Detection of objects in noisy images and site percolation on square lattices

Detecti on of ob jects in nois y images and site p erc olation on square lattices Mikhail Lango v o y ∗ Mikhail L angovoy, T e chnische Univ ersiteit Eindhoven, EURANDOM, P.O. Box 513 , 5600 MB, Eindhoven, The Netherlands e-mail: langovoy@e urandom.tue. nl Phone: (+31) (40) 247 - 8113 F ax: (+31) (40) 247 - 8190 and Olaf Wittic h Olaf Wittich, T e chnische Univ ersiteit Eindhoven and EURANDOM, P.O. Box 513 , 5600 MB, Eindhoven, The Netherlands e-mail: o.witt ich@tue.nl Phone: (+31) (40) 247 - 2499 Abstract: W e propose a no vel probabilistic method for dete ction of ob- jects in noisy images. The method uses results from percolation and random graph theories. W e present an algorithm that all o ws to detect ob j ects of un- kno wn shap es in the presence of random noise. Our pr o cedure substant ially differs from wa velets-based algorithms. The algorithm has linear complex- ity and exponen tial accuracy and is appropriate for real-time systems. W e prov e results on consistency and algorithmic complexit y of our pro cedure. Keywords and phrases: Image analysis, signal detection, image recon- struction, percolation, noisy image. 1. In tro duction In this paper, we prop ose a new efficien t tec hnique for quic k detection of ob jects in noisy imag es. Our approach us es mathema tical p erc o lation theory . Detection of o b jects in noisy ima ges is the mo st basic pr o blem of image analy- sis. Indeed, when one lo oks at a noisy image, the first question to ask is whether there is any ob ject at all. This is also a primary question of interest in such diverse fields as, for example, cancer detection (Ricci-Vitiani et al. (20 07)), au- tomated urban analysis (Negri e t al. (2006)), detection of cracks in buried pip es (Sinha and Fieguth (200 6)), a nd other p o ssible applications in astr o nomy , elec- tron microscopy and neurolo gy . Moreov er, if ther e is just a random noise in the picture, it do esn’t make sens e to run computationa lly intensiv e pro c edures for image reco nstruction for this particular picture. Surpr is ingly , the v ast ma jority of image analys is metho ds, b o th in statistics and in engineer ing, skip this s tage and sta rt immediately with image reconstruction. The cr ucial difference o f our metho d is that we do not imp ose any shape or smo othness a ssumptions on the b oundary of the ob ject. This p ermits the ∗ Corresp onding author. 1 imsart-g eneric ver. 2007/04/13 file: Randomized_Algo rithms_and_Percolation_Square_3.tex date: September 12, 2018 M. L angovoy and O. Witt ich/Dete c tion in noisy images and p er c olation. 2 detection o f nonsmoo th, irregula r or disco nnected ob jects in noisy images, under very mild assumptions on the ob ject’s interior. This is e s pe c ia lly suitable, for example, if one ha s to detect a highly irregular non-conv ex o b ject in a noisy image. Although our detectio n pro c e dur e works for regular ima ges as w ell, it is precisely the class of ir regular images with unknown s ha p e wher e o ur metho d can be very adv antageous. Many mo dern metho ds of o b ject detection, especia lly the ones that ar e used by pra ctitioners in medical image analysis requir e to p erform a t leas t a pr elim- inary reconstruction of the image in or der for an ob ject to b e detected. This usually makes suc h metho ds difficult for a rigor ous analysis of p erfo rmance and for er ror con trol. Our approach is free from this drawback. Even though so me pap ers w ork with a s imila r s etup (se e Aria s-Castro et al. (2005)), b o th our ap- proach and our re s ults differ substantially from this and other studies of the sub ject. W e also do not use any wa velet-based techniques in the pres e n t pap er. W e view the ob ject detection problem as a nonpar ametric hypothesis testing problem within the class of dis crete s tatistical inverse pr o blems. In this pap er , we prop os e an algo rithmic so lution for this nonparametric hy- po thesis testing problem. W e prov e that o ur algorithm has linear complexity in terms of the num b er of pix e ls on the screen, and this pro cedure is not o nly asymptotically consis tent, but o n top of that has accura cy that grows expo ne n- tially with the ”num b er of pixels” in the ob ject of detection. The alg o rithm has a built-in data-dr iven stopping r ule, so there is no need in human as sistance to stop the a lgorithm at an appropriate s tep. In this pap er , we assume that the orig inal image is black-and-white and that the noisy image is grayscale. While our fo cusing o n gr ayscale imag es could have bee n a serio us limitation in ca s e of ima ge r econstruction, it essentially do e s not affect the sco pe of applications in the case of ob ject detection. Indeed, in the v ast ma jority of problems, an ob ject that has to b e detected either has (on the picture under ana lysis) a color that differs from the bac kgr ound colours (for example, in ro a ds detection), or ha s the s ame colour but of a very differ ent int ensity , or a t leas t a n ob ject has a relatively thick b oundary that differs in colour from the background. Moreover, in pr actical applications one often has some pr ior informa tion ab o ut colour s of both the ob ject o f interest and of the background. When this is the case, the method o f the presen t pap er is applica ble after simple res caling of colour v alues. The pap er is organize d as follows. In Sec tion 2 we describ e so me general assumptions that o ne makes in s tatistical pro ces sing of digita l imag e s. The s ta- tistical mo del itself is describ ed in details in Section 3. Suitable thr e sholding for noisy image s is crucial in our metho d and is dev elop ed in Section 4. Our algo- rithm for ob ject detection is presented in Sectio n 5. T heo rem 1 is the ma in res ult ab out consistency and co mputational c omplexity o f the algorithm. Section 6 is devoted to the pro of of the main theorem. 2. Basic framew ork In this sectio n we dis cuss s ome natural and very basic assumptions that we impo se on our mo del. Suppo se w e h av e an analogo us tw o -dimensional image. F or numerical or graphical pro cess ing of images on computers, the image alwa ys has to be dis- cretized. This is achieved via a pixelization pro cedure. imsart-g eneric ver. 2007/04/13 file: Randomized_Algo rithms_and_Percolation_Square_3.tex date: September 12, 201 8 M. La ngovoy and O. Witt ich/Dete c tion in noisy images and p er c olation. 3 A typical example of pixelization pro cedure is as follows. Consider an N × N grid on the squa re containing the imag e, and c o lor black those and only those pixels for whose the pixel’s interior has common p oints with the original image. The result is called a pixelized picture. After certain pro cedure o f pixelization, ea ch pixel gets a rea l num b er attached to it. This assumption means that o ur o nly data av ailable a re given as an array of N 2 real num b ers { Y ij } N i,j =1 . In order to perfor m s ta tistical imag e analysis, we will use only these N 2 nu mbers plus our mo del assumptions. This lea ds us to the following basis assumption. h A1 i Assume that we hav e a square s creen of N × N pix e ls, where we observe pixelized images. W e a ssume that we are getting o ur information abo ut the image from this scr een alone. In the present paper we are in terested in detection of ob jects that hav e a known colo ur. And this colour is differen t from the colo ur of the bac kgro und. Mathematically , this can b e roug hly for m ulated as follows. h A2 i The true (non-nois y) image s ar e black-and-white. Indeed, we are free to a ssume that all the pixels that belong to the meaningful ob ject within the digitalized image have the v alue 1 attached to them. W e ca n call this v alue a black c olour . Additionally , assume that the v alue 0 is attached to thos e and only those pixels that do not b elo ng to the o b ject in th e true image. If the num ber 0 is a ttached to the pixel, we ca ll this pixel white . In this paper we always assume that we observe a noisy image. The nois e itself can b e caus e d, for example, by channel defects or other tra nsmission erro r s, distortions, e tc. Medical scans pr ovide a classic example of noisy images: the scan is alwa ys made indirectly , through the bo dy . In astronomy , one also o bserves noisy imag e s: there ar e optical a nd technological er rors, atmos pher e conditions, etc. The o bserved v a lues o n pixels could b e different from 0 and 1. T his means that we will actually always ha ve a g r eyscale image in the b eginning of our analysis. h A3 i On ea ch pixel we hav e random nois e that has the distribution function F , where F has mean 0 and known v ariance σ > 0; the noise at each pix el is completely independent from nois es o n other pixels. An imp o rtant specia l case is when the noise is normally distributed, i.e. F = N (0 , σ ), where N (0 , σ ) stands for the normal distribution with mean 0 and known v a riance σ > 0. How ever, in general, the nois e do esn’t need to be smooth, symmetric or even contin uous. R emark 1 . W e limit ourselves to tw o - dimensional images only . How ever, our metho d mak es it possible to analyze also n − dimensional imag es, for ea ch n ≥ 1. R emark 2 . It doesn’t really matter if the screen is squar e or rectang ular. W e consider a squa re scre e n only to simplify o ur notation. R emark 3 . It is behind the scop e of the present pa p e r to discuss v arious wa ys to pixelize real-world analogous ima ges. One po ssible metho d of pixelization is often used in literature (see Arias-Castro et al. (2005) and related r eferences). imsart-g eneric ver. 2007/04/13 file: Randomized_Algo rithms_and_Percolation_Square_3.tex date: September 12, 201 8 M. L angovoy and O. Witt ich/Dete c tion in noisy images and p er c olation. 4 The way o f pixelizatio n plays an impo rtant role mostly for those problems in image ana lysis where one needs to give a symptotic estimates f or the b oundary of the ob ject. F or example, consider pixelizing a plane curve. Then o ne ca n cons ider also colouring blac k those pixels that hav e at lea st 4 common po int s with the curve (not necessa r ily 4 interior p oints), etc. How ever, for simpler pro blems like image detection (or crude estimation of o b ject’s shape or in terior ) it is often not impo rtant how many pixels y ou colour at the bounda ry of your ima ge. R emark 4 . Our metho d works not o nly for normal or i.i.d. noise. W e hav e chosen this t y p e of no ise in o rder to ac hieve relatively simple and explicit results. The metho d itself allows the treatment of more co mplicated situations, s uch as singular noise, discrete noise, pixel colo ur flipping, different noise in different screen areas, a nd depe nden t noise. 3. Statistical Mo del Now we are able to formulate the mo del more for mally . W e hav e an N × N array of obser v ations, i.e. w e obser ve N 2 real num b er s { Y ij } N i,j =1 . Denote the true v alue on the pixel ( i , j ), 1 ≤ i, j ≤ N , b y I m ij , and the corresp onding noise by σ ε ij . Therefore, by our mo del assumptions, Y ij = I m ij + σ ε ij , (1) where 1 ≤ i , j ≤ N , σ > 0 and, in accor dance with a ssumption h A 2 i , I m ij =  1 , if ( i , j ) be longs to the ob ject; 0 , if ( i , j ) do es no t belong to the o b ject. (2) T o stress the dependence on the no ise level σ , w e write assumption h A 3 i in the following wa y: ε ij ∼ F, E ε ij = 0 , V ar ε ij = 1 . (3) The noise here doesn’t need to be smo o th, symmetric or even contin uous. More- ov e r, all the r esults b elow ar e ea sily transfer r ed to the even mor e gener al c ase when the no ise has ar bitr ary but known distribution function F gen ; it is not necessary that the no is e has mea n 0 and finite v ar iance. The only adjustment to be made is to replace in a ll the statemen ts quan tities of the form F  · /σ  by the quantities F gen ( · ). The Algorithm 1 b elow and the main Theor em 1 ar e v alid without any changes for a general noise distribution F gen satisfying (8) and (9). Now we can pr o ceed to preliminary quantitativ e estimates. If a pixel ( i, j ) is white in the o riginal imag e, let us denote the corr esp onding pro ba bilit y distr ibu- tion of Y ij by P 0 . F or a bla ck pixel ( i , j ) we denote the c o rresp onding distribution of Y ij by P 1 . W e are free to omit dep endency of P 0 and P 1 on i and j in our notation, since a ll the noises are independent and iden tically distributed. imsart-g eneric ver. 2007/04/13 file: Randomized_Algo rithms_and_Percolation_Square_3.tex date: September 12, 201 8 M. La ngovoy and O. Witt ich/Dete c tion in noisy images and p er c olation. 5 Lemma 1. Su pp ose pixel ( i, j ) has white c olour in the original image. Then for al l y ∈ R : P 0 ( Y ij ≥ y ) = 1 − F  y σ  , (4) wher e F is the di stribution function of t he standar dize d n oise. Pr o of. (Lemma 1): By (3), P 0 ( Y ij ≥ y ) = 1 − P ( σ ε ij < y ) = 1 − F  y σ  . Lemma 2. Supp ose pixel ( i, j ) has black c olour in the original image. Then for al l y ∈ R : P 1 ( Y ij ≤ y ) = F  y − 1 σ  . (5) Pr o of. (Lemma 2): By (3) again, w e have P 1 ( Y ij ≤ y ) = P (1 + σ ε ij ≤ y ) = P ( σ ε ij ≤ y − 1) = F  y − 1 σ  . 4. Thresholding and Graphs of Images Now we a re ready to describe one o f the main ingredients of our method: the thr esholding . The idea of the thresholding is as follows: in the noisy grayscale image { Y ij } N i,j =1 , we pick some pixels that lo ok as if their real colour was blac k. Then w e colo ur a ll those pixels blac k, irresp ectively of the exact v alue of g r ey that was observed on them. W e take int o a ccount the int ensity of gr ey observed at those pixels only once, in the b eg inning of our pro cedures. The idea is to think that some pixel ”seems to hav e a bla ck co lour” when it is not v ery likely to obta in the observed grey v alue when adding a ”reasonable” no ise to a white pixel. W e colour white all the pixels that weren’t coloured black at the previous step. A t the end of this pro cedur e, w e would have a transformed v ector of 0’s and 1’s, call it { Y i,j } N i,j =1 . W e will b e a ble to analyse this transfo r med picture by using certain r esults from the mathematical theory o f perco lation. This is the main g oal of the present paper . But firs t we ha ve to give more details ab out the thresholding pr o cedure. Let us fix, for each N , a rea l num b e r α 0 ( N ) > 0, α 0 ( N ) ≤ 1, such that there exists θ ( N ) ∈ R satisfying the following condition: P 0 ( Y ij ≥ θ ( N ) ) ≤ α 0 ( N ) . (6) imsart-g eneric ver. 2007/04/13 file: Randomized_Algo rithms_and_Percolation_Square_3.tex date: September 12, 201 8 M. L angovoy and O. Witt ich/Dete c tion in noisy images and p er c olation. 6 Lemma 3 . Assume that (6) is satisfie d for some θ ( N ) ∈ R . Then for the smal lest p ossible θ ( N ) satisfying (6) it holds that F  θ ( N ) σ  = 1 − α 0 ( N ) . (7) Pr o of. (Lemma 3): Obvious by Lemma 1. In this pap er w e will a lwa y s pic k α 0 ( N ) ≡ α 0 for a ll N ∈ N , for s ome constant α 0 > 0. But we will need to hav e v a rying α 0 ( · ) for our future r esearch. W e ar e prepar ed to describ e our thresho lding principle for mally . Let p site c be the critic al pr ob ability for site p er c olation on Z 2 (see Gr immett (19 9 9) for definitions). As a first step, we transfo r m the observed noisy image { Y i,j } N i,j =1 in the following wa y: for all 1 ≤ i , j ≤ N , 1. If Y ij ≥ θ ( N ), s et Y ij := 1 (i.e., in the transformed picture the c o rre- sp onding pixel is colo ur ed black). 2. If Y ij < θ ( N ), s et Y ij := 0 (i.e., in the transformed picture the c o rre- sp onding pixel is colo ur ed white). Definition 1. The a b ove tra ns formation is called thr esholding at the level θ ( N ). The resulting vector { Y i,j } N i,j =1 of N 2 v alues (0’s and 1’s) is called a t hr esholde d pictur e . Suppo se for a moment that we ar e given the or iginal black and white image without noise. One can think of pixels fro m the original picture as of vertices of a planar graph. F urther more, let us colour these N 2 vertices with the same colours as the corres po nding pixe ls o f the o riginal ima ge. W e obtain a graph G with N 2 black or white vertices and (so far) no edges. W e add edg es to G in the following wa y . If any tw o black vertices ar e neigh- bo urs (i.e. the corr esp onding pixels hav e a common side), we connect these tw o vertices with a black edge. If a ny tw o white vertices ar e neighbo urs, we co nnect them with a white edge. W e will not add any e dges betw een non- neighbouring po int s, and w e will not connect v ertices of different co lours to each other. Finally , w e s e e that it is pos sible to view our blac k and white pixelized picture as a collec tio n of black and white ”clusters” on the very sp ecific planar graph (a square N × N subset of the Z 2 lattice). Definition 2. W e ca ll gr aph G the gr aph of the (pur e) pictur e . This is a very sp ecial planar graph, so ther e a re many efficient algorithms to work with blac k and white components of the gra ph. Poten tially , they could be used to efficient ly pro cess the picture. How ever, the ab ov e repres e n tation of the imag e as a graph is lost when one considers no isy images: because of the presence of random noise, w e get man y gray pixels . So, the a b ove cons tr uction do esn’t make sense anymore. W e ov ercome this obstac le with the help of the ab ov e thres holding pro cedur e. W e make θ ( N ) − thres ho lding of the noisy image { Y i,j } N i,j =1 as in Definition 1, but with a very specia l v alue of θ ( N ). O ur goal is to choo se θ ( N ) (a nd corres p o nding α 0 ( N ), see (6)) such tha t: imsart-g eneric ver. 2007/04/13 file: Randomized_Algo rithms_and_Percolation_Square_3.tex date: September 12, 201 8 M. La ngovoy and O. Witt ich/Dete c tion in noisy images and p er c olation. 7 1 − F  θ ( N ) σ  < p site c , (8) p site c < 1 − F  θ ( N ) − 1 σ  , (9) where p site c is the critica l pro ba bilit y for site per colation o n Z 2 (see Grimmett (1999), Kesten (1982)). In case if both (8 ) and (9) are satisfied, what do w e get? After applying the θ ( N ) − thresholding on the no is y picture { Y i,j } N i,j =1 , we obtained a (rando m) black-and-white image { Y i,j } N i,j =1 . Let G N be the graph of this image, as in Definition 2. Since G N is a random, we actually o bserve the so -called site p er c olation o n black vertices within the subset of Z 2 . F ro m this p oint, we can us e results fro m per colation theory to predict formation of black and white clusters on G N , as well as to estimate the n umber of clusters a nd their s iz es a nd shap es. Relations (8) and (9) are crucial here. T o explain this mor e formally , let us split the se t of vertices V N of the gra ph G N int o to groups: V N = V im N ∪ V out N , where V im N ∩ V out N = ∅ , and V im N consists of those and only thos e v ertices that co rresp ond to pixels b elong ing to the orig ina l ob ject, while V out N is left for the pixels from the ba ckground. Denote G im N the subgraph of G N with vertex set V im N , and denote G out N the subgraph of G N with vertex set V out N . If (8 ) and (9 ) are satisfied, w e will obser ve a so-called sup er critic al p er c olation of black clusters on G im N , and a sub critic al p erco lation of black clusters on G out N . Without g oing into m uch details on p erco la tion theory (the necess a ry introduc- tion can b e found in Grimmett (19 9 9) or Kesten (1982)), we men tio n that there will b e a high pr obability of forming relatively la rge black cluster s o n G im N , but there will be o nly little and sca rce black cluster s on G out N . The difference b etw een the two regio ns will be striking , and this is the main comp onent in our imag e analysis metho d. In this pa p er , mathematical p erc olation theory will b e used to der ive quan ti- tative results on behaviour o f clusters for b oth cases. W e will apply those results to build efficien t randomized algorithms that will b e able to detect and estimate the ob ject { I m i,j } N i,j =1 using the difference in p ercola tion phases on G im N and G out N . If the noise level σ is not too large, then (8) and (9) are s atisfied for some θ ( N ) ∈ (0 , 1). Indeed, one simply has to pick θ ( N ) close enough to 1. On the other hand, if σ is r elatively la r ge, it may happ en that (8) and (9) cannot b oth be satis fie d at the s a me time. Definition 3. In the framework defined b y relations (1)-(2) a nd assumptions h A 1 i - h A 3 i , we s ay that the no ise lev el σ is smal l enough (or 1-smal l ), if the system o f inequalities (8) and (9) is satisfied for some θ ( N ) ∈ R , fo r a ll N ∈ N . A very imp ortant practica l iss ue is that of choosing an optimal thr eshold v alue θ . F ro m a purely theoretica l p oint of view, this is not a big issue: o nce (8) and (9) holds for some θ , it is g uaranteed that after θ − thresho lding w e will observe qualitatively different b ehaviour of bla ck and white clusters in or outside of the true ob ject. W e will make use of this in what follo ws. How ever, for practical computations, especially for moderate v alues of N , the imsart-g eneric ver. 2007/04/13 file: Randomized_Algo rithms_and_Percolation_Square_3.tex date: September 12, 201 8 M. L angovoy and O. Witt ich/Dete c tion in noisy images and p er c olation. 8 v alue of θ is imp or tant. Since the goal is to make pe r colations on V im N and V out N lo ok as differen t as p os sible, one has to make the cor resp onding p ercolatio n probabilities for bla ck co lour, namely , 1 − F  θ ( N ) σ  and 1 − F  θ ( N ) − 1 σ  , as differen t as po ssible b oth from each other and from the critical probability p site c . There can b e several reasonable wa ys for c ho osing a suitable thr eshold. F or example, we can pro po se to c ho ose θ ( N ) a s a maximizer of the following function:  1 − F  θ ( N ) σ  − p site c  2 +  1 − F  θ ( N ) − 1 σ  − p site c  2 , (10) provided tha t (8) a nd (9) ho lds. Alterna tively , we can pr op ose to use a maximizer of sig n  1 − F  θ ( N ) − 1 σ  − p site c  + sig n  p site c − 1 + F  θ ( N ) σ  . (11) 5. Ob ject detection W e either obser ve a blank white scree n with accidental noise or there is a n actual ob ject in the blurr ed picture. In this section, we prop ose a n algorithm to make a decisio n o n whic h of the t wo poss ibilities is true. This algorithm is a statistical testing pro cedure. It is desig ned to so lve the questio n of testing H 0 : I ij = 0 for all 1 ≤ i , j ≤ N versus H 1 : I ij = 1 for some i, j . Let us c ho ose α ( N ) ∈ (0 , 1) - the pr ob ability of false dete ction o f an ob ject. More formally , α ( N ) is the ma ximal probability that the a lgorithm finishes its work with the decision that there w as an ob ject in the picture, while in fact there was just noise. In statistical terminolo gy , α ( N ) is the probability of an error of the first kind. W e allow α to dep end on N ; α ( N ) is connected with complexity (and ex- pec ted working time) of our ra ndomized a lgorithm. Since in our metho d it is crucial to observe some kind of p ercolatio n in the picture (at least within the image), the image has to b e ” not too small” in order to b e detectable b y the algorithm: o ne can’t o bserve a nythin g p er colation-a like on just a few pixe ls . W e will use perc olation theory to determine ho w ”large” precisely the ob ject has to be in or der to b e detectable. Some size assumption has to b e present in any detection pr oblem: for example, it is hop eles s to detect a single p oint ob ject on a v ery large screen even in the case of a mode r ate noise. F or an e a sy star t, we make the following (wa y to o strong) la rgeness assump- tions ab o ut the image: h D1 i Assume that the imag e { ( i, j ) | 1 ≤ i, j ≤ N , I ij = 1 } con tains a completely black square with the side of size at least ϕ im ( N ) pixels, where lim N →∞ log 1 α ( N ) ϕ im ( N ) = 0 . (12) imsart-g eneric ver. 2007/04/13 file: Randomized_Algo rithms_and_Percolation_Square_3.tex date: September 12, 201 8 M. La ngovoy and O. Witt ich/Dete c tion in noisy images and p er c olation. 9 h D2 i lim N →∞ ϕ im ( N ) log N = ∞ . (13) F urthermo re, we as s ume the obvious c onsistency assumption ϕ im ( N ) ≤ N . (14) Assumptions h D 1 i and h D 2 i are sufficient c onditions for our algorithm to work. They are wa y to o strong for our purpos es. It is poss ible to relax (13) a nd to replace a squa re in h D 1 i b y a triang le - shap ed figure. Although the a b ove tw o co nditions a re o f a symptotic character , most o f o ur estimates below are v alid for finite N as w ell. Nev ertheless , it is imp or tant to remark here tha t a symptotic results for N → ∞ a lso have int eres ting practical consequences. More specifica lly , assume tha t ph ysica lly we always hav e scr e ens of a fixed size, but the reso lution N 2 of our cameras can grow unboundedly . When N tends to infinity , w e see that the same physical ob ject that has, say , 1mm in width a nd in length, contains more and more pixe ls o n the pixelized image. Therefor e, for high-resolution pictures, our alg orithm could detect fine structures (lik e nerves etc.) that are not directly visible b y a human eye. Now w e ar e ready to for mu late o ur D et e ction Algo rithm . Fix the false detec- tion rate α ( N ) be fo re r unning the a lg orithm. Algorithm 1 (Detection). • Step 0. Find an optimal θ ( N ). • Step 1. Perform θ ( N ) − thresho lding of the noisy picture { Y i,j } N i,j =1 . • Step 2. Until {{ Black cluster of size ϕ im ( N ) is found } or { all blac k cluster s a r e found } } , Run depth-fir st sea rch (T arjan (19 72)) on the graph G N of the θ ( N ) − thresholded picture { Y i,j } N i,j =1 • Step 3. If a blac k cluster o f size ϕ im ( N ) was fo und, rep or t that an o b ject was detected • Step 4. If no blac k cluster was la r ger than ϕ im ( N ), rep or t that there is no ob ject. A t Step 2 o ur alg orithm finds and stores not only sizes of black clusters, but also co ordina tes o f pixels constituting each cluster. W e remind tha t θ ( N ) is defined as in (6), G N and { Y i,j } N i,j =1 were defined in Sectio n 1, and ϕ im ( N ) is any function sa tisfying (12). The depth-first sea rch algorithm is a standard pro cedure used for searching connected comp onents on g raphs. This pro cedur e imsart-g eneric ver. 2007/04/13 file: Randomized_Algo rithms_and_Percolation_Square_3.tex date: September 12, 201 8 M. L angovoy and O. Witt ich/Dete c tion in noisy images and p er c olation. 10 is a deterministic algorithm. The detailed description a nd rig orous complexity analysis can b e fo und in T ar jan (1972), or in the classic b o o k Aho et al. (197 5), Chapter 5. Let us prove that Algorithm 1 works, and determine its complexity . Theorem 1. L et σ b e 1-smal l. Supp ose assumptions h D 1 i and h D 2 i ar e satis- fie d. Then 1. Algorithm 1 fi nishes its work in O ( N 2 ) steps, i.e. is line ar. 2. If ther e was an obje ct in the picture , Algorithm 1 dete cts it with pr ob ability at le ast (1 − exp( − C 1 ( σ ) ϕ im ( N ))) . 3. The pr ob ability of false de te ction do esn ’t exc e e d min { α ( N ) , exp( − C 2 ( σ ) ϕ im ( N )) } for al l N > N ( σ ) . The c onstants C 1 > 0 , C 2 > 0 and N ( σ ) ∈ N dep end only on σ . R emark 5 . Dep endence o n σ implicitly means dep endence o n θ ( N ) as well, but this do esn’t sp oil Theo rem 1. Remember that we can c o nsider θ ( N ) to b e a function of σ in v iew of our comments b efor e (10 ) a nd (11). Theorem 1 means tha t Algor ithm 1 is of quick est po ssible or de r : it is line ar in the input s ize. It is difficult to think o f a n algorithm working q uick er in this problem. Indeed, if the ima ge is v ery small and loca ted in a n unknown place on the s creen, or if there is no image a t all, then any algo rithm solving the detection problem will hav e to at lea s t upload informa tion ab o ut O ( N 2 ) pixels, i.e. under general ass umptions of Theore m 1, a ny detection alg orithm will ha ve at least linear complexity . Another imp or tant point is that Algorithm 1 is not o nly co ns istent , but that it ha s ex p onential r ate o f a ccuracy . 6. Pro ofs This section is devoted to pro vide complete pro ofs of the above results. So me crucial estimates from per c olation theory are also pr e s ented for the reader ’s conv enience. Pr o of. (Theorem 1): Part I. Fir st w e prove the complexity result. Finding a suitable (approximate, within a pr edefined error ) θ fro m (10 ) or (11) takes a constant n umber of ope r - ations. See, for example, Krylov e t al. (1976). The θ ( N ) − thresho lding giv es us { Y i,j } N i,j =1 and G N in O ( N 2 ) oper ations. This finishes the analysis of Step 1. As for Step 2, it is known (see, fo r exa mple, Aho et al. (1975), Chapter 5, or T arjan (19 7 2)) that the standard depth-fir st sea rch finishes its work also in O ( N 2 ) steps. It tak es not mo re than O ( N 2 ) op era tions to save p ositions o f all pixels in all cluster s to the memor y , since one has no more than N 2 po sitions and clusters. This completes ana lysis of Step 2 and shows that Algorithm 1 is linear in the size of input data. imsart-g eneric ver. 2007/04/13 file: Randomized_Algo rithms_and_Percolation_Square_3.tex date: September 12, 201 8 M. La ngovoy and O. Witt ich/Dete c tion in noisy images and p er c olation. 11 Part I I. Now we prove the b ound on the probability of false detection. Deno te p out ( N ) := 1 − F  θ ( N ) σ  , (15) a probability of erroneously marking a white pixel outside of the imag e as black. Under assumptions of Theorem 1, p out ( N ) < p site c . W e prov e the following a dditional theor em: Theorem 2. Supp ose that 0 < p out ( N ) < p site c . Ther e exists a c ons t ant C 3 = C 3 ( p out ( N )) > 0 such that P p out ( N ) ( F N ( n )) ≤ ex p( − n C 3 ( p out ( N ))) , for al l n ≥ ϕ im ( N ) . (16) Her e F N ( n ) is the event that ther e is an erroneo usly marked black cluster of size gr e ater or e qual n , lying in the squar e of size N × N c orr esp onding to the scr e en. (An err one ously marke d black clu s t er is a black cluster on G N such that each of the pixels in the cluster was wr ongly c olour e d bla ck after the θ − thr esholding.) Before pro ving this result, we state the following theore m ab out sub critical site perc olation. Theorem 3 . (Aize nman-N ewman) Consider site p er c olation with pr ob ability p 0 on Z 2 . Ther e exists a c onstant λ site = λ site ( p 0 ) > 0 such that P p 0 ( | C | ≥ n ) ≤ e − n λ site ( p 0 ) , for al l n ≥ 1 . (17) Her e C is the op en cluster c ontaining the origin. Pr o of. (Theorem 3): See Bollob´ as and Riordan (2006). T o conclude Theor em 2 from Theorem 3, we will use the celebrated FKG inequality (see F ortuin et al. (1971), or Grimmett (1999), Theorem 2.4, p.34; see a lso Grimmett’s b o ok for some explana tion o f the terminology). Theorem 4. If A and B ar e b oth incr e asing (or b oth de cr e asing) events on the same me asur able p air (Ω , F ) , then P ( A ∩ B ) ≥ P ( A ) P ( B ) . Pr o of. (Theorem 2): Denote by C ( i, j ) the lar gest clus ter in the N × N scre e n containing the pixel with co ordinates ( i, j ), a nd by C (0) the larg est bla ck clus ter on the N × N screen con taining 0 . By Theor em 3, for a ll i , j : 1 ≤ i , j ≤ N : P p out ( N ) ( | C (0 ) | ≥ n ) ≤ e − n λ site ( p out ) , (18) P p out ( N ) ( | C ( i, j ) | ≥ n ) ≤ e − n λ site ( p out ) . Obviously , it only he lp ed to inequalities (17) and (18) that we have limited our clusters to only a finite subset instead of the who le lattice Z 2 . On a side note, there is no sy mmetry anymore b etw een arbitra ry points of the N × N finite square; luckily , this do esn’t a ffect the present pro of. imsart-g eneric ver. 2007/04/13 file: Randomized_Algo rithms_and_Percolation_Square_3.tex date: September 12, 201 8 M. L angovoy and O. Witt ich/Dete c tion in noisy images and p er c olation. 12 Since { | C (0) | ≥ n } and { | C ( i, j ) | ≥ n } are increasing events (on the mea- surable pair corres po nding to the standard random-g r aph mo del on G N ), w e hav e that { | C (0 ) | < n } and { | C ( i, j ) | < n } are decr easing ev ents for all i , j . By FKG inequality for decreasing even ts, P p out ( N ) ( | C ( i, j ) | < n fo r a ll i, j, 1 ≤ i, j ≤ N ) ≥ Y Y 1 ≤ i,j ≤ N P p out ( N ) ( | C ( i, j ) | < n ) ≥ (b y (18)) ≥  1 − e − n λ site ( p out )  N 2 . W e denote below by C a b the ” a o ut of b ” binomia l coe fficie n t. It follows that P p out ( N ) ( F N ( n )) = P p out ( N )  ∃ ( i, j ) , 1 ≤ i, j ≤ N : | C ( i , j ) | ≥ n  ≤ 1 −  1 − e − n λ site ( p out )  N 2 = 1 − N 2 X k =0 ( − 1) k C k N 2 e − n λ site ( p out ) k = N 2 X k =1 ( − 1) k − 1 C k N 2 e − n λ site ( p out ) k = N 2 e − n λ site ( p out ) + o  N 2 e − n λ site ( p out )  , bec ause we assumed in (16) tha t n ≥ ϕ im ( N ), and log N = o ( ϕ im ( N )). More- ov e r, w e see immediately that Theorem 2 follows now w ith some C 3 such that 0 < C 3 ( p out ( N )) < λ site ( p out ( N )). The expo nent ial b ound on the pro bability o f fals e detection follows from Theorem 2. Part II I. It rema ins to prove the low er b ound on the pr obability o f true detection. First w e prov e the following theorem: Theorem 5 . Consider site p er c olation on Z 2 lattic e with p er c olation pr ob ability p > p site c . L et A n b e the event that ther e is an op en p ath in the r e ctangle [0 , n ] × [0 , n ] joining some vertex on its left s ide to some vertex on its right side. L et M n b e the maximal numb er of vertex-disjoint o p en left-right cr ossings of the r e ctangle [0 , n ] × [0 , n ] . Then ther e exist c onstant s C 4 = C 4 ( p ) > 0 , C 5 = C 5 ( p ) > 0 , C 6 = C 6 ( p ) > 0 such that P p ( A n ) ≥ 1 − n e − C 4 n , (19) P p ( M n ≤ C 5 n ) ≤ e − C 6 n , (20) and b oth ine qualities holds for all n ≥ 1 . imsart-g eneric ver. 2007/04/13 file: Randomized_Algo rithms_and_Percolation_Square_3.tex date: September 12, 201 8 M. La ngovoy and O. Witt ich/Dete c tion in noisy images and p er c olation. 13 Pr o of. (Theorem 5): One proves this by a sligh t mo dification of the cor resp ond- ing result for bond pe r colation on the sq uare lattice. See pro of of L e mma 11.22 and pp. 294 - 295 in Grimmett (1999). Now supp ose that we have an o b ject in the picture that s atisfies assump- tions of Theorem 1. Consider any ϕ im ( N ) × ϕ im ( N ) squar e in this image. After θ − thresholding of the picture by Algor ithm 1 , w e observe on the selected square site perc olation with probability p im ( N ) := 1 − F  θ ( N ) − 1 σ  > p site c . Then, by (19) o f The o rem 5, there exists C 4 = C 4 ( p im ( N )) such that there will be at le ast one cluster of size no t less than ϕ im ( N ) (for ex a mple, one could take any of the existing left-rig ht crossing s as a pa rt of such cluster ), provided that N is bigger than certain N ( p im ( N )) = N ( σ ); and all that happens with probability at least 1 − n e − C 4 n > 1 − e − C 3 n , for s ome C 3 : 0 < C 3 < C 4 . Theorem 1 is prov ed. Ac knowledgmen ts. The authors would like to thank Laurie Da vies, Remco v an der Ho fstad, Artem Sap o zhniko v , Shota Gugushvili a nd Geurt Jongblo ed for helpful disc us sions. References Alfred V. 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