯ x||2 2= n X i=1 ¯x2 i= n X i=1 ¯xi= n X i=1 yi=kpk1 k.(3) 6 Obse ing ha n X i=1 yi≤ n X i=1 y2 i=||y||2 2,(4) wi h equali y i and only i yis bina y, yields he esul . Lemma 3. Le ¯ x∈¯ SW(p). Then k¯ x−x∗k2=qkpk1 k− kx∗k2 2. P oo . F om he de ini ion o x∗we ha e (¯ x−x∗)∈ N(W), and x∗⊥(¯ x−x∗). Applying Py hago as’ Theo em and Lemma 1 yields k¯ x−x∗k2 2=kpk1 k− kx∗k2 2.(5) De ine R=qkpk1 k− kx∗k2 2. We will use his cons an h oughou he emainde o his a - icle, and e e o Ras he cen al adius. Acco ding o Lemma 3, any bina y solu ion o he econs uc ion p oblem is on he hype sphe e cen e ed in x∗wi h adius R. Supposing he exis ence o a leas wo di e en bina y solu ions, Lemma 3 allows us o de i e an uppe bound o he numbe o pixel di e ences be ween hose solu ions. Theo em 4. Le ¯ x,¯ y∈¯ SW(p)and pu a(1) =4R2. Then d(¯ x,¯ y)≤a(1). P oo . Acco ding o Lemma 3, we ha e k¯ x−x∗k2=k¯ y−x∗k2=R. The e o e, k¯ x−¯ yk2≤ k ¯ x−x∗k2+k¯ y−x∗k2=2R. As ¯ xand ¯ ya e bina y, we ha e d(¯ x,¯ y)=k¯ x−¯ yk1=k¯ x−¯ yk2 2. Using he iangle inequali y, a simple bound can also be gi en o he dis ance be ween any bina y image and a solu ion o he econs uc ion p oblem, as ollows: Co olla y 5. Le ¯ ∈ {0,1}nbe a gi en bina y image and pu b(1) =(R+k¯ −x∗k2)2. Then o any bina y image ¯ x∈¯ SW(p), we ha e d(¯ x,¯ )≤b(1). 4.2. Bounds based on ounding he cen al econs uc ion The ac ha all elemen s o ¯ SW(p) ha e equal dis ance o he cen al econs uc ion x∗, com- bined wi h he ac s ha bina y solu ions a e he sho es solu ions among all in ege solu ions (Lemma 2) and ha x∗is he sho es eal- alued solu ion, sugges s ha bina y solu ions can o en be ound nea x∗. I is he e o e na u al o conside he image ha is ob ained by ound- ing each en y o x∗ o he nea es bina y alue. In his sec ion, we will de i e se e al bounds based on he numbe o di e ences be ween a bina y solu ion o he econs uc ion p oblem and a bina y image ob ained by ounding x∗. Fo α∈R, le bin(α)=min(|α|,|1−α|). Pu T=qPn i=1bin2(x∗ i), i.e., he Euclidean dis ance om x∗ o he nea es bina y ec o . We will use his cons an h oughou his pape and e e o Tas he cen al ounding dis ance. 7 Co olla y 6. I R <T, hen ¯ SW(p)=∅. I R =T, hen all solu ions in ¯ SW(p)can be ob ained by ounding he alues in x∗ o he nea es bina y alues, and a ia ions a e only possible o he en ies i whe e x∗ i=1 2. Le T={¯ ∈ {0,1}n:k¯ −x∗k2=T}and le ¯ ∈ T , i.e., ¯ is among he bina y ec o s ha a e nea es o x∗in he Euclidean sense. I R>Tand R−Tis small, i is possible o say ha a ac ion o he ounded alues a e co ec , i.e., o p o ide an uppe bound on he numbe o pixel di e ences be ween any solu ion in ¯ SW(p) and ¯ . In mos cases we can no say which ounded alues a e co ec . Lemma 7. Le ¯ ∈ T and le ¯ ∈ {0,1}nbe any bina y ec o . Then k¯ −x∗k2 2=T2+Pi∈D(¯ ,¯ )|2x∗ i−1|. P oo . We ha e he ollowing iden i ies: k¯ −x∗k2 2=k¯ −¯ +¯ −x∗k2 2 =k¯ −x∗k2 2+2h¯ −x∗,¯ −¯ i+h¯ −¯ ,¯ −¯ i =T2+2h¯ −¯ ,x∗i+k¯ k2 2− k¯ k2 2 =T2+2 n X i=1 (¯ i−¯ i)x∗ i+ n X i=1 ¯ i− n X i=1 ¯ i =T2+ n X i=1 (¯ i−¯ i)(2x∗ i−1) =T2+X i∈D(¯ ,¯ ) |2x∗ i−1|. Lemma 7 can be in e p e ed as ollows: conside he se o en ies whe e ¯ and ¯ a e di e en . I we ans o m ¯ in o ¯ by pe o ming a sequence o single-en y changes (ei he om 0 o 1, o om 1 o 0), each ime an en y io ¯ is changed he squa ed Euclidean dis ance om he cu en ec o o x∗inc eases by si=|2x∗ i−1|. Le πbe a pe mu a ion o {1,...,n}such ha sπ(1) ≤sπ(2) ≤. . . ≤sπ(n), which can be ob ained by so ing he en ies siin inc easing o de . Co olla y 8. Le ¯ ∈ T and le ¯ ∈ {0,1}nbe any bina y ec o . Then k¯ −x∗k2 2≥T2+P` i=1sπ(i), whe e `=d(¯ ,¯ ). P oo . Acco ding o Lemma 7 we ha e k¯ −x∗k2 2=T2+X i∈D(¯ ,¯ ) si≥T2+ ` X i=1 sπ(i). As he Euclidean dis ance om x∗ o any ¯ x∈¯ SW(p) is R, a bound can now be de i ed on he maximal numbe o pixels in ¯ ha mus be changed o mo e om ¯ o ¯ x. 8 Theo em 9. Le ¯ ∈ T ,¯ x∈¯ SW(p). Pu c(1) =max        0≤`≤n: ` X i=1 sπ(i)≤R2−T2       . Then d(¯ x,¯ )≤c(1). P oo . As ¯ x∈¯ SW(p), we ha e k¯ x−x∗k2 2=R2.Applying Lemma 7, we ind ha R2−T2=X i∈D(¯ x,¯ ) si≥ d(¯ x,¯ ) X i=1 sπ(i), which implies ha d(¯ x,¯ )≤c(1). The p oo o Theo em 9 can be in e p e ed as ollows: conside he se o en ies whe e ¯ and ¯ xa e di e en . I we ans o m ¯ in o ¯ xby pe o ming a sequence o single-en y changes (ei he om 0 o 1, o om 1 o 0), each ime an en y io ¯ is changed he squa ed Euclidean dis ance om he cu en ec o o x∗inc eases by si=|2x∗ i−1|. As all bina y solu ions o he econs uc ion p oblem a e on a hype sphe e cen e ed in x∗wi h adius R, we know ha once we ha e c ossed he bounda y o his hype sphe e, a bina y solu ion can no longe be ob ained by changing he alues o addi ional en ies ha ha e no ye been changed. An uppe bound on he numbe o di e ences be ween ¯ and ¯ xcan be ob ained by coun ing he numbe o s eps equi ed o c oss he hype sphe e, each ime choosing a pixel which esul s in he minimal inc ease o he dis ance o x∗. The ollowing wo Co olla ies ollow di ec ly om Theo em 9: Co olla y 10. Le ¯ ∈ T ,¯ x,¯ y∈¯ SW(p)and le a(2) =2c(1) wi h c(1) de ined as in Theo em 9. Then d(¯ x,¯ y)≤a(2). Co olla y 11. Le ¯ ∈ T and le ¯ ∈ {0,1}nbe a gi en bina y image and le b(2) =c(1) +d(¯ ,¯ ) wi h c(1) de ined as in Theo em 9. Then d(¯ x,¯ )≤b(2). In ac , he bound om Co olla y 10 can be sha pened by no ing ha we can assume ha he se s D(¯ ,¯ x) and D(¯ ,¯ y) a e disjoin , as en ies ha occu in bo h se s do no con ibu e o he numbe o di e ences be ween ¯ xand ¯ y. This obse a ion leads o he ollowing Theo em: Theo em 12. Le ¯ ∈ T ,¯ x,¯ y∈¯ SW(p). Pu a(3) =max        0≤`≤n: ` X i=1 sπ(i)≤2(R2−T2)       . Then d(¯ x,¯ y)≤a(3). P oo . De ine ˆ xby ˆxi=¯ ii ¯xi=¯yi, and ˆxi=¯xio he wise. De ine ˆ yanalogously. Then d(ˆ x,ˆ y)=d(¯ x,¯ y), kˆ x−x∗k2 2≤R2, and kˆ y−y∗k2 2≤R2. Hence, 2R2≥ k ˆ x−x∗k2 2+kˆ y−y∗k2 2=2T2+ Pi∈D(¯ ,ˆ x)si+Pi∈D(¯ ,ˆ y)si.As D(¯ ,ˆ x) and D(¯ ,ˆ y) a e disjoin , we ha e 2R2−2T2≥Pd(¯ ,ˆ x)+d(¯ ,ˆ y) i=1sπ(i). This implies ha d(¯ x,¯ y)=d(ˆ x,ˆ y)≤d(¯ ,ˆ x)+d(¯ ,ˆ y)≤a(3). A simila bound can be de i ed o he case whe e a pa icula bina y image ¯ , no neces- sa ily a solu ion o he omog aphy p oblem, is gi en. Fo his, we ans o m ¯ in o ¯ and hen we pe o m a sequence o single-en y changes in ¯ wi h he exclusion o he pixels ha di e be ween ¯ and ¯ because hey ha e al eady been coun ed as w ong pixels o ¯ . 9 w ong wi h espec o he unknown phan om image. 6. Discussion o he esul s Despi e he ac s ha he ou phan oms ha e s ong di e ences in shape and mo phology, and ha he g id and s ip models a e qui e di e en , he esul s shown in Figs. 6–9 a e su p isingly consis en h oughou all expe imen s. Mos o he bounds become smalle as he numbe o p ojec ion di ec ions is inc eased bu mono onici y is no a p ope y o all he bounds p esen ed in Sec ion 4. F om he di e ence be ween he bounds p esen ed in Sec ion 4.1 and he bounds based on he ounded cen al econs uc ion, we see ha in mos cases he phan om ¯ xis subs an ially close o ¯ han o x∗. In Figs. 6 and 8, i can be obse ed ha he ue ac ion o pixel di e ences be ween he phan om image ¯ xand he ounded cen al econs uc ion ¯ , deno ed by Ec, is o en app oxima ed qui e well by he bound c, in pa icula o he g id model. This indica es ha wi h espec o ¯ , he bounds p esen ed in his a icle can be qui e sha p. In Fig. 8(a), pa s o he g aphs o he bounds aand c, o mo e han 6 p ojec ions, a e missing. In ac , in his case all o hem a e ze o, such ha hey canno be displayed in he loga i hmic scale. This illus a es ha ou heo ems o bounding he dis ance be ween any wo bina y solu ions can be used o p o e uniqueness o a bina y solu ion, e en when he co espond- ing eal- alued sys em o equa ions is unde de e mined. In mos o he expe imen s, he g aphs o a(2) and a(4) a e almos indis inguishable and he bound a(3) is smalle han a(4) bu i can happen, as in Fig. 9(a), ha his is no he case. In Fig. 9(a) he bound a(4) p o es uniqueness om 6 p ojec ion di ec ions o mo e while he bound a(3) does no o 12, 14 and 16 p ojec ions. Al hough a(3) is he sha pes bound in mos o he expe imen s, his is no ue in gene al and all bounds should be used o ob ain he bes esul . The bounds a(2), a(3) and a(4) use simila p inciples and hei compu a ion can be pe o med simul aneously. 7. Ou look and conclusions In his a icle, we ha e p esen ed a ange o gene al bounds on he accu acy o econs uc- ions in bina y omog aphy, wi h espec o he unknown o iginal objec . The bounds can be compu ed wi hin easonable ime and gi e gua an ees on he numbe o pixels ha can be di - e en be ween any wo bina y solu ions o he omog aphy p oblem, on he di e ence be ween an image ob ained by ounding he cen al econs uc ion and any bina y solu ion, and on he di e ence be ween any bina y image and any bina y solu ion. The expe imen al esul s show ha by using hese bounds, one can p o e ha he numbe o di e ences be ween bina y econ- s uc ions mus be e y small, e en when he co esponding eal- alued sys em o equa ions is se e ely unde de e mined. In o de o make hese bound p ac ically use ul, ou esul s will ha e o be ex ended o deal wi h noisy p ojec ion da a, which we will inco po a e in u u e esea ch. Acknowledgemen s L.H. was suppo ed by he OTKA g an s K67580 and K75566, and by he T ´ AMOP 4.2.1./B- 09/1/KONV-2010-0007 p ojec . The p ojec is implemen ed h ough he New Hunga y De el- 16 opmen Plan, co inanced by he Eu opean Social Fund and he Eu opean Regional De elop- men Fund. W.F. acknowledges suppo om he E asmus Mundus p og am o he Eu opean Union. K.J.B. was suppo ed by he Ne he lands O ganisa ion o Scien i ic Resea ch (NWO), p og amme 639.072.005. [1] Alpe s, A., 2003. Ins abili y and S abili y in Disc e e Tomog aphy. Ph.D. hesis. Technische Uni e si ¨ a M¨ unchen. Shake Ve lag, ISBN 3-8322-2355-X. 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Ma h. 156, 2359–2367. 1234567 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a c Ec (a) Phan om 1, 32×32 0 5 10 15 20 25 30 10−4 10−3 10−2 10−1 Numbe o angles F ac ion o pixels a c Ec (b) Phan om 1, 128×128 0 50 100 150 200 10−6 10−5 10−4 10−3 10−2 10−1 Numbe o angles F ac ion o pixels a c Ec (c) Phan om 1, 512×512 0 2 4 6 8 10 12 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a c Ec (d) Phan om 2, 32×32 0 10 20 30 40 50 60 10−4 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a c Ec (e) Phan om 2, 128×128 0 50 100 150 200 10−5 10−4 10−3 10−2 10−1 Numbe o angles F ac ion o pixels a c Ec ( ) Phan om 2, 512×512 0 5 10 15 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a c Ec (g) Phan om 3, 32×32 0 10 20 30 40 50 60 10−4 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a c Ec (h) Phan om 3, 128×128 0 50 100 150 200 10−6 10−4 10−2 100 Numbe o angles F ac ion o pixels a c Ec (i) Phan om 3, 512×512 0 5 10 15 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a c Ec (j) Phan om 4, 32×32 0 20 40 60 80 10−5 10−4 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a c Ec (k) Phan om 4, 128×128 0 50 100 150 200 10−6 10−4 10−2 100 Numbe o angles F ac ion o pixels a c Ec (l) Phan om 4, 512×512 Figu e 6: G id model: compu ed bounds as a unc ion o he numbe o p ojec ion di ec ions. 18 0 5 10 15 20 10−3 10−2 10−1 100 101 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) (a) Phan om 1, 32×32 0 20 40 60 80 10−5 10−4 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) (b) Phan om 1, 128×128 0 50 100 150 200 10−5 10−4 10−3 10−2 10−1 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) (c) Phan om 1, 512×512 0 5 10 15 20 10−4 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) (d) Phan om 2, 32×32 0 20 40 60 80 10−4 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) (e) Phan om 2, 128×128 0 50 100 150 200 10−4 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) ( ) Phan om 2, 512×512 0 5 10 15 20 10−4 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) (g) Phan om 3, 32×32 0 20 40 60 80 10−5 10−4 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) (h) Phan om 3, 128×128 0 50 100 150 200 10−4 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) (i) Phan om 3, 512×512 0 5 10 15 20 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) (j) Phan om 4, 32×32 0 20 40 60 80 10−4 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) (k) Phan om 4, 128×128 0 50 100 150 200 10−4 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) (l) Phan om 4, 512×512 Figu e 7: G id model: compu ed bounds as a unc ion o he numbe o p ojec ion di ec ions. No e ha he g aphs o a(2) and a(4) ha e s ong o e lap and can ha dly be dis inguished. 19 0 2 4 6 8 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a c Ec (a) Phan om 1, 32×32 0 20 40 60 80 10−5 10−4 10−3 10−2 10−1 Numbe o angles F ac ion o pixels a c Ec (b) Phan om 1, 128×128 0 50 100 150 200 10−6 10−5 10−4 10−3 10−2 10−1 Numbe o angles F ac ion o pixels a c Ec (c) Phan om 1, 512×512 0 5 10 15 20 10−2 10−1 100 Numbe o angles F ac ion o pixels a c Ec (d) Phan om 4, 32×32 0 20 40 60 80 10−2 10−1 100 Numbe o angles F ac ion o pixels a c Ec (e) Phan om 4, 128×128 0 50 100 150 200 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a c Ec ( ) Phan om 4, 512×512 Figu e 8: S ip model: compu ed bounds as a unc ion o he numbe o p ojec ion di ec ions o Phan oms 1 and 4. 0 5 10 15 20 10−4 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) (a) Phan om 1, 32×32 0 20 40 60 80 10−4 10−3 10−2 10−1 100 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) (b) Phan om 1, 128×128 0 50 100 150 200 10−4 10−3 10−2 10−1 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) (c) Phan om 1, 512×512 0 5 10 15 20 10−0.7 10−0.5 10−0.3 10−0.1 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) (d) Phan om 4, 32×32 0 20 40 60 80 10−2 10−1 100 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) (e) Phan om 4, 128×128 0 50 100 150 200 10−2 10−1 100 Numbe o angles F ac ion o pixels a(1) a(2) a(3) a(4) ( ) Phan om 4, 512×512 Figu e 9: S ip model: compu ed bounds as a unc ion o he numbe o p ojec ion di ec ions o Phan oms 1 and 4. No e ha he g aphs o a(2) and a(4) ha e s ong o e lap and can ha dly be dis inguished. 20