The output distribution of important LULU-operators

Calculating the output distribution of st a c k filters that are erosion-dilati o n cascades, in particular LU LU -filters R. Anguelo v 1 , P .W. Butler 2 , C.H. Rohw er 2 , M. Wild 2 1 Departmen t of Mathematics and Applied Mathematics, Univ ersity of Pretoria Pretoria 0002 , South Africa 2 Departmen t of Mathematical Sciences, Univ ersit y o f Stellen b osc h Priv ate Bag X1, Matieland 7602 , South Africa Abstract Two pro cedures to compute the output distribution φ S of certain stack filters S (so called er o sion-dilation cascades) are given. One res ts on the disjunctive normal form of S and a lso yie lds the ra nk selection pro babilities. The o ther is based on inclusion- exclusion and e.g. y ields φ S for some imp ortant LU LU - op erator s S . Prop er ties o f φ S can b e use d to characterize smo othing prop er ties of S . 1 In tro d uction The LULU op erators are w ell kn o wn in the nonlinear multiresol u tion analysis of sequences. The notation for th e basic op erators L n and U n , where n ∈ N is a parameter r elated to the windo w size, has giv en rise to the name LULU for the theory of th ese op erators and their comp ositions. Since the time they w ere in tr o duced nearly th irt y ye ar ag o, w hile also b eing u sed in practical problems, they slo wly led to the dev elopment of a new fr amew ork for characte r izing, ev aluating, comparing and designing non lin ear smo others. Th is framew ork is based on concepts lik e idemp otency , co-idemp otency , trend preserv ation, total v ariation p reserv ation, consisten t decomp osition. As opp osed to th e deterministic natur e of the ab o ve pr op erties, the fo cus of this pap er is on prop erties o f the LULU op er ators in the s etting of random s equ ences. More pr ecisely , this setting can b e d escrib ed as follo ws: Supp ose th at X is a bi-infi nite sequence of random v aria b les X i ( i ∈ Z ) whic h are indep enden t and with a common (cum u lativ e) distrib ution function F X ( t ) from L 1 ([0 , 1] , [0 , 1]). Let S b e a smo other. Then we consider the follo w ing t w o questio ns: 1. Find a map φ S : [0 , 1] → [0 , 1] suc h that the common output distribution F S X ( t ) of ( S X ) i ( i ∈ Z ) equals F S X = φ S ◦ F X . The function φ S is also called d i stribution tr ansfer . 2. Characterize the smo othing effect an op erator S h as on a random sequence X in terms of the prop erties of the common distribution o f ( S X ) i ( i ∈ Z ). 1 With regard to the first question we pr esen t a new tec h n ique which one ma y call ”expansion calculus” which uses a shorthand notation for the probab ility of comp osite ev ent s and a set of rules for manipulation. Usin g this tec hnique w e pro vide new elegan t pr o ofs of the earlier results in [1 ] for the distribution transfer of the operators L n U n and (dually) U n L n . The p o wer of this approac h is further demonstrated by deriving the distrib ution transfer maps for the alternating sequen tial filters C n = L n U n L n − 1 U n − 1 ...L 1 U 1 and F n = U n L n U n − 1 L n − 1 ...U 1 L 1 . With regard to the second question, we ma y note that it is reasonable to exp ect that a smo other should reduce the standard deviation of a random sequence. Indeed, for simple distributions (e.g. uniform ) and fi lters with small wind o w size (three p oint a v erage, M 1 , L 1 U 1 , U 1 L 1 ) when the computations can b e carried out a sig nifican t redu ction o f the standard deviati on is observ ed (for the u niform distribution the men tioned filters redu ce the standard deviation resp ectiv ely b y factors of 3, 5/3, 1.293, 1.293 ). Ho we ver, in general, obtaining suc h results is to a large exten t practic ally imp ossible d ue to the tec hnical co mp lexit y particularly w hen nonlinear filters are concerned. In this p ap er we pr op ose a new ∗ concept of r obustness which c haracterizes the probabilit y of the o ccurr ence of outliers rather then considering the standard deviation. Upp er robu stness characte r izes the probability of p ositiv e outliers while the lo w er r obustness c haracterizes the p robabilit y of n egativ e outliers. In general, th e higher the ord er of robustness of a smo other the lo w er the probabilit y of o ccur rence of outliers in the output sequence. In te rms of this concept it is ea sy to c haracterize a smo other giv en its distribution transfer function. The pap er is stru ctured as follo ws. In the next section we giv e th e defin itions of the LULU op erators with some fu ndamenta l pr op erties. The concept of robustness is defined and studied in Section 3. In Section 4 we sho w h ow the inclusion-exclusion p rinciple helps to obtain the distribution tr an s fer fun ction of erosion-dilation cascades, a kind of op erator frequently used in Mathematica l Morph ology . Th is metho d is considerably refined in Section 5 wh ere it is a pplied to LULU-op erators. They are particular cases of su c h cascades. F ormulas for the ma j or LULU- op erators are obtained explicitly or recursivel y . Using these r esults the robustness of these op erators is also analyzed. Sectio n 6 prop oses to sub stitute inclusion-exclusion b y some nov el principle of exclusion whic h excels for erosion-dilation cascades that don’t allo w the refinemen ts of inclusion-excl usion p ossible f or LU LU -op erators. 2 The basics of the LU LU theo ry Giv en a b i-infinite sequence x = ( x i ) i ∈ Z and n ∈ N the b asic LULU op erators L n and U n are defined a s follo ws ( L n x ) i := ( x i − n ∧ x i − n +1 ∧ · · · ∧ x i ) ∨ ( x i − n +1 ∧ · · · ∧ x i +1 ) ∨ · · · ∨ ( x i ∧ · · · ∧ x i + n ) (1) ( U n x ) i := ( x i − n ∨ x i − n +1 ∨ · · · ∨ x i ) ∧ ( x i − n +1 ∨ · · · ∨ x i +1 ) ∧ · · · ∧ ( x i ∨ · · · ∨ x i + n ) (2) where α ∧ β := min( α, β ), and α ∨ β := max( α, β ) for all α, β ∈ R . C en tral to the theory is the co n cept of separator , whic h w e define b elo w. F or ev ery a ∈ Z the op erator E a : R Z → R Z giv en b y ( E a x ) i = x i + a , i ∈ Z , is called a shift op erator . ∗ Related concepts of robu stness exist. W e touch up on one of them in Section 6. 2 Definition 1 An op er ator S : R Z → R Z is c al le d a separator if (i) S ◦ E a = E a ◦ S, a ∈ Z ; (horizonta l shift invarianc e) (ii) P ( f + c ) = P ( f ) + c, f , c ∈ R Z ; c - c onstant function (vertic al shift invarianc e); (iii) P ( αf ) = αP ( f ) , α ∈ R , α ≥ 0 , f ∈ R Z ; (sc ale invarianc e) (iv) P ◦ P = P ; (Idemp o tenc e) (v) ( id − P ) ◦ ( id − P ) = id − P . (Co-idemp ot e nc e) The fir st t wo axioms in Definition 1 and p artially the third one w ere first introdu ced as requir ed prop erties of nonlinear smo others b y Mallo ws, [4 ]. Roh wer further made the concept of a smo other more precise by u sing the prop erties (i)–(iii) as a definition of this concept. Th e axiom (iv) is an essen tial requirement for what is called a morpholo gic al filter , [11], [12], [13]. In fact, a morphological filter is exactly an increasing op erator which satisfies (iv). Th e co-idemp otence axiom (v) in Definition 1 w as introdu ced b y Rohw er in [8], where it is also sho wn that it is an essen tial requirement for op erators extracting signal f rom a sequence. More p recisely , axioms (iv) and (v) provide for consisten t separation of noise from s ignal in the follo w ing sense: Ha ving extracted a signal S x from a sequence x , the add itiv e residu al ( I − S ) x , the noise, should con tain no signal left, that is S ◦ ( I − S ) = 0. Similarly , the signal S x should con tain no noise, that is ( I − S ) ◦ S = 0. I t w as sho w n in [8] that L n , U n and their comp ositions L n U n , U n L n are separators. The smo othin g effect of L n on an input sequence is the remo v al of picks, while the smo othing effect of U n is th e r emo v al of p its. The comp osite effect of the tw o LU - op er ators L n U n and U n L n is that the output sequen ce con tains neither picks nor p its wh ich will fit in the win do w of th e op erators. These are the so called n -monotone sequences, [8]. Let u s recall that a sequence x is n -monotone if an y s ubsequence of n + 1 consecutive elemen ts is monotone. F or v arious tec hn ical reasons th e analysis is t ypically restricted to th e set M 1 of absolutely summab le sequences. Let M n denote the set of all sequences x ∈ M 1 whic h are n -monoto n e. T hen M n = Rang e ( L n U n ) = Ran g e ( U n L n ) is the set of signals. The p ow er of th e LU -op erators as separators is further d emons trated by their trend preserv ation prop erties. Let us recall, see [8], that an op erator is called neighbor trend preserving if ( S x ) i ≤ ( S x ) i +1 whenev er x i ≤ x i +1 , i ∈ N . An op erator S is fully trend preserving if b oth S and I − S are neighb or trend preserving. The op erators L n , U n and all their comp ositions are fully trend preservin g. With the total v ariation of a sequen ce, T V ( x ) = X i ∈ N | x i − x i +1 | , x ∈ M 1 a generally accepted m easure for the amount of con trast presen t, since it is a semi-norm on M 1 , an y separation ma y only increase the total v ariation. More p recisely , for an y op erator S : M 1 → M 1 w e ha ve T V ( x ) ≤ T V ( S x ) + T V (( I − S ) x ) . (3) 3 All o p erators S that are fully trend preserving ha v e v ariation p reserv ation, in that T V ( x ) = T V ( S x ) + T V (( I − S ) x ) . (4) W e mention these prop erties b ecause they p ro vide b ut f ew of the motiv ation f or stu dying the robustness of op erators, when the p opular medians are optimal in that resp ect. W e inte nd to sho w that s ome LU LU -comp osition are nearly as go o d as th e medians, but hav e su p eriorit y most imp ortant asp ects. An op erator S satisfying prop ert y (4) is called total v ariat ion preserving , [6]. As mentioned already , the LU -op erators are to tal v ariation p reserving. 3 Distribution transfer and degree of robustness of a smo other Supp ose that X is a bi-infinite s equence of random v ariables X i ( i ∈ Z ) whic h are indep end ent and with a common (c u m u lativ e) distrib ution f unction F X . L et S b e a smoother. As stated in the in tro du ction we seek a function φ S : [0 , 1] → [0 , 1], called a distribution tr ansfer fu nction suc h that F S X = φ S ◦ F X (5) is the common distribution of ( S X ) i ( i ∈ Z ). W e should n ote that for an arbitrary smo other the existence of suc h a d istribution trans f er f unction is not ob vious. Ho wev er, for the smo others t ypically considered in nonlinear signal pro cessing (i.e. stac k filters of wh ic h the LULU op erators are particular cases) su c h a function d o es not only exist but it is a p olynomial. F or example, it is sh o wn in [5] that the distribution transfer f u nction of the r ank ed order op erators ( R nk x ) i = the k th smallest v alue of { x i − n , ..., x i + n } is give n b y φ R nk ( p ) = 2 n +1 X j = k  2 n + 1 j  p j (1 − p ) 2 n +1 − j . (6) The p opu lar median smoothers M n , n ∈ N , are p articular cases of the ranked o r der op erators, namely M n = R n,n +1 . Hence we h a v e φ M n ( p ) = 2 n +1 X j = n +1  2 n + 1 j  p j (1 − p ) 2 n +1 − j . (7) Note that in terms of (5) the common distrib u tion fu nction of ( M n X ) i , i ∈ Z , is F M n X ( t ) = 2 n +1 X j = n +1  2 n + 1 j  F j X ( t )(1 − F X ( t )) 2 n +1 − j , t ∈ R . Using that d dz φ M n ( p ) = (2 n + 1)  2 n n  p n (1 − p ) n (8) its densit y is f M n X ( t ) = d dz φ M n ( F X ( t )) f X ( t ) = (2 n + 1)  2 n n  F n X ( t )(1 − F X ( t )) n f X ( t ) 4 where f X ( t ) = d dt F X ( t ), t ∈ R , is the common d ensit y of X i , i ∈ Z . The distribution of the output sequence of the basic smo others L n and U n is derive d in [8]. Equ iv alentl y these results can be form ulated in terms of distrib ution tran s fer. More precisely we ha ve φ L n ( p ) = 1 − ( n + 1)(1 − p ) n +1 + n (1 − p ) n +2 (9) φ U n ( p ) = ( n + 1) p n +1 − n p n +2 (10) A primary aim of the pro cessing of signals throu gh n onlinear smo others is the remo v al of im- pulsive noise. Therefore, th e p o wer of suc h a smo other can b e characte r ized by h o w well it eliminates outliers in a random sequence. The c oncepts of robustness of a smo other introdu ced b elo w are aimed at suc h c haracterization. Definition 2 A smo other S : R Z → R Z is c al le d low er robust of or der r if ther e exists a c onstant α > 0 such that for eve ry b i-infinite se qu e nc e X o f identic al ly distribute d r andom variables X i ( i ∈ Z ) ther e exists t 0 ∈ R such that P ( X i < t ) < ε implies P (( S X ) i < t ) < αε k for al l t < t 0 and ε > 0 . Similarly, a smo other S : R Z → R Z is c al le d upp er robust of or der r i f ther e exists a c onstant α > 0 su c h that for every b i -infinity se quenc e X of identic al ly distribute d r andom variables X i ( i ∈ Z ) ther e exists t 0 ∈ R such that P ( X i > t ) < ε implies P (( S X ) i > t ) < αε k for al l t > t 0 and ε > 0 . A smo other which is b oth lower r obust of or der r and upp er r obust of or der r is c al le d robust of or der k . The reasoning b eh in d these concepts is simp le: If a d istribution density is hea vy tail ed , there is a probabilit y ε that the size of a random v ariable is excessivel y large (la r ger than t ) in absolute v alue. Using a non-linear smo other we w ould aim to restrict this to an acceptable probabilit y αε k that such an excessiv e v alue can app ear in S X , by c ho osing a smo other w ith the order of robustness k . Clearly th ere is a general problem of smo othing: a trade-off to b e m ade b et wee n m aking a smo other more robust, and the (inevitable) damage to the un derlying signal preserv ation. (A smo other clearly cannot create information, b ut only selectiv ely discard it.) This is fun damen tal. There are tw o main reasons f or using one-sided robustness: Firstly , the unr easonable pulses often are only in one direction, as in th e case of ”glin t” in signals r eflected from ob jects with pieces of p erfect reflectors, and there clea r ly are no reflections of negativ e in tensit y p ossible. S econdly , w e ma y c h ose smo others that are not symmetric, as are the LU -op erators, for reasons that are of primary imp ortance. I n this c ase t he robustness is determined fr om the sig n of the impulse. The robu stness of a sm o other can be c h aracterized through its distribu tion transfer fun ction as stated in the theorem b elo w. Theorem 3 L et the smo other S have a distribution tr ansfer function φ S . Then a) S is lower r obust of or der r if and only if φ S ( p ) = O ( p r ) as p → 0 . 5 b) S is upp er r obust of or der r if and only if φ S (1 − p ) − 1 = O ( p r ) as p → 0 . Pro of. Po ints a) an d b) are pr o ve d using similar arguments. Hence we pro ve only a). Let φ S ( p ) = O ( p r ) as p → 0. This means that th ere exists α > 0 and δ > 0 suc h that φ S ( p ) < αp k for all p ∈ [0 , δ ). Let X b e a sequence of identica lly distr ib uted rand om v ariables with c ommon distribution fun ction F X . S ince lim t →−∞ F X ( t ) = 0, there exists t 0 suc h th at F X ( t 0 ) < δ . Let t < t 0 and ε > 0 b e such that P ( X i < t ) < ε . The monotonicit y of F X implies that F X ( t ) ∈ [0 , δ ). Then P (( S X ) i < t ) = F S X ( t ) = φ S ( F X ( t )) < α ( F X ( t )) k < αε k , whic h pro ves th at S is low er r obust of order k . It is easy to see that the argument can b e rev ersed so th at th e sta ted condition is also necessary . In the common case wh en the distr ib ution transfer function is a p olynomial, conditions a) and b) can b e form ulated in a m uc h simpler w a y as giv en in the n ext corolla r y . Corollary 4 L et the distribution tr ansfer f u nction of a smo other S b e a p olynomial φ S . Then a) S is lower r obust of or der r if and only if p = 0 is a r o ot of or der r of φ S . b) S is upp er r obust of or der r if and only if p = 1 is a r o ot of or der r of φ S − 1 . Using the distrib ution transfer functions giv en in (9) and (10) it f ollo ws from Corollary 4 th at U n is lo wer r ob u st of order n + 1 and that L n is u pp er robu st of order n + 1. The robustness of the median filter M n can b e ob tained fr om (7). Obviously p = 0 is a root of order n + 1. F urtherm ore, φ M n (1) = 1. Th en usin g also that p = 1 is a ro ot of ord er n of d dz φ M n , see (8), we ob tain that p = 1 is a ro ot of order n + 1 of φ M n − 1. Therefore, M n is r obust of order n + 1. Clearly with symmetric smo others, in that S ( − x ) = − S ( x ), the concepts of lo w er and up p er robustness are not n eeded, as is the case for example with M n . Ho wev er, w e ha ve to recall in this r egard that the op erators L n , U n and their comp ositions, w h ic h are the primary sub ject of our in v estigatio n , are n ot symmetric. A usefu l feature of the lo we r and upp er r ob u stness is that it can b e indu ced through the p oin t-wise defined partial order b et w een the op erators. Let us recall that giv en the maps A, B : R Z → R Z , the r elation A ≤ B m eans that Ax ≤ B x for all x ∈ R Z . Theorem 5 L et A, B : R Z → R Z b e smo others. If A ≤ B then φ B ≤ φ A . Pro of. Let X b e a sequence of indep enden t r andom v ariables X i ( i ∈ Z ) un iformly distributed on [0 , 1] . Let p ∈ [0 , 1]. If t is suc h that p = F X ( t ) then φ B ( p ) = φ B ( F X ( t )) = F B X ( t ) = P (( B X ) i ≤ t ) ≤ P (( AX ) i ≤ t ) = F AX ( t ) = φ A ( F X ( t )) = φ A ( p ) . 6 As a direct co n sequence of Th eorem 5 and Theorem 3 we obtain the follo wing theorem. Theorem 6 L et A, B : R Z → R Z b e smo others such that A ≤ B . Then a) If A is lower r obust to the or der k , then so is B . b) If B is upp er r obust to the or der k , then so is A . Using Theorem 6 one can d eriv e s tatemen ts ab ou t th e low er robustn ess and the upp er robustn ess of the LU -op erators: U n L n ≤ M n ≤ L n U n (11) Therefore, U n L n inherits the upp er-robustness of M n , while L n U n inherits the lo wer-robustness of M n . More precisel y • U n L n is u pp er robust of order n + 1; (12) • L n U n is low er r ob u st of order n + 1 One may exp ect that, since L n is u pp er robust of order n + 1 and U n is low er robust of ord er n + 1, their comp ositions sh ould b e b oth lo wer and upp er robu st of order n + 1. Ho wev er, as w e will see later, this is n ot the case. The problem is the follo wing. The definition of robustness requires th at the random v ariables in th e sequen ce X are identica lly distributed b ut they are n ot necessarily indep enden t. Ho we ver, the distribution transfer fu nctions φ L n and φ U n are deriv ed under the assump tion of such ind ep endence. Noting that en tr ies in the sequences L n X are not indep end en t, it b ecomes clear that the common distribution of U n L n X cannot b e obtained b y applying φ U n to F L n X . More generally , since the distribution transfer functions are derive d for sequences of indep enden t identic ally distribu ted random v ariables the equalit y φ AB = φ A ◦ φ B do es not hold for arbitrary op erators A and B . Th er efore the order of r obustness of B is not necessarily preserv ed b y the comp osition AB . Observe that another concept of robu stness is int r o duced in [10]. Other than Definition 2 it only applies to stac k filters. Th e co n cept is similar in that it also based on certa in p robabilities (in this case “se lection probabilities”). 4 The ou tput distribution of arbitrary erosion-dilatio n cascades Here we p resen t a metho d for obtaining output distributions of so called erosion-dilation cas- cades (defined b elo w). It essential ly uses th e inclusion-exclusion principle for the pr obabilit y o f sim ultaneous ev en ts. F or con ve n ience we recall this principle b elo w. F or n = 2 the easy pro of will b e gi ven alo ng the wa y . Lemma 7 F or any r andom variables Z 1 , Z 2 · · · Z n it holds that 7 P ( Z 1 , · · · , Z n ≤ t ) = 1 − n X i =1 P ( Z i > t ) + X 1 ≤ i t ) − X 1 ≤ i t ) + · · · + ( − 1) n P ( Z 1 , · · · , Z n > t ) Let us recall th at in the general setting of Mathematic al Morph ology [12] the b asic op erators L n and U n are morphological op ening and closing resp ectiv ely . As su c h they are comp ositions of an erosion and a dilat ion. More precisely , for a sequence x = ( x i ) i ∈ Z w e ha ve ( L n x ) i := ( W n ( V n x )) i ( U n x ) i := ( V n ( W n x )) i where ( V n x ) i := x i − n ∧ x i − n +1 ∧ · · · ∧ x i is an er osion with structural elemen t W = {− n, − n − 1 , ..., 1 , 0 } and ( W n x ) i := x i ∨ x i +1 ∨ · · · ∨ x i + n is a dilation with structur al element W ′ = { 0 , 1 , ..., n } . Generalizing the LU -op erators L n U n and U n L n , call a LU LU - op er ator any comp osition of the basic smo others L n and U n , such as L 3 U 4 L 2 U 1 U 5 . In particular, eac h LU LU -op erator is a comp osition of dilations and erosions, that is, an er osion-dilation c asc ade (EDC). More generally , eac h alternating sequen tial fi lter (ASF), whic h by defin ition [3] is a comp osition of morphological openin gs and closings with structural elemen ts of increasing size, is a EDC w ith the extra pr op ert y of featuring the same n u m b er of erosions and dilations. W e will d emonstrate our metho d on tw o examples of EDC’s - the first in one dimension, the second in t wo dimen sions. This method is considerably refined in the next section. Example 1. Consider S := W 1 V 2 W 3 . It is a cascade of an erosion V 2 and dilations W 1 , W 3 (but not an ASF). T o compute the distribution transfer of S , let X b e a b i-infinite sequence of indep end en t id en tically distribu ted random v ariables X i . Put Y i = 3 _ X ! i , Z i := 2 ^ Y ! i , A i := 1 _ Z ! i . (13) Th us Y , Z , A are again bi-infinite sequences of identica lly distribu ted (though d ep endent) random v ariables. Let t ∈ R and p = F X ( t ). Then φ S ( p ) = F S X ( t ) = F A ( t ) = P ( A 0 ≤ t ) = P ( Z 0 ∨ Z 1 ≤ t ) = P ( Z 0 ≤ t and Z 1 ≤ t ) In order to reduce the Z i ’s to the Y i ’s we switc h all ≤ t to > t by usin g exc lu sion-inclusion (the case n = 2 in Lemma 7): P ( Z 0 , Z 1 ≤ t ) = P ( Z 0 ≤ t ) − P ( Z 0 ≤ t, Z 1 > t ) 8 = P ( Z 0 ≤ t ) − ( P ( Z 1 > t ) − P ( Z 1 , Z 0 > t ) ) = 1 − P ( Z 0 > t ) − P ( Z 1 > t ) + P ( Z 1 , Z 0 > t ) Since our Z i ’s are identic ally distrib u ted we ha ve P ( Z 0 > t ) = P ( Z 1 > t ) and hence φ S ( p ) = 1 − 2 P ( Z 0 > t ) + P ( Z 1 , Z 0 > t ) = 1 − 2 P ( Y − 2 ∧ Y − 1 ∧ Y 0 > t ) + P ( Y − 1 ∧ Y 0 ∧ Y 1 , Y − 2 ∧ Y − 1 ∧ Y 0 > t ) = 1 − 2 P ( Y − 2 , Y − 1 , Y 0 > t ) + P ( Y − 2 , Y − 1 , Y 0 , Y 1 > t ) = 1 − 2 P ( Y 0 , Y 1 , Y 2 > t ) + P ( Y 0 , Y 1 , Y 2 , Y 3 > t ) By the dual of Lemma 7 and b ecause e.g. P ( Y 0 , Y 1 ≤ t ) = P ( Y 1 , Y 2 ≤ t ) = P ( Y 2 , Y 3 ≤ t ) we get φ S ( p ) = 1 − 2 ( 1 − 3 P ( Y 0 ≤ t ) + 2 P ( Y 0 , Y 1 ≤ t ) + P ( Y 0 , Y 2 ≤ t ) − P ( Y 0 , Y 1 , Y 2 ≤ t ) ) + ( 1 − 4 P ( Y 0 ≤ t ) + 3 P ( Y 0 , Y 1 ≤ t ) + 2 P ( Y 0 , Y 2 ≤ t ) + P ( Y 0 , Y 3 ≤ t ) − 2 P ( Y 0 , Y 1 , Y 2 ≤ t ) − 2 P ( Y 0 , Y 1 , Y 3 ≤ t ) + P ( Y 0 , Y 1 , Y 2 , Y 3 ≤ t ) ) = 2 P ( Y 0 ≤ t ) − P ( Y 0 , Y 1 ≤ t ) + P ( Y 0 , Y 3 ≤ t ) − 2 P ( Y 0 , Y 1 , Y 3 ≤ t ) + P ( Y 0 , Y 1 , Y 2 , Y 3 ≤ t ) = 2 P ( X 0 , X 1 , X 2 , X 3 ≤ t ) − P ( X 0 , X 1 , X 2 , X 3 , X 4 ≤ t ) + P ( X 0 , X 1 , · · · , X 6 ≤ t ) − 2 P ( X 0 , X 1 , · · · X 6 ≤ t ) + P ( X 0 , X 1 , · · · , X 6 ≤ t ) = 2 p 4 − p 5 + p 7 − 2 p 7 + p 7 = 2 p 4 − p 5 Example 2. Let S b e an op enin g on R Z × Z with d efining structural element a 2 × 2 square. Let no w X b e an infin ite 2-dimensional arra y of ind ep endent iden tically distributed random v ariables X ( i,j ) where ( i, j ) ranges o ver Z × Z . In order to derive the outp ut d istribution of S w e put 9 Y ( i,j ) := X ( i,j ) ∧ X ( i − 1 ,j ) ∧ X ( i,j +1) ∧ X ( i − 1 ,j +1) Z ( i,j ) := Y ( i,j ) ∨ Y ( i +1 ,,j ) ∨ Y ( i,j − 1) ∨ Y ( i +1 ,j − 1) Let t ∈ R and p = F X ( t ). Th e output distribution of S is φ S ( p ) = P ( Z (0 , 0) ≤ t ) = P ( Y (0 , 0) , Y (1 , 0) , Y (0 , − 1) , Y (1 , − 1) ≤ t ) F ollo wing [1], whic h in tro d uced that hand y notation in the 1-dimensional case, we abb r eviate the lat ter as ((0 , 0) , (1 , 0) , (0 , − 1) , (1 , − 1)) Y If sa y (( (0 , 0) , (1 , 0) , (0 , − 1) ) Y means P ( Y (1 , 0) ≤ t, Y (0 , 0) , Y (0 , − 1) > t ) , then it follo w s from Lemma 7 and from tr an s lation in v ariance (e.g. ( (0 , 0) , (1 , 0) ) = ((0 , − 1) , (1 , − 1) ) that φ L ( p ) = ((0 , 0) , (1 , 0) , (0 , − 1) , (1 , − 1)) Y = 1 − 4 (0 , 0) Y + 2((0 , 0) , (1 , 0) ) Y + 2((0 , 0) , (0 , − 1) ) Y + ((1 , 0) , (0 , − 1) ) Y +( (0 , 0) , (1 , − 1) ) Y − ((0 , 0) , (0 , − 1) , (1 , − 1) ) Y − ((0 , 0) , (1 , 0) , (1 , − 1) ) Y − ( (0 , 0) , (0 , − 1) , (1 , 0) ) Y − ((1 , 0) , (0 , − 1) , (1 , − 1) ) Y + ((0 , 0) , (1 , 0) , (0 , − 1) , (1 , − 1) ) Y According to the definition of Y ( i,j ) w e e.g. hav e ( (0 , 0) , (0 , − 1) ) Y = ( (0 , 0) , ( − 1 , 0) , (0 , 1) , ( − 1 , 1) , (0 , − 1) , ( − 1 , − 1) , (0 , 0) , ( − 1 , 0) ) X = ( (0 , 0) , ( − 1 , 0) , (0 , 1) , ( − 1 , 1) , (0 , − 1) , ( − 1 , − 1) ) X Putting q = 1 − p = P ( X (0 , 0) > t ) the latter con tributes a term q 6 to φ L ( p ) = 1 − 4 q 4 + 2 q 6 + 2 q 6 + q 7 + q 7 − q 8 − q 8 − q 8 − q 8 + q 9 = 1 − 4 q 4 + 4 q 6 + 2 q 7 − 4 q 8 + q 9 10 5 F o rm ulas for the distribution transfer of the ma jor LU L U - op erators As it wa s already done in the pr eceding S ection it is often conv enien t to u se the notation q = 1 − p . F or example the outpu t distribu tion of M n , L n and U n giv en in (7), (9) and (10) resp ectiv ely can be written in the follo wing shorter form: φ M n ( p ) = 2 n +1 X j = n +1  2 n + 1 j  p j q 2 n +1 − j , (14) φ L n ( p ) = 1 − ( n + 1) q n +1 + n q n +2 , (15) φ U n ( p ) = ( n + 1) p n +1 − np n +2 . (16) Theorem 11 b elow deals with the output distrib ution of L n U n and U n L n . They we re fir st derived in [2], bu t the statemen t of the theorem was also indep enden tly pro ve d by Bu tler [1]. In 5.1 w e present a pr o of usin g Butler’s ”expansion calculus”. In 5.2 this metho d is applied to more complicated situati on s . 5.1 The output distribution of the LU -op erators First, obs erv e that instead of win ding u p with full blo wn in clus ion-exclusion w h en switc h ing al l inequalities > t to ≤ t (dual of Lemma 7), one can b e economic and only switch some inequalities: ( 0 , 1 , · · · , n ) X = (0 , · · · n − 1) X − (0 , · · · , n − 1 , n ) X (17) = (0 , · · · , n − 2) X − (0 , · · · , n − 2 , n − 1) X − (0 , · · · , n − 1 , n ) X . . . = ( 0) X − (0 , 1) X − (0 , 1 , 2) X − · · · − (0 , · · · , n − 1 , n ) X = 1 − (0) X − n − 1 X i =0 ( 0 , · · · , i, i + 1) X Lemma 8 [1, Cor ol lary 4]: L e t X b e a bi-i nfinite se quenc e of r andom variables. Then ( 0 , 1 , · · · , n ) X = 1 − [ n + 1](0) X + n − 1 X i =0 [ n − i ](0 , 1 , · · · , i, i + 1) X Note that for i = 0 w e get the summand n (0 , 1) X . 11 Pr o of: F rom ( k + 1) X = ( k + 1 , k ) X + ( k + 1 , k ) X = ( k + 1 , k ) X + ( k + 1 , k , k − 1) X + ( k + 1 , k , k − 1 ) X = · · · = ( k + 1 , k ) X + ( k + 1 , k , k − 1) X + · · · + ( k + 1 , k , · · · , 1 , 0) X + ( k + 1 , k , · · · , 0) X follo ws, b y trans lation inv ariance, that ( 0 , · · · , k , k + 1) X = ( k + 1) X − ( k , k + 1) X − k − 1 X i =0 ( k − i − 1 , k − i, · · · , k , k + 1) X = (0) X − (0 , 1) X − k − 1 X i =0 (0 , 1 , · · · , i + 1 , i + 2) X Using (1 7) on e deriv es for (sa y) n = 4 that ( 0 , 1 , 2 , 3 , 4) X = 1 − (0) X − 3 X k =0 ( 0 , · · · , k , k + 1) X = 1 − (0) X − 3 X k =0 " (0) X − (0 , 1) X − k − 1 X i =0 (0 , 1 , · · · , i + 1 , i + 2) X # = 1 − 5(0) X + 4(0 , 1) X +(0 , 1 , 2) X +(0 , 1 , 2) X + (0 , 1 , 2 , 3) X +(0 , 1 , 2) X + (0 , 1 , 2 , 3) X + (0 , 1 , 2 , 3 , 4) X = 1 − 5(0) X + 3 X i =0 [4 − i ](0 , 1 , · · · , i, i + 1) X  Unsurp r isingly , for dep endently d istributed random v ariables B i certain com bin ations of B i ’s b eing ≤ t and simulta neously other B j ’s b eing > t , are imp ossible, i.e. hav e probab ility 0. More sp ecifically: Lemma 9 [1, The or em 10]: L et A b e a bi - infinite identic al ly distribute d se quenc e of r andom variables and let B = W r A . Then (0 , 1 , · · · , n − 1 , n ) B =    0 , n ≤ r + 1 (0 , · · · , r, r + 1 , n − 1 , n, · · · , n + r ) A , r + 1 < n < 2 r + 4 12 F or instance, for n = 5 , r = 1 w e ha ve r + 1 < n < 2 r + 4, and so (0 , 1 , 2 , 3 , 4 , 5) B = (0 , 1 , 2 , 4 , 5 , 6) A Let us giv e an ad ho c argument whic h con v eys the spirit of the pro of. In view of B i = A i ∨ A i +1 one e. g. has that B 5 ≤ t ⇔ A 5 , A 6 ≤ t . Using inclusion-exclusion we get (0 , 5 , 2 , 4 , 1 , 3 ) B = (0 , 5 , 2 , 4) B − (0 , 5 , 2 , 4 , 1) B − (0 , 5 , 2 , 4 , 3) B + (0 , 5 , 2 , 4 , 1 , 3) B = P ( A 0 , A 1 , A 5 , A 6 ≤ t, B 2 , B 4 > t ) − P ( A 0 , A 1 , A 2 , A 5 , A 6 ≤ t, B 2 , B 4 > t ) − P ( A 0 , A 1 , A 3 , A 4 , A 5 , A 6 ≤ t, B 2 , B 4 > t ) + P ( A 0 , A 1 , · · · A 6 ≤ t, B 2 , B 4 > t ) Since A 4 , A 5 ≤ t is incompatible with B 4 = A 4 ∨ A 5 > t , the last t wo terms are 0. F urthermore, giv en that A 5 ≤ t , the statemen t B 4 > t amoun ts to A 4 > t . Ditto, giv en that A 2 ≤ t , the statemen t B 2 > t amounts to A 3 > t . Hence (0 , 5 , 2 , 4 , 1 , 3 ) B = P ( A 0 , A 1 , A 5 , A 6 ≤ t, A 4 > t, B 2 > t ) − P ( A 0 , A 1 , A 2 , A 5 , A 6 ≤ t, A 4 > t, A 3 > t ) = ( P ( A 0 , A 1 , A 5 , A 6 ≤ t, A 4 > t ) − P ( A 0 , A 1 , A 5 , A 6 ≤ t, A 4 > t, A 2 ≤ t, A 3 ≤ t ) ) − P ( A 0 , A 1 , A 5 , A 6 ≤ t, A 4 > t, A 2 ≤ t, A 3 >t ) = P ( A 0 , A 1 , A 5 , A 6 ≤ t, A 4 > t ) − P ( A 0 , A 1 , A 5 , A 6 ≤ t, A 4 > t, A 2 ≤ t ) = (0 , 1 , 2 , 4 , 5 , 6) A Dualizing Lemma 9 yiel ds: Lemma 10 [1, Cor ol lary 11]: L et B = V r A . Then ( 0 , 1 · · · , n − 1 , n ) B =    0 , n ≤ r + 1 ( 0 , · · · , r , r + 1 , n − 1 , n , · · · , n + r ) A , r + 1 < n < 2 r + 4 Theorem 11 The distribution tr ansfer functions of L n U n and U n L n ar e: φ L n U n ( p ) = p n +1 + n p n +1 q + p 2 n +2 q + 1 2 ( n − 1)( n + 2) p 2 n +2 q 2 , (18) φ U n L n ( p ) = 1 − φ L n U n ( q ) = 1 − q n +1 − n pq n +1 − pq 2 n +2 − 1 2 ( n − 1)( n + 2) p 2 q 2 n +2 . (19) 13 Pr o of: Since L n U n = ( W n V n )( V n W n ) = W n V 2 n W n w e put A = n _ X, B = 2 n ^ A, C = n _ B and ca lculate φ L n U n ( p ) = P ( C 0 ≤ t ) = (0) C = (0 , · · · , n ) B = 1 − [ n + 1]( 0) B + n − 1 X i =0 [ n − i ]( 0 , 1 , · · · , i, i + 1) B (dual of Lemma 8 ) = 1 − [ n + 1](0 , · · · , 2 n ) A + n (0 , · · · , 2 n + 1 ) A + n − 1 X i =1 0 (Le mma 10 , r = 2 n ) = 1 − [ n + 1] " 1 − [2 n + 1](0) A + 2 n − 1 X i =0 [2 n − i ](0 , 1 , · · · , i, i + 1) A # + n " 1 − [2 n + 2](0) A + 2 n X i =0 [2 n + 1 − i ](0 , 1 , · · · , i, i + 1) A # (Lemma 8 ) = [ n + 1](0) A + 2 n X i =0 [ i − n ](0 , 1 , · · · , i , i + 1) A = [ n + 1](0) A − n (0 , 1) A + n X i =1 [ i − n ](0 , 1 , · · · , i , i + 1) A +(0 , 1 , · · · , n + 1 , n + 2) A + 2 n X i = n +2 [ i − n ](0 , 1 , · · · , i, i + 1) A = [ n + 1](0 , · · · , n ) X − n (0 , · · · , n + 1) X + 0 + (0 , · · · , n , n + 1 , n + 2 , · · · , 2 n + 2) X + 2 n X i = n +2 [ i − n ](0 , · · · , n, n + 1 , i, i + 1 , · · · , i + 1 + n ) X (Lemma 9) = ( n + 1) p n +1 − n p n +2 + p 2 n +2 q + 2 n X i = n +2 ( i − n ) p 2 n +2 q 2 = p n +1 + n p n +1 − n p n +1 p + p 2 n +2 q + (2 + 3 + · · · + n ) p 2 n +2 q 2 = p n +1 + n p n +1 q + p 2 n +2 q + 1 2 ( n − 1)( n + 2) p 2 n +2 q 2  F rom (18 ) it is clear that p n +1 is the h ighest p o w er of p dividing φ L n U n . An easy calc u lation confirms that, as a p olynomial in p , the righ t hand side of (19) is (2 n + 3) p 2 + ( · · · ) p 3 + · · · . F rom Corollary 4 hence follo w s th at L n U n is lo wer robust of order n + 1, b ut upp er robust only of order 2. 14 5.2 The output distributions of the LU LU -op erators C n and F n W e consider n ext the sp ecific, mutually d ual LULU-op erators C n = L n U n L n − 1 U n − 1 · · · L 1 U 1 , F n = U n L n U n − 1 L n − 1 · · · U 1 L 1 . In view of C n − 1 = _ n − 1 ^ 2 n − 2 _ n − 1 C n − 2 C n = _ n ^ 2 n _ n C n − 1 = _ n ^ 2 n _ 2 n − 1 ^ 2 n − 2 _ n − 1 C n − 2 w e define the follo wing doubly infin ite sequen ces of iden tically distributed random v ariables. Starting with a sequence X of i.i.d. random v ariables, put A := _ n − 1 C n − 2 X B := ^ 2 n − 2 A C ′ := _ n − 1 B C := _ 2 n − 1 B D := ^ 2 n C E := _ n D Theorem 12 [1, The or e m 14] With A, B as define d ab ove the output distribution φ C n of C n c an b e c ompute d r e cu rsively as fol lows: φ C n = φ C n − 1 + n ( G 2 n − G 2 n − 1 ) , wher e G 2 n := (0 , · · · , 2 n − 1 , 2 n, 2 n + 1 , · · · , 4 n ) B G 2 n − 1 := ( 0 , · · · , 2 n − 2 , 2 n − 1 , 2 n , · · · , 4 n − 2) A 15 Pr o of: First, o ne calculates φ C n − 1 = (0) C ′ = (0 , · · · , n − 1) B = 1 − n ( 0) B + n − 2 X i =0 [ n − 1 − i ]( 0 , 1 , · · · , i, i + 1) B (dual of Lemma 8) = 1 − n (0 , · · · , 2 n − 2) A + [ n − 1](0 , · · · , 2 n − 1) A + n − 2 X i =1 0 (Lemma 10 , r = 2 n − 2) = 1 − n ( 0 , · · · , 2 n − 2) A + [ n − 1](0 , · · · , 2 n − 1) A (20) The e xpansion of φ C n is d riv en a b it further: φ C n = (0) E = (0 , · · · , n ) D = 1 − [ n + 1](0) D + n − 1 X i =0 [ n − i ]( 0 , 1 , · · · , i, i + 1) D (dual of Lemma 8) = 1 − [ n + 1](0 , · · · , 2 n ) C + n (0 , · · · , 2 n + 1) C + n − 1 X i =1 0 (Lemma 10 , r = 2 n ) = 1 − [ n + 1] " 1 − [2 n + 1](0) C + 2 n − 1 X i =0 [2 n − i ](0 , 1 , · · · , i, i + 1) C # + n " 1 − [2 n + 2](0) C + 2 n X i =0 [2 n + 1 − i ](0 , 1 , · · · , i, i + 1) C # (Lemma 8 ) = [ n + 1](0) C − 2 n X i =0 [ n − i ](0 , 1 , · · · , i, i + 1) C (easy arithmetic) = [ n + 1](0 , · · · , 2 n − 1) B − n (0 , · · · , 2 n ) B − 2 n − 1 X i =1 0 + n (0 , · · · , 2 n − 1 , 2 n, 2 n + 1 , · · · , 4 n ) B (Lemma 9 , r = 2 n − 1) = [ n + 1](0 , · · · , 2 n − 1) B − n (0 , · · · , 2 n ) B + n G 2 n (21) This yields φ C n − nG 2 n = [ n + 1](0 , · · · , 2 n − 1) B − n (0 , · · · , 2 n ) B (b y (21)) = [ n + 1] " 1 − 2 n ( 0) B + 2 n − 2 X i =0 [2 n − 1 − i ]( 0 , 1 , · · · , i, i + 1) B # − n " 1 − [2 n + 1]( 0) B + 2 n − 1 X i =0 [2 n − i ]( 0 , 1 , · · · , i, i + 1) B # (dual of Lemma 8) 16 = 1 − n (0) B + 2 n − 1 X i =0 [ n − 1 − i ]( 0 , 1 , · · · , i, i + 1) B (easy arithm etic) = 1 − n (0 , · · · , 2 n − 2) A + [ n − 1](0 , · · · , 2 n − 1) A + 2 n − 2 X i =1 0 − n ( 0 , · · · , 2 n − 2 , 2 n − 1 , 2 n , · · · , 4 n − 2) A (Lemma 1 0 , r = 2 n − 2) = φ C n − 1 − n G 2 n − 1 (b y (20)) whic h , up on a d ding nG 2 n on b oth sid es, giv es the claime d form u la for φ C n .  As an example, let us compute th e output distr ibution of C 2 . F rom A = W 1 C 0 X = W 1 X foll o ws B = V 2 A = V 2 W X . Using expans ion ca lculu s the reader ma y v er if y that G 2 n = G 4 = (0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8) B = p 4 q 2 [ p + p 2 q ] 2 Similarly one gets G 2 n − 1 = G 3 = p 2 q 2 (1 − p 2 ) 2 . Therefore φ C 2 = φ C 1 + 2( G 4 − G 3 ) = 3 p 3 + 3 p 4 − 9 p 5 + 4 p 6 + 4 p 7 − 10 p 8 + 4 p 9 + 8 p 10 − 8 p 11 + 2 p 12 As to r obustness, from the ab o ve repr esen tation of φ C 2 and b y using Corollary 4 w e obtain that C 2 is lo w er r obust o f order 3 lik e U 2 . Similar to L 2 U 2 discussed in 5.1, the upp er robustness of C 2 is n ot in herited f rom L 2 . Ind eed, w e ha ve φ C 2 − 1 = q 2 (2 p 10 − 4 p 9 − 2 p 8 + 4 p 7 + 4 p 4 − p 3 − 3 p 2 − 2 p − 1) whic h imp lies that C 2 is u pp er rob u st only of order 2. Ho wev er, u pp er r obustness is not con- stan tly 2; these results w ere obta in ed from Th eorem 12: n 1 2 3 4 5 6 lo we r r obustness 2 3 4 4 5 6 upp er robustness 2 2 3 3 4 4 W e ment ion that s ome nice close d formula for G 2 n and G 2 n − 1 is v erified for small n and conjec- tured to hold univ ersally in [1, Section 4.5.6]. 17 6 Using exclusion instead of inc lusion-exclusion As witnessed by 5.2, one can sometimes exploit symmetry to tame the inherent exp onential complexit y of inclusion-exclusion. Ho we ver, without the p ossibility to clump together man y iden tical terms, the n u m b er of sum mands in Lemma 7 is 2 n , whic h is infeasible already for n = 20 o r so. In [14] on the other hand , some multi- purp ose principle of exclusion (POE) is emplo y ed whic h had b een useful in other situations b efore. When POE is aimed at calculating the output distribution of a stac k fi lter S , a prerequisite is that the stac k fi lter † S b e given as a d isjunction of conju nctions K i , i.e. in disjunctive normal form (DNF). T he POE then b egins with the calculatio n of the s et Mo d 1 of all bitstrin gs that satisfy K 1 , then from Mo d 1 it excludes all bitstrings that v iolate K 2 . Th is yields Mo d 2 ⊆ Mo d 1 , from whic h all bitstrings are excluded that violate K 3 , and so on. The feasibilit y of the POE h inges on th e compact represen tation (using w ildcards) of the set s Mo d i . The details b eing give n in [14], here we address the question of ho w one gets the DNF in the first place. Sp ecifically w e consider the fr equen t case that our stac k filter S is a EDC (Section 4) whose structural elemen ts are provided. Let us go in med ias res b y rew orking S = W 1 V 2 W 2 of Example 1: ( S X ) 0 = A 0 = Z 0 ∨ Z 1 (DNF) = ( Y − 2 ∧ Y − 1 ∧ Y 0 ) ∨ ( Y − 1 ∧ Y 0 ∧ Y 1 ) (blo wup ) = Y 0 ∧ Y − 1 ∧ ( Y − 2 ∨ Y 1 ) (get CNF) = ( X 0 ∨ X 1 ∨ X 2 ∨ X 3 ) ∧ ( X − 1 ∨ X 0 ∨ X 1 ∨ X 2 ) (blo wup ) ∧ (( X − 2 ∨ X − 1 ∨ X 0 ∨ X 1 ) ∨ ( X 1 ∨ X 2 ∨ X 3 ∨ X 4 )) = ( X 0 ∨ X 1 ∨ X 2 ∨ X 3 ) ∧ ( X − 1 ∨ X 0 ∨ X 1 ∨ X 2 ) (condense) ∧ ( X − 2 ∨ X − 1 ∨ X 0 ∨ X 1 ∨ X 2 ∨ X 3 ∨ X 4 ) = ( X 0 ∨ X 1 ∨ X 2 ∨ X 3 ) ∧ ( X − 1 ∨ X 0 ∨ X 1 ∨ X 2 ) (condense fu rther) = X 0 ∨ X 1 ∨ X 2 ∨ ( X 3 ∧ X − 1 ) (get DNF) The last line is the sough t DNF of S . It is obtained b y s tarting with the DNF Z 0 ∨ Z 1 . This gets “blo wn up” to a DNF in terms of Y i ’s (us in g defin ition (13) of Z 0 and Z 1 ). This DN F needs to b e switc hed ‡ to CNF (= conjunctiv e normal form). This in turn is blo wn up to a CNF in terms of X i ’s. Usually th e result can and must b e condensed in ob vious w ays (“condense further” mean t that only the inclusion-minimal ind ex sets carry o ver). Contin uing lik e this one take s † More precisely , th e p ositive Boolean function that un derlies the stack filters m ust b e giv en in DNF. ‡ How one switc h es b etw een DNF and CNF of a p ositive Bo olean fun ction is a well researched t opic which we w on’t persue here. 18 turns switc hing DNF’s with CNF’s, and blo wing up e x p ressions. This is d one a s often as there are structural elements. As a “side pro duct” the s o calle d r ank sele c tion pr ob abilities RS P [ i ] are calculate d. The latter is defined as the probabilit y that the filter selects the i -th smallest pixel in the w -element sliding window. F or instance here w = 5 and RS P [1] = R S P [2] = RS P [3] = 0 , RS P [4] = 0 . 4 , RS P [5] = 0 . 6. The four th author h as written a Mathematica 9.0 pr ogram § whic h , giv en the structur al elemen ts of an y EDC’s (also 2-dimensional), first calculates the DNF of S and from it the output distri- bution φ S ( p ). (Alternativ ely the DNF of an y stac k filter, whether EDC or not, can b e fed in directly .) Alb eit Wild’s algo r ithm is multi-purp ose, it managed to calculate φ C n ( p ) up to n = 5, and the result agreed with Butler’s. W ritten out as EDC w e ha ve C 5 = W 5 V 10 W 9 · · · V 2 W and C 5 has a slid in g windo w of length 61. The corresp onding structural elemen ts { 0 , 1 , 2 , 3 , 4 , 5 } , { 0 , − 1 , · · · , − 10 } , { 0 , 1 , · · · , 9 } and so forth triggered the calculation of a DNF comprising a plen tifu l 12018 conjun ctions (time: 168224 sec). F rom this φ C 5 ( p ) w as calculated in 45069 sec. Here it is: 12 p 5 + 7 x 6 − 23 p 7 + 19 p 8 − 130 p 9 + 194 p 10 − 59 p 11 − 142 p 12 + 460 p 13 − 787 p 14 + 715 p 15 − 7 p 16 − 1030 p 17 +1959 p 18 − 2216 p 19 + 208 p 20 + 3711 p 21 − 6748 p 22 + 8412 p 23 − 7587 p 24 + 2023 p 25 + 4680 p 26 − 7903 p 27 + 8839 p 28 − 13540 p 29 + 30009 p 30 − 51715 p 31 + 50159 p 32 − 7686 p 33 − 51417 p 34 +78198 p 35 − 50589 p 36 + 6900 p 37 − 7680 p 38 + 56330 p 39 − 86905 p 40 + 43710 p 41 + 49540 p 42 − 114680 p 43 + 103390 p 44 − 40555 p 45 − 15370 p 46 + 33955 p 47 − 25460 p 48 + 11790 p 49 − 3645 p 50 +740 p 51 − 90 p 52 + 5 p 53 As to the rank selection probabilities of C 5 one has RS P [1] = · · · = RS P [4] = 0 , RS P [5] = 0 . 000002 , RS P [6] = 0 . 00001 , · · · , RS P [37] = 0 . 04701 , RS P [38] = 0 . 04703 , RS P [39] = 0 . 04643 , · · · , R S P [58] = 0 . 00012 , R S P [59] = RS P [60] = RS P [61] = 0, the maximum b eing RS P [38]. 7 Conclusion As men tioned in the int ro duction, concepts of robu st smo others related to ours ha ve b een previously co n sidered. T o q u ote f rom th e abstract of [10]: In this p ap er we fo cu s on r ank sele ction pr ob abilities (RSPs) as me asur e s of r obustness as i t is wel l known that other statistic al char acterization of stack filters, such as output distributions, br e akdown pr ob abilities and output distributional influenc e functions c an b e r epr esente d in terms of RSPs. § It is av ailable u p on sending an email to mwild@sun.ac.za 19 While w e agree with this praise of RSPs we don’t share the opinion on page 1 642 of [10 ]: Efficient sp e ctr al algorithms exist for the c omputation of the sele ction pr ob abilities of stack filters. The a r ticle cited in [10] is [15] whic h offers Bo olean deriv ativ es and weig hted Cho w p arameters but no computational evid en ce of the feasibilit y of the p rop osed in tricate metho d. Similarly [16] whic h reduces the calculations of the RSPs of stac k fi lters of wind o w size n to the corresp ondin g problem for size n − 1 offers no computational data; its complexit y in theory (and lik ely in practise) is O ( n !). In the same vein no numerical exp erimen ts are carried out in [17]. It is eviden t that none of the a p proac hes [1 5], [16], [17] (and in fact none that h an d les the mo d els of a B o olean function one b y one) scales up to [14] . References [1] P .W. Butler, The transfer of d istributions of LULU smo others, MSc Thesis, Un iversit y of Stellen b osc h 2008 (directed by C. Roh wer an d M. Wild). Accessible with the Searc h F unction at http:// s c holar.sun.ac.za/ [2] W. J. Conradie, T de W et and M Janko w itz, Exact and asymptotic d istributions of L ULU smo others, Journa l of Computational and Applie d Mathematics 1 86 (2006) 253 -267. [3] H.J.A.M. Heijmans, Comp osing m orp hological filters, IEEE T r ansactions on Image Pr o- c essing 6 (1997) 713-72 3. [4] C.L. Mallo ws, Some theory of nonlinar smoothers, Annals of Statistics 8 (1980) 69 5–715. [5] T.A. No des, N.C. Gallagher, Median filters: Some pr op erties and their mo difications, IEEE T ransactions on Acoustics, Sp eac h , Signal Pro cessing 30 (1982) 739– 746. [6] C.H. Roh wer, V ariation reduction and LU LU -smo othing, Quaestiones Mathematicae 25 (2002 ) 163–17 6. [7] C.H. Roh w er, F ully tr en d preserving op erators, Quaestiones Mathematicae 27 (2004) 217– 230. [8] C.H. Roh wer, Nonlinear Smo others and Multiresolution Analysis, Birkh ¨ auser, 2005. [9] C. Roh we r , M. Wild, LULU-theory , idemp oten t stac k filters, and the mathematics of vision of Marr, Adv ances in Imaging and Electron P hysics 1 46 (2007) 5 7–162. [10] I. Sh m u levic h, O . Yli-Harja, J . Astola, A. Korshuno v, On the robustness of the class of stac k filters, I EEE T ransactions on Signal Pr o cessing 50 (2002) 164 0-1649. [11] J. Serra, Image Analysis and Mathematica l Morphology , Academic Press, London, 198 2. [12] J. Serra, I mage Analysis and Mathematical Morph ology , V olume I I : Theoretical Adv ances, J. S erra (Ed .), Academic Press, London, 1988. [13] P . Soille, Morp hological Image Analysis, Springer 1999. [14] M. Wild, Comp u ting the output distribu tion and selection pr obabilities of a stac k filter fr om the DNF of its p ositiv e Boolean function, Jour nal of Mathematical Imagi ng and Vision 46 (2013 ) 66-73. 20 [15] K. E giazarian, P . Kuosmanen and J. Asto la, “Bo olean d eriv ative s, weigh ted c how parame- ters and selection probabilities of s tack filters,” IEEE T ransactions Signal Pr o cessing, vol.4 4, pp.1634-16 41, July 1 996. [16] L. Lin, G.B. Adams, E.J. Co yle, Stac k filter lattices, Signal Pro cessing 38 (1994) 277-297. [17] M.K. Prasad, Stac k filter d esign us ing selection probabilities, IEEE T ransactions on signal pro cessing, v ol.53, no.3, Marc h 2005 21