Embedability theory of two-dimensional supports, with applications

Z. Mou-yan and R. Unbehauen Vol. 13, No. 2 /February 1996 /J. Opt. Soc. Am. A 243

Embedability theory of two-dimensional supports, with applications

Zou Mou-yan* and Rolf Unbehauen

Lehrstuhl fur Allgemeine und Theoretische Elektrotechnik,Universitat Erlangen-Nurnberg, D-91058 Erlangen, Germany

Received November 29, 1994; revised manuscript received July 28, 1995; accepted August 29, 1995

We establish a theory of embedability of supports of two-dimensional sequences. We found that the embed-ability of a support plays an essential role in the reducibility or irreducibility of the support. The theory canoffer a new insight and a geometric visuality to the support problems. For example, the irreducibility of aclass of supports including Eisenstein’s support can be checked by use of a simple lemma and by inspection.Based on convolution and embedability arguments, a new description of support irreducibility is suggested.As an application, we propose a new method for determining a reliable and tight bound on object support fromits autocorrelation support. 1996 Optical Society of America

1. INTRODUCTIONIt has been established that almost all polynomials intwo or more variables are irreducible.1 However, inmany fields of research such as image recovery throughphase retrieval and blind deconvolution and in multi-dimensional system theory, a problem remains if the mul-tidimensional polynomial in question can be factorable.2 – 4

This problem stems originally from the investigationof phase-retrieval problems. In that context, if a two-dimensional (2-D) sequence has an irreducible z trans-form, it will be uniquely defined by the magnitude of itsFourier transform.2

In image processing, an image is expressed by a dis-crete 2-D sequence. The set consisting of the coordinatesof all nonzero pixels of the 2-D sequence is called the sup-port of the sequence. In a study of phase retrieval ofimages, Bruck and Sodin first noted that the irreducibil-ity of 2-D sequences will reduce the retrieval ambiguity.5

The process of phase retrieval shows that if the objectsupport is known, the reconstruction of the object by theiterative method6 – 8 is relatively easy. This directly mo-tivates the study of the support problem. There are twoquite interesting questions: (1) what kind of supportscan guarantee the irreducibility of the sequences definedon the support (i.e., the uniqueness of the phase-retrievalsolution) and (2) how to estimate a tight support of theobject from its autocorrelation. Fiddy et al. found thata special kind of support, i.e., the so-called Eisensteinsupport, can ensure that the 2-D z-transform polyno-mials of sequences are irreducible.9 Nieto-Vesperinasand Dainty gave a proof of Eisenstein’s irreducibilitycriterion10 in terms of a result from integer polynomialtheory by van der Waerden.11 These results embodythe concept that a class of supports can indeed guar-antee the irreducibility of sequences and therefore theuniqueness of the phase-retrieval solution. Since thesolution is unique for a class of supports, it is reason-able to find a closed phase-retrieval algorithm. Someinteresting results have been reported by Fienup,12

Crimmins,13 and Brames.14 It is clearly worthwhile to

0740-3232/96/020243-10$06.00

enlarge the class of irreducible supports. Brames inves-tigated this problem and proposed some useful concepts,such as P irreducibility and C irreducibility.15 He alsosuggested several algorithms for testing irreducibility,although these algorithms need to be improved. Fienupet al.,16 Crimmins et al.,17 and Brames18 proposed severalmethods that reflect considerable progress in approachingthe support estimation problem.

In this paper we present a new theory of the supportproblem. We have found that the embedability of sup-ports plays an essential role in the reducibility or ir-reducibility of the supports. The idea of embedabilitycomes from the mathematical morphological method.19

It can offer a new insight and a geometric visuality tothe support problems.

The paper is organized as follows. In Section 2 weintroduce the embedability of supports as a fundamen-tal property of supports of deconvolvable sequences. InSections 3 and 4 we discuss convex and nonconvex sup-ports, respectively. We propose a new description of sup-port irreducibility based on convolution and embedabilityarguments. We show that the embedability theory canprovide effective methods for checking the irreducibilityof supports. For example, the irreducibility of a class ofsupports including Eisenstein’s support can be checkedby use of a simple lemma and by inspection. As an ap-plication of the embedability theory, in Section 5 we pro-pose a new method for determining a reliable and tightbound on the object support from its autocorrelation sup-port. Section 6 contains our conclusion.

2. CONVOLUTION AND THEEMBEDABILITY OF THE SUPPORTTo introduce the idea of convolution and the embedabil-ity of the supports, we recall the operation of convolu-tion. Conventionally, the convolution of two sequencesincludes several steps: folding and shifting of one of twosequences and calculating the sum of the products of theoverlapping part of the two sequences. In fact, a differ-ent method of convolution is much more elementary. In

1996 Optical Society of America

244 J. Opt. Soc. Am. A/Vol. 13, No. 2 /February 1996 Z. Mou-yan and R. Unbehauen

the discrete case the 2-D convolution with finite supportcan be written as

ysm, nd : xsm, nd p hsm, nd

:M21Pk0

N21Pl0

xsk, ldhsm 2 k, n 2 ld . (1)

In this expression we have assumed that both xsm, ndand hsm, nd are M 3 N sequences. Consequently, theconvolution, ysm, nd, has size s2M 2 1d 3 s2N 2 1d. Interms of the unit sample sequence,

usm, nd

(1 if m n 00 otherwise

, (2)

Equation (1) may be rewritten as

ysm, ndM21Pk0

N21Pl0

"hsk, ld

M21Pp0

N21Pq0

xsp, qd

3 usm 2 k 2 p, n 2 l 2 qd

#. (3)

For fixed sk, ld,

M21Pp0

N21Pq0

xsp, qdusm 2 k 2 p, n 2 l 2 qd

means that the array xsm, nd is shifted by the amountsk, ld. That is, xs0, 0d is shifted to sk, ld, xsp, qd is shiftedto sk 1 p, l 1 qd, and so forth. From Eq. (3) the convolu-tion of xsm, nd and hsm, nd may be done in the followingway: the array xsm, nd is shifted by the amount sk, ld,and each of its elements is multiplied by hsk, ld. Theprocedure runs over all the nonzero hsk, ld. The super-position of all the resultant arrays is the convolution. Weuse a simple example to show the operation in Fig. 1.

Note that if the convolution is calculated in this way,no folding operation of arrays is needed. We assume thatxsm, nd and hsm, nd are positive and real. Therefore nonumerical cancellation occurs in the convolution. In sucha case it is clear that the support of xsm, nd must be embe-dable into the support of ysm, nd. Owing to the commu-tativity of convolution, the support of hsm, nd also must beembedable into the support of ysm, nd. Although our ar-gument has been done for discrete convolution problems,the embedability property of supports holds also for con-tinuous convolution problems. A basic constraint is thatthe convolution factors be real and positive functions.

Fig. 1. Example of the calculation of convolution.

In this paper we discuss mainly the case that the 2-Dsequence ysm, nd has a finite support D , S , where S isa rectangular region of the integer grid: S hsm, nd [Z2jsm1 # m # m1 1 M 2 1, n1 # n # n1 1 N 2 1j, withtwo given m1, n1 and M, N being sufficiently large.

For simplicity, in the discrete case we use the term “anarea” to express a finite point set of coordinates. Let A

be an area in S . We can assign a reference point to A.The reference point need not lie on A, but the relativeposition of A and its reference is fixed. Therefore theshift of A may be expressed by the shift of its reference.Conveniently, we use A or As0, 0d for the original area,and Asm, nd means an area shifted by the amount sm, nd.We use psm, nd to denote a point located at sm, nd. Con-veniently, psm, nd is used also for the set containing onlyone point psm, nd.

Definition 1. An area De is said to be an embedablefactor of D if

1. De can be completely shifted into D ; i.e., there existamounts sm, nd such that Desm, nd , D ;

2. D can be completely covered by shifting De inD ; i.e., for an arbitrary point psm, nd [ D there ex-ist amounts sm, nd such that psm, nd [ Desm, nd andDesm, nd , D .

Completely covered is a key term in the definition.To completely cover D , we should shift De in D . Inthis case we can arbitrarily assign a point on De andrecord the shifting trace. The shifting trace will form afinite area in D . Such a finite area in D will be calledthe embeding trace of De and will be denoted by Dt.Conveniently, we regard ps0, 0d as the reference of bothDe and Dt. We have from the definition,

D [

;psm, nd[Dt s0, 0dDesm, nd . (4)

Note that we can write

Desm, nd [

;psm, nd[De s0, 0dpsm 1 m, n 1 nd . (5)

Thus we have

D [

;psm, nd[Dt s0, 0d

[;psm, nd[De s0, 0d

psm 1 m, n 1 nd

[

;psm, nd[De s0, 0d

[;psm, nd[Dt s0, 0d

psm 1 m, n 1 nd

[

;psm, nd[De s0, 0dDtsm, nd . (6)

From Eq. (6) we have immediately theorem 1:

Theorem 1. If De is an embedable factor of D , thenits embedding trace is also an embedable factor of D .

Thus an embedable factor and its embedding trace forma pair of complementary embedable factors (CEF’s) of D .We use an example to explain theorem 1 as follows.

Example 2.1. A support with its CEF’s is shown inFig. 2.

In Fig. 2 the area consisting of all ≤ and ? points isan embedable factor, and the area consisting of all ±


Fig. 2. Support and its CEF’s.

Fig. 3. Support as in Fig. 2 but with different CEF’s.

and 1 points embodies another embedable factor of thesupport. The two factors form a pair of complementaryembedable factors of the support. The complementarityof the two embedable factors can be understood from thecommutativity of the convolution of two sequences definedon the two factors. If we consider the two components ofa convolution on an equal basis, the complementarity oftheir supports must be true.

Clearly, for arbitrary D , D is self-embedable, and itsembedding trace is a single point. Such types of CEFare trivial.

Definition 2. For a given support D , if there exist nonontrivial CEF’s of D , we say that D is nondecomposableof the first kind; otherwise D is decomposable.

In Ref. 15 Brames defined a class of P irreducible sup-ports by means of characteristic functions and convolu-tion. The nondecomposable supports of the first kind arein fact the same as the P irreducible supports. We pre-fer the new definition because it can offer a geometric in-sight. From the convolution operation discussed above,the following fact must be true:

Theorem 2. Let ysm, nd be a 2-D sequence with afinite support D . For ysm, nd to be deconvolvable withthe nonnegativity constraint, a necessary condition is thatits support D be decomposable.

The decomposable supports have many interestingproperties. If D is decomposable, the CEF’s usuallyare not unique. In example 2.1 we can easily get sev-eral different CEF’s that may be similar or dissimilar tothe shown CEF’s. In Fig. 3 the support is the same asthat in Fig. 2, but the shown CEF’s in these two figuresare different. The nonuniqueness of a pair of CEF’s canoccur if at least one factor of the CEF’s is decomposable.This can be easily understood from the fact that if A isan embedable factor of B , and B is an embedable factorof D , then A must be an embedable factor of D . It has

been found that the decomposability of CEF’s is not thesole cause of the nonuniqueness. In Fig. 4 we illustratethis problem with an example.

We remark that definitions 1 and 2 and theorems 1 and2 can be generalized to the continuous case if D andDe are finite areas in R2 and ysm, nd is replaced by acontinuous distribution ysu, vd defined on D .

3. CONVEX SUPPORTS ANDCONVEX BOUNDARY

A. Convex Hull and Convex Enclosure

Definition 3. For a given support D , S , the convexhull of D , D , is defined by

D h y [ Sjy P

;xi[D

lixi; ;0 # li # 1,P

li 1j .

(7)

Example 3.1. Calculation of the convex hull.We assume that D consists of three points: x1 s1, 1d,

x2 s1, 3d, and x3 s4, 3d as shown by the three ≤ inFig. 5. The convex hull of D , D , clearly consists of sevenelements: (1, 1), (1, 2), (2, 2), (1, 3), (2, 3), (3, 3), and (4,3). All elements in D , may be calculated in terms of x1,x2, and x3:

s1, 2d 1/2x1 1 1/2x2, s2, 3d 2/3x2 1 1/3x3 ,

s3, 3d 1/3x2 1 2/3x3, s2, 2d 1/2x1 1 1/2s1/3x2 1 2/3x3d .

(8)

Definition 4. A support D is said to be convex if itis equal to its convex hull, i.e., D D . If D , D , wesay that D is nonconvex.

Clearly, the convex hull of a support D is the minimumconvex support that can contain D .

Let D be a support, convex or nonconvex. This sup-port actually consists of a set of lattice points on the

Fig. 4. Decomposable support A0 and its two different pairsof CEF’s sA11, A12d and sA21, A22d. Note that none of theembedable factors is decomposable.

Fig. 5. Example of the calculation of the convex hull.


Fig. 6. Example showing that on the discrete grid the set sumof two convex sets need not be convex. D1 and D2 are convex,but D1 1 D2 is not convex. The convex hull D D1 1 D2 is Creducible but nondecomposable of the first kind. The enclosuressatisfy the Titchmarsh–Lions theorem20: D D1 1 D2.

continuous plane. On R2 we can define the minimumconvex polygon that contains the convex hull D . Thiscan be achieved if we simply connect all vertex points ofD . In such a case the maximum area enclosed by all theconnecting lines can form a convex polygon. This polygonis called the convex enclosure of D and is denoted by D .Clearly, the convex enclosure is convex in R2. Conve-niently, the boundary of D is called the convex boundary(CB) of D . The CB consists of many edges. The edgesof the CB are called the edges of D . The orientation ofan edge includes the slope of the edge and the directionof its inner side. The length of an edge is the number ofpoints included in the edge minus 1. An edge is said tobe elementary if its length is 1.

We remark that our definition of the convex hull differsfrom that of other researchers; for example, see Ref. 15.

B. Convex Boundary PropertyTo clarify the implication of the discussion in what fol-lows, we mention a few interesting facts. In distributiontheory,20 Titchmarsh and Lions proved that in the contin-uous field R2 if D1 and D2 are convex hulls of the sup-ports of two distributions x and h and D is the convexhull of the support of x p h, then D D1 1 D2, wherethe set sum is defined as D1 1 D2 h pi 1 pj j;pi [D1, pj [ D2j. In the discrete grid Z2, however, exam-ples can be easily constructed such that D fi D1 1 D2.Moreover, in the discrete grid the set sum of two convexsupports need not be convex. A simple example is shownin Fig. 6. In the discrete grid it is even difficult to definean actual boundary of a support if the support is not con-vex. Therefore, as has been done previously, for a givensupport D , the boundary of its convex enclosure is calledthe CB of D . The convex enclosures D , D1, and D2 areconvex in R2. Hence Titchmarsh and Lions’s theorem isapplicable to convex enclosures: D D1 1 D2.

Since a convex enclosure is a convex polygon that can becompletely described by its boundary, an explicit relationamong the boundaries of D , D1, and D2 would be prefer-able. In Ref. 18 Brames discussed this problem. Todescribe convex polygons, he proposed using phasor func-tions of edges of polygons. Furthermore, if B , A1, andA2 are three convex polygons in R2, then B A1 1 A2

iff the phasor function of B is equal to the sum of thephasor functions of A1 and A2. Brames called this thesupport reduction theorem and gave an outline of theproof.18 Here we restate this theorem and offer a com-plete proof in terms of the embedability argument. Wenote that the relation B A1 1 A2 can be written as

B [

;psm, nd[A1

A2sm, nd . (9)

We consider the process of deriving B . Suppose that A1

is fixed and A2 is shifted. Consider an arbitrary edge ofA1, denoted by ei. Its two end points are pi1 and pi2.The extension line of ei will divide the plane into twohalves. A specified point vi on A2 is chosen so that whenthis point is shifted on ei, A2 and A1 are on the sameside of the plane divided by ei. This is clearly realizable,since A2 is convex. Figure 7 is an illustration of thisarrangement. When vi runs over ei from pi1 to pi2, theshifted A2 will cover an area in the half-plane that is apart of B . Then (1) ei will be reserved in B because it isimpossible to form an area of B that can cross the line ei

and enter into another half-plane. (2) The inner side ofei for A1 remains the inner side for B . (3) When vi runsalong the two neighboring edges of ei, the area covered bythe shifted A2 also cannot cross the line ei and enter intoanother half-plane, because A1 is convex. The formedtwo new edges and ei will create two vertex angles of B

that are not larger than 180±. Since the form of B isindependent of the choice of vi, the argument is true forall edges of A1. This implies that all edges of A1 will bereserved in B and that the related vertex angles of B arenot larger than 180±. By virtue of the commutativity orcomplementarity of A1 and A2, all edges of A2 will alsobe reserved in B , and the related vertex angles of B willalso not be larger than 180±. (4) Let the numbers of theedges of A1 and A2 be n1 and n2, respectively. Whenvi runs over ei, the shifted A2 will cover a polygon withat most n2 1 2 edges. When vi runs along the peripheryof A1, for each edge of A1 the convex enclosure of thepolygon covered by the shifted A2 will produce at mostone additional edge; or, when the shifted A2, crosses overthe extension line of another neighboring edge of ei, it willdecrease its number of edges by one. Thus the maximumnumber of edges of B will be n1 1 n2. We can summarizethe results and obtain theorem 3:

Theorem 3. Let A1 and A2 be two convex polygonsin R2 and let B be the composed polygon defined byEq. (9). Then (1) the boundary of B encompasses a con-vex polygon, and (2) the boundary of B is formed by allshifted edges of A1 and A2, with the orientation of theedges conserved.

The second part of theorem 3 includes the special casethat A1 and A2 have parallel edges on the same side.In such a case an edge will occur in B that is parallel tothe two parallel edges of A1 and A2, and its length isthe sum of the lengths of the two parallel edges.

Theorem 3 may be viewed as a restatement of the sup-port reduction theorem presented by Brames18 with dif-

Fig. 7. Illustration of the proof of theorem 3.


ferent descriptions. An advantage of theorem 3 is itsgeometric visuality.

Corollary. Let D1 and D2 be two convex supports inthe discrete grid Z2. D is defined by

D [

;psm, nd[D1

D2sm, nd . (10)

Then the CB of D is formed by all shifted edges of theCB’s of D1 and D2 with the orientation of the edgesconserved. The corollary can be proved directly fromtheorem 3 and the fact that D D1 1 D2.

C. An Application: Convex Boundary EstimationFrom the support reduction theorem, Brames18 proposeda method for testing the irreducibility of discrete sup-ports and finding all object supports that can generatea given discrete autocorrelation support. His methodworks when the autocorrelation support is convex. Fornonconvex autocorrelation supports the method can offera bound of object support. Suppose that a convex polygonA is given. It may be the convex enclosure of a discretesupport in our definition or a convex hull in Brames’s defi-nition. Brames’s method is to test whether the phasorfunction of A can be partitioned into two phasor func-tions, each of them corresponding to a convex polygon. Ifno such partition exists, the discrete support must be irre-ducible. If the phasor function of a centrosymmetric A

can be partitioned into two phasor functions with the cor-responding two convex polygons being centrosymmetricwith each other, then one of the two convex polygons willcorrespond to a possible object support that can generatethe given autocorrelation support. In many cases theremay be several possible object supports that can generatethe same autocorrelation support. Brames’s method in-cludes centroiding each possible object support and thenforming the union of these supports. The result yields acompact bound of all possible supports.

Since many practical autocorrelation supports are non-convex, we consider that Brames’s method can be used asthe first step in determining a bound of object supports.More precisely, Brames’s method can offer a bound of theCB of object supports. One may obtain a tighter bound ofobject supports by making use of embedability arguments.This problem will be discussed in Section 5. Here wecontinue to discuss the problem of determining the CBof object supports. From the support reduction theoremor theorem 3, the determination of a bound of the CB of anobject support from the convex enclosure A of its autocor-relation support should include the following procedures:(1) A procedure that enumerates all possible partitions ofthe edges of A under certain constraints. A partitionhere means two complementary sets of ordered edge seg-ments; each edge segment may be an edge or a part of anedge of A. If an edge segment belongs to one of the twosets, its centrosymmetric edge segment of A must belongto the complementary set. Each set of ordered edge seg-ments constitutes just half of the periphery of A. (2) Aprocedure for checking the closeness of each selected setof ordered edge segments. If the set forms a closed con-vex polygon, such a polygon will correspond to a possibleCB of the object support. (3) A procedure that centroidseach possible bound of the object support and forms theunion of these bounds.

Our method differs from Brames’s method only in theprocedure for checking the closeness. Brames deriveda closure relation in terms of phasor functions18 andused it for checking the closeness of the selected sets ofedge segments described by edge vectors and integers.In his procedure, integer multiplications are needed. Inour method, although no closed relation is introduced,checking the closeness requires only integer additions andsubtractions.

We suppose that a set of ordered edge segments is de-noted by d1, d2, . . . , dN . We begin with fixed d1 and se-quentially shift d2, d3, . . . , dN so that the first end pointof each succeeding one coincides with the second end pointof the preceding one. The set forms a closed polygon, iffthe final second end point coincides with the first endpoint of d1. If each edge segment di is described by useof the coordinates of its end points, i.e., sxi1, yi1, xi2, yi2d,the shift of di requires only integer additions andsubtractions.

An example of estimating the bound of the CB of anobject support is given in Section 5. In the general caseno efficient algorithm can be found for the CB estimationbecause of the complexity of the problem. In many casesthe convex hulls of object supports are irreducible. Insuch a case, by virtue of the geometric visualization, is itpossible to select a correct set of edge segments by onlya few trials.

D. A Property of Convex SupportsLemma 3.D. If D is convex and De is an embedablefactor of D , then the convex hull of De, D e, is also anembedable factor of D .

Proof. Under the assumption of the lemma, D canbe completely covered by shifting De. We denote theembedding trace of De by Dt. At each point of Dt, theshifted De will be bounded by the bound of D . It is truethat the shifted D e will also be bounded by the boundof D , since D is convex. Otherwise, the convexity ofD cannot hold in that region occupied by the shiftedDe. D e can clearly cover D completely. Therefore thelemma must be true.

E. Nondecomposable Convex SupportsThis kind of support was the most attractive for manyyears. The convexity and nondecomposability imply thatany sequence defined on the support must have an ir-reducible z polynomial. In phase retrieval this propertyguarantees the uniqueness of the solution. Eisenstein’ssupport is a typical one and has been generalized andused in phase-retrieval problems.9,10,12,15

In Ref. 15 Brames defined a type of C reduciblesupport. According to his definition, a support D isC reducible if we can find two supports D1 and D2

such that D D 1 1 D 2. In Fig. 6, D is C reduciblebut convex and nondecomposable. This shows that thedefinition of C irreducibility and the convex nondecom-posability are not exactly the same.

The nondecomposability test can be implemented by theembedability test. Usually the test needs an enumera-tion procedure that is a computationally complex problem.Although no method has been found that is computation-ally efficient, some practical methods exist that are highlyeffective and can solve many practical problems.


Lemma 3.E. Let D be a convex support. If thereexists an elementary edge of D , ei, such that no otherposition of D can be found that is embedable by ei, thenD must be nondecomposable.

Proof. Under the assumption of lemma 3.E., if D isdecomposable then it has a pair of CEF’s, denoted hereby D1 and D2. We can assume that they are convex. Ifthey are nonconvex, from lemma 3.D they can be replacedby their convex hulls, since their convex hulls must alsobe a pair of CEF’s of D . From theorem 3 the elementaryedge ei must be an edge of D1 or D2. On the other hand,D1 or D2 cannot contain ei. Otherwise, for example, ifD1 contains e1, the shift of D1 will imply that there existsat least a different position of D that is embedable by ei.The contradiction implies that the lemma must be true.

This formally simple lemma makes it possible to checkthe irreducibility of supports by inspection. The use ofthis lemma demonstrates that Eisenstein’s support is ob-viously nondecomposable. An Eisenstein support con-tains a rectangle defined by 0 # j # J 2 1 and 1 # k # Kand a single point at sJ, 0d. According to the assumptionin Ref. 9, the points at sJ, 0d and s0, 1d should be nonzero.The edge containing these two points is an elementaryedge that satisfies the condition of the lemma 3.E.

4. NONCONVEX SUPPORTS

A. Decomposability of Nonconvex SupportsIn applications the supports of most objects and their au-tocorrelations are nonconvex. The embedability theoryis valid for nonconvex supports, although the situation ismuch more complicated than that with convex supports.In many cases the support of an object consists of sev-eral separate parts. The convex hull of the support maybe decomposable, but this might be meaningless in ap-plications, because such a decomposability probably doesnot reflect any usable property of the object. Frequentlywe may investigate whether the object is deconvolvableby use of a factor such as a point-spread function. Thefollowing lemma relates to this question.

Lemma 4.A. If a given support consists of two ormore separate parts, then it is decomposable with a con-vex embedable factor iff there exists a common convexembedable factor of all parts.

Proof. The sufficiency part of the assertion is obvious.We need to prove the necessity part. Suppose that D

consists of N parts; then it can be expressed by

D N[

i1Di, Di

\Dj 0y, ;i fi j . (11)

If D is decomposable, it is expressible as

D [

;psm, nd[Dt

Desm, nd , (12)

where De is assumed to be convex. For an arbitraryisolated Di to be covered, there must exist a set of pointsDti such that

Di [

;psm,nd[Dti

Desm, nd . (13)

Thus De is a common embedable factor of all parts.Lemma 4.A and theorem 3 can offer a visual under-

standing of the decomposability of a type of nonconvex

support. If a given support D consists of several iso-lated convex subsupports, one can obtain the convex em-bedable factor(s) of D , if they exist, by checking all edgesof all subsupports. If a set of edge segments can be foundthat is common to all subsupports and that forms a closedconvex polygon, we can check to see whether such a con-vex polygon corresponds to an embedable factor of allsubsupports.

B. Testing the Decomposability of Nonconvex SupportsA nonconvex support may be by chance decomposablewith nonconvex embedable factors. Testing the decom-posability of nonconvex supports is generally tediouswork. In Ref. 15 Brames proposed a method for test-ing the P irreducibility of supports including nonconvexsupports. Suppose that a support D is given. In theP irreducibility test, one must test each possibly differ-ent subsupport of D , Di. According to Brames, each teststep involves calculations of the correlation and convolu-tion of a few characteristic functions. Such a method isfeasible only for very small supports. By virtue of theembedability theory we have a different method: in eachtest step we need to examine whether the subsupport Di

is an embedable factor of D . According to the definition,for each point p [ D we try to find a shift sm, nd suchthat p [ Dism, nd , D . If such a sm, nd is found, wecan continue to check other p’s. If no such sm, nd canbe found for a p, Di is not an embedable factor of D .In practice, not all points p [ D need to be checked.We need to check only those points on the border of D .Here, border can be understood in a conventional way,and any isolated point or line, if it exists, belongs to theborder. For a specified p [ D , only a shift sm, nd forwhich the borders of Dism, nd and D are consistent inthe neighborhood of p needs to be tested. The relatedshift and set operations can be accomplished in terms ofinteger and binary algebra.

In many cases the decomposability of a given nonconvexsupport can be checked by inspection. For the nonconvexsupport A shown in Fig. 8, we can try to cut a border partof A and test it to see whether the cut subsupport is anembedable factor of A. In order for A to be covered com-pletely, the cut subsupport cannot be chosen arbitrarily.The edges of the cut subsupport must be consistent witha set of edges of A. Thus there are not many cut situa-tions to be tested. As can be seen, we cannot find a cutsubsupport that is an embedable factor of A. ThereforeA must be nondecomposable of the first kind.

Fig. 8. Nonconvex support.


Fig. 9. A0 is nonconvex and is irreducible of the second kind. A1 is the convex hull of A0; it has a pair of CEF’s, A1e andA1t . B0 is similar to A0.

Clearly, for handling a more complicated nonconvexsupport efficiently, further work is needed in developingsome new criteria and algorithms for checking the embe-dability of supports.

C. Irreducibility of the Second KindFrom theorem 2 we know that if a support D is non-decomposable of the first kind, any positive sequencewith this support is irreducible. There exists a classof supports such that any sequence with such supportswill be irreducible. This class of supports is said to benondecomposable of the second kind. The convex andnondecomposable supports of the first kind are also non-decomposable of the second kind. However, not all non-decomposable supports of the second kind are convex.For example, the supports A0 and B0 shown in Fig. 9 arenonconvex and irreducible supports of the second kind.In fact, it is impossible to produce this kind of support byconvolution and numerical mergence.

Assume that D is nonconvex and nondecomposable ofthe first kind. Its convex hull is denoted by D . It issometimes possible to modify D by adding some pointswithin its CB so that the resultant support will be decom-posable, although D can remain nonconvex.

Consider a nonconvex support as shown in Fig. 10. Weassume that D consists of all ≤. It can be checked thatD is nonconvex and nondecomposable of the first kind.The convex hull of D , D , will include all points markedby ≤, ±, and ?. Note that if the point marked by ? isadded to D , the resultant support remains nonconvexbut decomposable.

In this example we can image the case that a sequencedefined on the modified support might be deconvolvablesince the modified support is decomposable. The van-ished point marked by ? may be produced by numeri-cal mergence in the convolution operation, provided thatthe two components of the convolution have both posi-tive and negative value points. This consideration canbe generalized. Some nonconvex and nondecomposable(of the first kind) supports can be modified by the ad-dition of some points so that the resultant supports aredecomposable. This implies that the sequences definedon such a nonconvex and nondecomposable (of the firstkind) support might be deconvolvable if the nonnegativ-ity constraint of the sequence is deleted.

Definition 5. Let D be a 2-D nonconvex support thatis nondecomposable of the first kind. Let D be the con-vex hull of D . If there exists an area C such that

1. C # D n D ,2. Da: C < D is decomposable, and3. There exists a (nontrivial) embedable factor of Da,

Dae, and at least two different shifts sm1, n1d and sm2, n2dso that the following relations hold simultaneously,

Daesm1, n1d < Daesm2, n2d # Da ,

C # Daesm1, n1d > Daesm2, n2d ,

then D is said to be generalized decomposable, the ex-tended support Da is called a pseudoconvex extension ofD , and such an embedable factor Dae of Da is called ageneralized embedable factor (GEF) of D .

We must give an explanation of the practical implica-tion of the definition. Point 3 ensures that C can be cov-ered by the GEF more than once. In fact, C contains allthe points added to D so that Da is decomposable. The2-D sequence that we are actually concerned with is de-fined on D . On C the values of the sequence must van-ish. The numerical mergence can occur only when C iscovered by the GEF more than once and the components ofthe convolution contain both positive and negative valuepoints. It can be checked that the supports A0 and B0

in Fig. 9 are not generalized decomposable, although theirconvex hull is decomposable.

If D is nonconvex but decomposable, it must be genera-lized decomposable with an empty C .

Definition 6. Let D be a 2-D support. If it is con-vex but not decomposable, or if it is nonconvex and notgeneralized decomposable, then we say that D is nonde-composable of the second kind.

By repeating the argument of theorem 2, we gettheorem 4:

Theorem 4. Let ysm, nd be a 2-D function with afinite support D . For ysm, nd to be deconvolvable, anecessary condition is that its support D be either con-vex and decomposable or nonconvex but generalizeddecomposable.

Fig. 10. Nonconvex support and its modification.


5. AN APPLICATION: DETERMINING ABOUND OF AN OBJECT SUPPORT FROMITS AUTOCORRELATION SUPPORT

A. Methods of Determining Bounds of Object SupportsThe application of phase-retrieval algorithms shows thata tighter support constraint typically results in faster con-vergence of the algorithm3,8 Many studies have been de-voted to determining a bound of an object support fromits autocorrelation support.16 – 18 Reference 18 describesa method that can be used to determine the convex hull ofan object. Reference 17 presents a set of rules that canoffer a tighter upper bound (single-sided locator set) onthe object’s support. The method in Ref. 18 can offer areliable bound of object supports. When an object sup-port is nonconvex, however, the obtained bound on theobject support is relatively loose. This is unfavorable forspeeding the convergence of phase-retrieval algorithms.The methods proposed in Ref. 17 can usually provide atighter bound on an object support. However, examplescan be constructed to show that, if a given autocorrela-tion support can be generated by several different ob-ject supports, the bound on the object support obtainedby use of the method in Ref. 17 might be incorrect. Bymeans of the embedability theory we can gain a new in-sight into support problems. In this section we propose anew method for determining a bound of an object supportfrom its autocorrelation support. We incorporate someideas from Ref. 18 and combine them with embedabilityarguments. As a result, the new method can produce atight and reliable bound on an object support.

Let D0 and Dr be the object and the autocorrelationsupports, respectively. Since autocorrelation is a kindof convolution, D0 must be embedable into Dr and, byshifting, can cover Dr completely. The complete cover-ing implies that at many locations the part of the bound-ary of D0 and Dr should be coincident, no matter whetherthis part of the boundary is convex. The same importantpoint is that the support of the autocorrelation of the con-vex enclosure of the object is equal to the convex enclosureof the support of the autocorrelation of the object. ThusD0 or D 0 can be determined from Dr or D r, as discussedin Subsection 3.C. By virtue of these results, we haveobtained a concept for determining a bound of an objectsupport.

We consider the example shown in Fig. 11. For thegiven convex hull D r (we show the faces of the convexenclosures in this figure for clarity), in Ref. 18 Bramespointed out three object supports that generate the sameautocorrelation support D r as shown in Fig. 11(a) (onlythe faces are shown). Since D r is a convex hull, theobtained D i, i 1, 2, 3, should be considered as possibleconvex hulls of object supports, and one of these convexhulls must be correct. We assume that D i is the correctconvex hull. Let pok be a vertex point of D i (i.e., a vertexpoint of D0). We embed and shift D i in D r and imaginethat D0 is shifted in Dr . There will be a set of borderpoints, including one or two vertex points of Dr thatcan be covered by the shifted pok. Such a set in Dr isdenoted by PsDr ,pok d. If prk [ PsDr ,pok d, then prk impliesan embedable location of D0 in Dr . This location can bedescribed by the coincidence of the point pair f pok; prkg,which is said to be a coincident point pair. The set that

consists of all possible coincident point pairs of D i and Dr

is denoted by QsDr ,Di d. Specifically, the set that consistsof all possible coincident vertex point pairs of D i and Dr

is denoted by QsDr ,Di ,vd.In Fig. 11 f po1; pr3g is a coincident point pair. The

coincidence of these two points determines an embedablelocation of D0 in Dr. This implies that when we shiftD 1 in D r so that these two points coincide with eachother, D0 must cover a part of Dr, including a part ofthe border of Dr. In other words, this part of the borderof Dr must coincide with a part of the border of D0 inthe neighborhood of these two points. The above analysiscan be described precisely by

D0 , D 1 > Drsf po1; pr3gd .

Generally, we have

D0 , D i > fff\

;f poj ;prj g[QsDr ,Di ,vd

Drsf poj ; prj gdggg . (14)

Now we suppose that pr1, pr2, pr3, pr4, and pr5 areon the border of Dr, as shown in Fig. 11(a). To coverDr by shifting D0, we can reach these points by po1

of D 1. This implies a set of available coincident pointpairs. Equation (14) can be generalized as

D0 , D i > fff\

;f poj ;prj g[QsDr ,Di d

Drsf poj ; prj gdggg . (15)

For a given Dr we can get a number of D i that gener-ate D r . To avoid a false result caused by some invalidchoices of D i, we must use

D0 ,[

;D i

hD i > fff\

;f poj ;prj g[QsDr ,Di ,vd

Drsf poj ; prj gdgggj (16)

or, in general,

D0 ,[

;D i

hD i > fff\

;f poj ;prj g[QsDr ,Di d

Drsf poj ; prj gdgggj . (17)

Equations (16) and (17) are the final formulas that canoffer a reliable and tight bound of the object support.

Fig. 11. (a) Faces of an autocorrelation support, (b) three objectsupports that generate the same autocorrelation support asin (a).


Fig. 12. (Ordering is from the top-left corner to the bot-tom-right corner.) (1) object support D0. (2) autocorrelationsupport Dr . (3) convex hull D r . (4) D11, the first possiblebound of the CB of the object support, determined fromD r . (5) D12, the bound of the object support, obtained fromD11 and Eq. (14); D12 D11 > s>6

i1 Drfffsf p0; pr gidggg, wheref p0; prg1,2,...,6 fs24, 1d; s47, 1dg, fs24, 24d; s47, 29dg, [(24, 24);(29, 47)], [(1, 24); (1, 47)], [(1, 19); (1, 19)], [(19, 1); (19,1)]. (6) D13, the bound of the object support, obtained fromD11 and Eq. (15). D13 D11 > fffffffff >14

i1 Dr sf p0; prgidggg, wheref p0; prg1,2,...,14 contain those point pairs in (5) and the following:>28

i25hfs24, 24d; s47, idg > fs24, 24d; si, 47dgj. (7) D21. (10) D31.(13) D41. These are other possible bounds of the CB of theobject support, determined also from D r . (8), (9) D22 and D23.(11), (12) D32 and D33. (14, (15) D42 and D43. These areresults similar to D12 and D13. (16) Db1 , the finally obtainedbound of the object support by the new method; Db1 >10

i1Di3.(17) Db2, the bound of the object support obtained by Brames’smethod.18 (18) An illustration used by Crimmins et al.17 toshow the necessity of their u-convex condition. (19) Db3, thebound of the object support by use of the two-point rule of Ref. 17.

B. Examples of Determining a Boundof an Object SupportOur method consists of the following steps:

1. Determine possible convex hulls of D0 from D r bymaking use of the methods introduced in Subsection 3.C.Generally, we can get a set of possible convex hulls D i,i 1, 2, . . . , M .

2. For each D i we determine a bound of the objectsupport Di by means of the method expressed by Eq. (14)or Eq. (15).

3. We centroid each possible bound of the object sup-port Di and form the union of these bounds as shown byEq. (16) or (17). The result would be a tight and reliablebound of the object support.

The first example is shown in Fig. 12. This exam-ple was used in Ref. 17 by Crimmins et al.. In Fig. 12,(1) is the original object support; (2) is the given autocor-relation support Dr; (3) is D r . (4), (7), (10), and (13)are four different D i that generate the same D r . Thetotal number of such D i is 10 for this example. Thecentroided union of these D i is shown in (17), which ex-presses the bound of the object support obtained by use ofBrames’s method. (5), (8), (11), and (14) are the boundsobtained from Eq. (14). (6), (9), (12), and (15) are thebounds obtained from Eq. (15). The centroided union of

these bounds gives our final result shown in (16). (18)was used in Ref. 17 by Crimmins et al. to show the ne-cessity of their u-convex condition. (19) is the bound ob-tained by use of the two-point rule of Ref. 17. The threeresults obtained by the three methods (ours, in Ref. 18,and in Ref. 17) can be compared from (16), (17), and (19),respectively.

In the second example the original object is similarto that used in Ref. 17 by Crimmins et al. The givenautocorrelation has the support shown in Fig. 13(a). Thevertex points of this support are marked by white points,which indicate the convex hull D r . For this examplewe can find only one possible convex hull of the objectsupport. We mark the vertex points of the convex hullof the object support by white points in Fig. 13(b). Theoriginal object support is also shown for comparison. Forthis example, only Eq. (14) is needed for determining thebound of the object support, since no additional coincidentpoint pair is available. The result is shown in Fig. 13(c).We find that the determined bound is tight and no pointis lost.

Fig. 13. Example of the estimation of the object’s support fromits autocorrelation’s support. (a) The given autocorrelation sup-port; the white points mark the vertex set of its convex enclosure.(b) The calculated vertex set of the convex enclosure of theobject support is marked by white points. For comparison, thetrue object support is also shown. (3) The obtained supportestimation after step 3 in the text.


C. RemarksIt would be interesting to indicate some relations be-tween our method and the method used by Crimminset al.17 We may say that the rules presented in Ref. 17can be interpreted in terms of the embedability theory.In fact, the two-point rule in Ref. 17 is nearly equivalentto Eq. (14), whereas the other rules in Ref. 17 includ-ing the totally asymmetric rule and the u-convex rule,closely relate to our Eq. (15). Our method begins withdetermining the CB’s of the object support and incorpo-rates the advantage of Brames’s method. As a result,the method offers a reliable bound on the object sup-port. In the same time, the method makes full use of theinformation available from the autocorrelation support.Consequently, the finally obtained bound on the objectsupport is tight. An additional advantage is that themethod has a regular form. It is simple, efficient, andgeneral enough for practical use.

In practical applications the autocorrelation supportcannot be exact because of the noise involved. To es-timate the autocorrelation support, we must use somehigher threshold so that the effect of low-level noise canbe reduced. In such a case the obtained support bound ofthe object is usually not sufficient. Enlargement of theobtained support bound is needed so that the constraintis not an inadvertently truncating part of the object, assuggested by Crimmins et al.17

6. CONCLUSIONSIn this paper it has been revealed that the embedabilityof supports has an essential importance in the investiga-tion of the reducibility or irreducibility of supports. Thetheory can offer a new insight and a geometric demonstra-tion of the support problems. From the theory we knowthat many 2-D objects with certain supports are neces-sarily nondeconvolvable; objects having separated partsare more likely to be nondeconvolvable because, with-out a special physical background such as a convolutionprocess with a common point-spread function, the possi-bility of the existence of a common embedable factor of allparts is small; objects having nonconvex supports can benondeconvolvable, because many nonconvex supports arenot decomposable. On the other hand, a 2-D sequenceysm, nd with a support may be deconvolvable or nonde-convolvable. If it is deconvolvable, the supports of itscomponents must be a pair of CEF’s (if the nonnegativityconstraint is imposed) or GEF’s (if no nonnegativity con-straint is imposed) of the support of ysm, nd. For blinddeconvolution the support of ysm, nd is known. By find-ing the CEF’s of the support we may gain more informa-tion about the solution. In many cases the pattern of a2-D sequence may be quite complicated. By inspection ofthe embedability of the pattern, we might obtain a concep-tion of the deconvolvability of the sequence and a roughestimation of the components of the sequence.

We have proved that the irreducibility of a class ofsupports including Eisenstein’s support can be checkedthrough a simple lemma and by inspection. We havesuggested a new description of support irreducibilitybased on convolution and embedability arguments. Asan application, we have proposed a new method for de-termining a reliable and tight bound on an object support

from its autocorrelation support. The method is simple,regular, effective, and general enough for practical use.

We think that embedability is a new subject in mathe-matical morphology. In this paper we have proposed afew primary ideas. Further exploration is clearly neededinto both theoretical and practical aspects of these ideas.

ACKNOWLEDGMENTSThis work was supported by the German Research As-sociation (Deutsche Forschungsgemeinschaft, DFG). Wethank the reviewers for carefully reading the manuscriptand giving many insightful comments that helped us toavoid some errors that existed in the earlier version ofthe manuscript.

*On leave from the Institute of Electronics, AcademiaSinica, Beijing, China.

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Embedability theory of two-dimensional supports, with applications

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