2796 ON 41, Locally Monotonic Regression...2796 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 41, NO. 9, SEPTEMBER 1993 Locally Monotonic Regression Alfred0 Restrepo, Member, IEEE,

2796 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 41, NO. 9, SEPTEMBER 1993

Locally Monotonic Regression Alfred0 Restrepo, Member, IEEE, and Alan C. Bovik, Senior Member, IEEE

Abstract-The concept of local monotonicity appears in the study of the set of root signals of the median filter and provides a measure of the smoothness of a signal. The median filter is a suboptimal smoother under this measure of smoothness, since a filter pass does necessarily yield a locally monotonic output; even if a locally monotonic output does result, there is no guarantee that it will possess other desirable properties such as optimal similarity to the original signal. Locally monotonic regression is a technique for the optimal smoothing of finite- length discrete real signals under such a criterion. A theoretical framework where the existence of locally monotonic regressions is proven and algorithms for their computation are given. Regression is considered as an approximation problem in R", the criterion of approximation is derived from a semimetric and the approximating set is the collection of signals sharing the property of being locally monotonic.

I. INTRODUCTION

HE rbnning median was conceived by Tukey as a tool T for the exploration of time series. It is used as a smoother of discrete signals, due to its ability to preserve monotonic segments, edges, and constant neighborhoods [ 11, 121, while eliminating short-duration pulses; since signal components of these types have overlapping frequency spectra, accomplishing the same task with an invariant linear filter is impossible.

The median filter has a simple and concise local definition; however, it is not easy to provide a precise global characterization of it. This becomes evident in the characterization of the set of its root signals [ I], [3], [4], which is not obvious. For the smoothing of signals, several al- ternatives to the median filter have been proposed, e.g., rank order smoothers 151, the recursive median filter [6], moving trimmed means [7], FIR-median filters [8] and others.

A property of signals that has relevance in the study of the median filter is local monotonicity [ 11. Local monotonicity provides a criterion of smoothness since it sets a restriction on how often changes of trend (increasing to

Manuscript received July 26, 1991; revised November 1 I , 1992. The associate editor coordinating the review of this paper and approving it for publication was Prof. Aggelos K. Katsaggelos.

A. Restrepo was with the Department of Electrical and Computer En- gineering, University of Texas at Austin, Austin, TX 78712-1084. He is now with the Departamento de Ingenieria Elictrica, Universidad de 10s Andes, A.A. 4976, Bogota, Colombia.

A. C. Bovik is with the Department of Electrical and Computer Engi- neering, University of Texas at Austin, Austin, TX 78712-1084.

IEEE Log Number 9210114.

decreasing and vice versa) may occur. In the evolution of this idea, it is natural to ask for optimal smoothers under a criterion of local monotonicity.

As pointed out in [9], the design of nonlinear smoothers is in many instances an art rather than a science. Locally monotonic regression is a direct, optimal approach for smoothing finite-length discrete signals, under a local monotonicity criterion.

The concept of local monotonicity is relatively unex- plored and has been mostly limited to the theory of a large class of nonlinear filters that includes median filters 113, [2], rank-order filters [6], and order-statistic filters [ 101-

The smoothness constraint of local monotonicity, on a discrete signal, does not limit the magnitude of the changes the signal makes in going from one coordinate point to the next one but it does limit the frequency of the oscillations of the signal. When a discrete signal increases its value from one coordinate to the next, it is said to have an increasing transition; analogously for a decreasing transition; in a locally monotonic signal, between an increasing transition and a decreasing transition there is always a constant segment of a specified minimum length [l] , [2]. Unlike other properties, such as linearity, local monotonicity is defined at the local level rather than at the global level: a locally monotonic signal is monotonic at the local level while at the global level it may be monotonic or not. Local monotonicity is a meaningful measure of signal shape in many instances; for example, the scan lines produced by a convex object in a digital image are often locally monotonic. Departures from local monotonicity may indicate the contamination of a signal with noise and, given a nonlocally monotonic signal, it may be desirable to find a signal that is both similar to the given signal and locally monotonic.

The similarity between two signals may be measured using a semimetric for R". The use of semimetrics instead of metrics is not wasteful; the approximation of signals under a semimetric criterion provides maximum likelihood estimators of signals embedded in very impulsive noise [ 131, [ 141. We consider a space of finite dimension where the Heine-Bore1 theorem [15] holds and the existence of regressions is easily proven.

Given a nonlocally monotonic signal v , a signal w in the set of locally monotonic signals that is closest to v is said to be a locally monotonic regression of U . In this paper, the existence of optimal approximations is shown; algorithms for computing the approximations are devised.

[121.

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RESTREPO AND BOVIK: LOCALLY MONOTONIC REGRESSION 2791

0 2 4 6 8 10

(a)

350

3w

250

200

150 0 2 6 8 1 0

(b) Fig. 1 . Comparison of 3-point median filtering and 101110-3 regression. The original signal is indicated by white squares. (a)

Three-point median filtered signal (black diamonds); (b) lomo-3 regression is shown in black diamonds.

Even for signals of relatively short duration, the algorithms are computationally expensive; here we define and explore a new smoothing technique; faster algorithms are the subject of further research [16].

To illustrate these ideas, consider the signal segment [224, 192,254,278,249, 312,259, 223,2571, shown in white squares in Fig. 1, taken from [17] (data on bank suspensions corresponding to the years 192 1 - 1929). When this signal is filtered with a (padding) median filter of window size 3 , the signal [224, 224, 254, 254, 278, 259,259,257,2571 results (black diamonds in Fig. l(a)); this signal is not smooth due to the peak at coordinate 5. A locally monotonic regression [208, 208, 254, 263.5, 263.5, 285.5, 285.5, 240, 2401 of the original signal is also shown (black diamonds in Fig. l(b)).

The theoretical framework developed here applies to regressions defined with respect to any nonempty closed subset of R". The set of locally monotonic signals is relatively large in R". For example, the set of (affine) linear signals is a proper subset of the set of locally monotonic signals; a larger approximating set provides more free- dom in the choice of a similar signal: a locally monotonic regression is at least as close to the signal being regressed as a linear regression is. The approximation of a signal from a set of signals having a given characteristic is an optimization problem. The study of algorithms that com-

pute regressions falls within the field of computational geometry [ 181 where properties of subsets of R" are studied with the help of computers; the design of algorithms that perform regression must be treated on a case-by-case basis.

The remainder of the paper is organized as follows. In Section 11, a collection of semimetrics is introduced; they are used as the distance measures under which regression is performed. Conditions for the existence of regressions as well as bounds on the cardinality of the set of regressions are developed. Section I11 develops the concept of local monotonicity, gives a characterization of the boundary of the set of locally monotonic signals, and deals with the concepts of constant regression and of constant semiregression, which are used to compute locally monotonic regressions. In Section IV, algorithms that compute locally monotonic regression are developed; after present- ing examples, the paper is concluded in Section V with suggestions for further research.

11. SIGNAL REGRESSION UNDER A FAMILY OF SEMIMETRICS

In this section, a particular collection of semimetrics is presented, the concept of regression (or projection) is ex- amined, and the existence and multiplicity of regressions are addressed.


2798 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 41, NO. 9 , SEPTEMBER 1993

An integer intervalla, bL where a and b are integer numbers, is defined as the set {c E 2: a I c I b} of integer numbers that are greater than or equal to a and smaller than or equal to b. An n-point signal (or a discrete signal of length n) is a real function x having as domain a nonempty integer intervalla, bl, where b - a = n - 1. Its graph is a subset o f l a , b / x R' and is usually drawn as a coordinate-domain plot (called its time series repre- sentation); alternatively, the signal may be thought of as a point in R" (Fig. 2). The origin [0, 01 of R" is denoted as 6. Given a subset S of R", Cl(S), Bd(S), and Int(S) , respectively, denote the closure, the topological boundary, and the interior of S , in the standard topology for R" [15].

A. A Family of Semimetrics A semimetric for R" is a positive definite, symmetric

function d : R" X R" + [0, 003. Such properties may be stated as follows:

V x, y E R"; d ( x , y) = 0 e x = y (positive definite)

V x, y E R"; d ( x , y) = d ( y , x)

Thus, many functions are semimetrics for R". For mea- suring the similarity between signals it is convenient that the semimetric be continuous, positive homogeneous and translation invariant:

V x, Y E R"; V y E R;

(symmetric).

d ( y x , y y ) = Iyl d ( x , y ) (positive homogeneous)

V x, y , z E R"; d ( x + z, y + z) = d ( x , y )

(translation invariant).

A metric is a semimetric that has the triangle-inequality property :

V x, y , z E R", d ( x , z) I d ( x , y ) + d ( y , z).

The collection of the p-semimetrics is a well-known collection of translation invariant, positive homogeneous continuous semimetrics indexed by the parameter p E (0, a]; given two signals x = [x,, - * , x,l and y = [VI ,

, ynl, define . . .

d,(x, y ) = max { ( x i - yi(: 1 I i 5 n } .

As a function of p , d p ( x , y ) is a nonincreasing [19] continuous function and, as p increases, for each x and y in R", d p ( x , y ) tends to d,(x, y ) . F o r p E [ I , 0 3 1 , dp is a metric; f o r p E (0, l), dp is a semimetric but is not a metric. The collection includes the Euclidean metric d2 , the square metric d l and the product metric d, .

For p E (0, 00) and x E R", the set B p ( x , p ) = { s E R": dp (x, s ) c p } is the open p-ball of radius p centered at x. The p-distance between a point x and a set S is given by

Dp(x, S) = inf { d p ( x , s): s E S } .

R " (a) (b)

Fig. 2 . A finite-length discrete signal U may be represented (a) as a point in R", or (b) as a time series, in its coordinate domain.

\

B. p-Regressions Signals are usually processed on the basis of a closed

formula that determines a transformation R" -+ R", e.g., circular convolution with a given signal. An alternate approach is used here; assume that a desirable property for a signal to have has been specified. Given a signal lacking the property, the problem is to find a signal from among the set of signals having the property, that is, closest to the given signal. Such a property is defined in terms of the coordinate domain of the signals and as such may be called a shape restriction. Let Q be the truth-value function that is true on signals meeting the constraint and false otherwise, and let A = {s E R": Q(s ) } be the set of signals with the required shape.

Suppose that A is a nonempty proper subset of R" and let x E R". The set

RA(x) = {U E A: d p ( x , U ) = Dp (x, A)}

is called the set of p-regressions (or p-projections) of x with respect to A.

Two-projections on subspaces of R" are projections in the usual sense of the word; that is, the difference between a point and its 2-projection on a subspace is orthogonal (given the standard inner product on R") to the subspace. The use of semimetrics different from the Euclidean metric generalizes the concept of projection in a metric way and provides several criteria to measure the similarity between signals. It turns out that it is useful to have these many criteria; for example, the maximum likelihood estimates of lomo signals contaminated with certain very impulsive noises are regressions of the noise signal under a semimetric that is not a metric; similarly, if the noise is uniform, the maximum likelihood estimates are regressions under the product metric [ 141, [20].

C. Existence and Uniqueness of Regressions For each p E (0, a)] and for each x E R", since the

p-semimetrics are continuous, the restriction dp (x, - ) of dp to {x} x R" is a continuous function. Given a nonempty, proper, and closed subset A of R", and points a E A and x E (R" - A), let U = d p ( x , a) . The closed ball = C1[Bp(x, U)] is nonempty and bounded and, from def- initions, it follows that it contains the set of p-regressions of x with respect to A. Since R" has the Heine-Bore1



Fig. 3 . p-balls of the same radius and center in R 2 , for p = 1 / 2 , I . 2, and W .

property [ 1 5 ] , is compact and since d,,(x, 0 ) is continuous, the image under d,,(x, e ) of B rl A is closed and contains its infimum D,, (x, A ) . Since the points in B n A that are mapped to D,,(x, A ) are the regressions of x, this proves the existence' of p-projections on nonempty, closed, proper subsets of R".

Moreover, if A is a proper subset of R", x is a point in (R" - A ) and p = Dp (x, A ) , the set of p-regressions RA (x of x with respect to A is given by Bd[B,,(x, p ) ] n A [20].

A related topic is that of the cardinality of the set of regressions, that is, the multiplicity of the solution set to the approximation problem. It depends on the type of convexity of both the p-balls and of the approximating set A under consideration. A subset S of R" is said to be convex if the line segment between each pair of points of S is a subset of S ; a subset S is said to be strictly convex if for each pair of points x and y in Bd(S), the open line segment ( a x + (1 - a ) y : a E (0, l)} is a subsct of Int(S).

For p E (1, 03), p-balls are strictly convex, for p = 1 and forp = 03, p-balls are convex but not strictly convex, and f o r p E (0, l), p-balls are not convex. The fact that, for p < 1 , p-balls are not convex and dp is not a metric is not a coincidence: in a large and general collection of semimetrics, those semimetrics that determine convex balls are metrics and those that determine nonconvex balls are not metrics [20]. Fig. 3 depicts p-balls in R2 for p = 0.5, 1 , 2 , and 00.

The following lemma says that if the set A is convex and Bp(O, 1 ) (the convexity of a p-ball is independent of its radius or center) is strictly convex then, for each x in R", RA@) is a singleton set, that is, regressions are unique. If Bp(O, 1 ) is only convex then regressions may or may not be unique. The proofs of all lemmas are given in Appendix A.

Lemma I: If A is a nonempty convex closed subset of R", p E (1, 03) and x is in R" - A , then the boundary of the p-ball centered at x with radius Dp (x, A ) intersects A at exactly one point.

In this section, the problem of signal regression with respect to a semimetric d,, has been defined within a general theoretical framework. From a mathematical stand- point, it generalizes the concept of projection of points on subsets of R"; from a signal processing perspective, it de- fines the concept of shaping a signal, given a shape re-

'The authors thank Prof. I . W. Sandberg for showing them this elegant proof of the existence of regressions.

striction. The problem of how to compute (if any) the elements of RA (x) depends on the particular choices of A and p and must be treated on a case-by-case basis. When A is the set of locally monotonic signals, locally monotonic regression results. When A is the set of constant signals, constant regression results; when A is the set of linear signals, which is the plane spanned by the signals 0 , [ 1 , 1, - n ] , linear regression results, etc. 11 and [ 1 , 2, *

111. LOCAL MONOTONICITY AND CONSTANT REGRESSION

In this section, the concepts of local monotonicity and of constant regression are explored. Monotonicity is a property of time series that has been successfully ex- ploited in the field of statistical estimation [21]. Local monotonicity is a property of one-dimensional signals that provides a measure of the smoothness of a signal; it con- strains the roughness of a signal in a particular way. It does so not by limiting the support of its Fourier trans- form, nor by limiting the number of its level crossings, but rather, by imposing a local constraint on how often a change of trend (increasing to decreasing or vice versa) may occur. In a sense, it limits the frequency of the oscillations that a signal may have, without restricting the magnitude of the changes the signal makes. Constant regression, or signal approximation with constant signals, is an important tool for the computation of locally monotonic regressions, and is used extensively in Section IV.

If u is an n-point signal and y is an integer less than or equal to n , then a segment of u of length y is a signal [v , + 1 , , v, +,I whose components are consecutive components of v ; it is the restriction of U to/i + 1, i + y 1 A signal U : / 1, n / -+ R is monotonic if either v 1 I v2 5 e - . I v , o r v l 1 v2 2 1 v, and strictly monotonic if either v 1 < v2 < - * < v,, or v l > v2 > . * > v,. If v l = v2 = - - . = v,, v is constant. Signals of length 1 are defined as monotonic. The concatenation u'1u2 of U' = [ v i , , vr+,]

, v,+,], of length r is the signal [ v i , *

+ s. A constant segment [ y o , * - - , ub] whose components take the value x is denoted ( / a , b/ , x), where the intervalla, b/is the width of the signal and the real number x is the level of the signal. A signal can be uniquely expressed as the concatenation of constant segments, where the level of each segment is different from the levels of its (at most two) neighbor segments. This is called the (canonical) segmentation of a signal into constant segments.

*

, v,] and v2 = [ v , + ~ , * *

3 V r , v r + l , * * * *

A . Local Monotonicity

A signal is locally monotonic of degree a , or lomo-a (or Zomo, if a is understood) if each of its segments of length a is monotonic. Except when stated otherwise, from this point on it is assumed that the length n and the degree of local monotonicity CY are given, with 3 I (Y I n. The set of signals of length n that are locally monotonic


2800

of degree CY is denoted as A and the collection of monotonic signals of length n is denoted as M.

If components xi and xi + of a signal x are such that xi < xi + I (x; > xi + it is said that x has an increasing (decreasing) transition at coordinate i. If a signal is lomo- CY and has an increasing transition at j and a decreasing transition at k then the signal has a constant segment of length at least a - 1, with coordinates between j and k [l] . Each signal in (A - M) has a constant segment of length at least CY - 1.

If (3 and y are natural numbers with 3 I p 5 y, a signal that is lomo-y is lomo-6 as well; thus, the lomotonicity of a signal is defined as the highest degree of local monotonicity that it possesses. (The minimal degree of local monotonicity any signal has is one; also, any signal of length at least two is lomo-2.)

Regression with respect to the set A is called locally monotonic regression; from the results in Section 11, locally monotonic regressions exist provided that A is closed.

Lemma 2: A is closed. Let I? be the set of signals having at least one constant

segment of length at least two. Lomo regressions of nonlomo signals lie on the boundary of A; Lemma 3 says that Bd(A) = A n I? and that Int(A) = A n r'.

Lemma 3: Bd(A) = A n r and Int(A) = A n I". From Lemma 3 we conclude that the lomo regressions

of nonlomo signals have constant segments of length at least two. This suggests a way in which lomo regressions may be computed.

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 41, NO. 9, SEPTEMBER 1993

B. Constant Regression Constant regression is regression with respect to the set

of constant signals 9 , which is the line spanned by 6' and [ l , 1, - * * , 11. As shown in Section IV, constant regressions are used when computing lomo regressions to replace segments of the signal being regressed.

In what follows, we use the order statistics v ( ~ ) , , . , v, [22], where

v(;) is the ith order statistic of U . If n is odd, v ( [ n + is the median of U . If n is even, the closed interval [v(,12), v ( [ , , / ~ ] + I)] bounded by the two central ranked values of v , is the set of medians of v . The midrange of v is [ v ( ~ ) + vcfl,]/2; the range of v is v(,) - v ( ~ ) .

For p E (1, oo), the strict convexity of p-balls and the convexity of 9 guarantee the uniqueness of constant regressions. a-balls are not strictly convex; nevertheless, constant oo-regressions are unique.

By definition, a constant p-regression of a signal v = [ V I , * * * , v,] is a constant signal of length n where each of its components has a value of x and its distance to v is minimal. For p = 2, the value of x is the average of the components of U . For p = 1, x is the median of v if n is odd, and any of the medians of v if n is even. For p = a, x is the midrange of v [20]. F o r p E (1, 00) a n d p f 2, there are known closed-form expressions for computing constant regressions; they can be numerically approx-

of a signal v with components v I , -

TABLE I MULTIPLICITY OF CONSTANT REGRESSIONS

p Convexity of p-ball Multiplicity of Constant Regressions

E (0, 1 ) Not Convex Finite (Possibly Multiple) 1 Convex, Not Strictly Unique or Uncountable E (1, w ) Strictly Convex Unique W Convex, Not Strictly Unique

TABLE I1 COMPUTATION OF CONSTANT REGRESSIONS

P Constant Regressions of [U,, . . . , U,]

E (0, 1) E { [ u 8 , U,, . . . , U,, U,]: i = / l , n / ) 1 2 € ( I , m) - (21

Median(s) of [U,, U?, . . . , U,] Average of [U,, . . . , u,J N o Known Closed-Form Expression Midrange of [U!, . . . , U"] 00

imated by minimizing the error function n

For p E (0, l), given a signal v = [vl, * * , v,], for each point in {vi: i E/ 1, n / 1, the function

/ I I \

has a local minimum in the variable x; each of the constant signals (/ 1, ./, vj), j E/ 1, n/that locally minimizes the error is a constant semiregression; the constant regressions of ZJ are its constant semiregressions at the smallest distance.

Tables I and I1 summarize the multiplicity and com- putability of constant regressions, while Fig. 4 shows constant regressions of the signal [6 , 15, 1 , 3 , 2, 51 f o r p = 1 /2, 1, 2 and 00. The values of the components of the regressions shown are, in respective order, 3 , 4, 5.33, and 8.

IV. COMPUTING LOCALLY MONOTONIC REGRESSIONS

In this section algorithms for computing lomo p-regressions are given. The set of lomo signals is a disjoint union of a large number of convex cones; computing lomo regressions is a complex problem. Two algorithms are de- scribed: the blotching algorithm and the tube algorithm.

A . The Complexity of the Problem The sign skeleton s of an n-point signal v is the (n -

1)-point trivalued signal that, for each i in/ 1, n - 1 has ith component si with a value of - 1, 0 or + 1, respectively, if (vi + - vi) is negative, null, or positive. A sign skeleton contains sufficient information to determine the lomotonicity of the signal it is derived from. The set of signals having a given sign skeleton is a convex cone (a



0 1 2 3 4 5 6 7

Fig. 4. A signal (white squares) and four constant regressions under the product metric (black squares), Euclidean metric (white diamonds), square metric (black squares), and 0.5-semimetric (black diamonds).

cone is a subset of R" closed under multiplication by positive real numbers); the collection of all such cones par- titions R" into 3"-' convex cones; let G = {Ci: i € / I , 3" - I/} denote this partition. The set 9 of constant signals is the cone in of signals having null sign skeleton; 9 is the only cone in G that is a subspace.

Clearly, a subset C' of G provides a partition of the set A of lomo signals: A is a finite union of convex cones, it is not a subspace, nor a convex set, nor a cone. The problem of finding 2-projections on algebraic subspaces or convex subsets of R" is solved [23]. Since the set of projections of a signal onto each of the cones in G' contains the projections of the signal on A, this number of cones provides a measure of the complexity of the problem of computing projections on A. For example, the set of nondecreasing signals and the set of nonincreasing signals of length n are each the union of 2"-' cones in G, and the set of monotonic signals is the union of 2" - 1 cones (constant signals are both nondecreasing and nonincreasing). The set of locally monotonic signals of degree 3 is the union of (3/2 + &)A;-* + (3/2 - &)A;-' cones of G, where X I = 1 + &and X3 = 1 - h ( s e e A p p e n - dix B for the derivation of this formula). Accordingly, for n = 3, 4 , 5 , 6, 7 , and 8, this number is 7 , 17, 41, 99, 239, and 577, respectively. For degrees of local monotonicity larger than 3 the complexity is somewhat smaller, but it is clear that these numbers grow very fast with n .

B. Blotched Signals The boundary of A consists of lomo signals that have

constant segments of length at least 2; each lomo 2-regression of a signal may be obtained by replacing some segments of the signal with their corresponding regressions [24]. This is also true for other values of p ; locally monotonic regressions are obtained by replacing certain segments with constant regressions if 1 I p < 03 [25], and with constant semiregressions if p < 1. For p = 03 some, but not necessarily all, lomo regressions are obtained in this way.

The procedure of replacing a collection of nonoverlap- ping segments of a signal with their corresponding con-

stant regressions or semiregressions is called blotching and the resulting signal is a blotched version of the original signal. The term "blotch" has been used in the lit- erature with a similar meaning to describe a sometimes undesirable effect of the median filter [26] when applied to images. In [3] the term block is used with the meaning of longest constant segment.

Consider the collection of the 2" - subsets of / 1, n - 1 1 partially ordered by set inclusion. Each such subset K, determines the subspace S, = {U: (V j E K,) v, = v, + of R" of the signals whose sign skeletons have null components (at least at) at the coordinates in K,; their collection is denoted S = { S I : i E/ 1, n - 1 /v, = v, + Moreover, to each subset K, there corresponds a set H, of disjoint open subintervals of / 1, n 1 whose union express K, U K: as a minimal union of integer intervals, where K : = { j : ( 3 k E K , ) j = k + l } . For example, to K = (1, 2, 5, 6) there corresponds H = { / l , 3 / , / 5 , 7/}.

For p E [ l , 001, given a set H, , a blotching transformation on a signal produces the signals that result from replacing the segments with coordinates indicated by H, with their constant regressions. Thus, for p E [ 1, 03) the blotched versions of a signal are its projections on SI (see the proof of Lemma 4 in Appendix A). For p E (1, 033

such blotched versions are unique. For p = 1, the number of blotched versions may be infinite, if the length of a segment to be replaced is even. For p = 00, all blotched versions are projections on S, but not all projections are blotched versions; the reason has to do with the particular way in which the product metric is defined, see the com- ment after the proof of Lemma 4 in Appendix B.

For p < 1, given a set H, , a blotching transformation produces signals that result from replacing the indicated segments with either constant regressions or constant semiregressions. Thus, all projections on SI are blotched but not all blotched versions are projections.

Summarizing, there are 1 - 1 correspondences between the sets K,, the sets H, , and the spaces S,; H, denotes the blotching transformation and S, its projecting set. For example, under the product metric, the H = { / 1, 3 1 /5, 6/} applied to [ 1, 2, 5 , 4, 3, 9 , 1, 6 , 71 implies the replacement of segments [ l , 2, 51 and [3, 9, 11 with con-



0 2 4 6 8 10

Fig. 5 . The 8-point signal 11, 2 , 5 , 4 , 3 , 9, 1 , 6, 71 (white squares) is blotched with {/ l , 3 / / 5 , 7 h , producing the blotched signal [3 , 3 , 3 , 4 , 2 , 2 , 2 , 6, 71 (black diamonds).

0 2 4 6 8 10

Fig. 6. A signal (white squares) and its lomo-5 1/2-regression (black squares) obtained by replacing the segments with coordinates in/2, 4/and in/5, 7/with constant semiregressions.

stant regressions, producing [3, 3, 3, 4 , 5 , 5, 5, 6, 71 (Fig. 5) .

For length-n signals, there are 2"-' blotching transformations (counting the "do-nothing" transformation given by the sets K = 6, H = {6} and the space S = R", and the "replace-the-whole-signal" transformation given by the sets K = / 1, n - 1 1 H = { / 1, n / } and the space S = 4'). The collection of blotched versions of v produced

Example: Consider the 8-point signal [0 , 7, 5, 7 , 4 , 6, 4, 111 shown in Fig. 6 and its lomo-5 (1 /2)-regression (which is monotonic) [0, 5 , 5 , 5, 6, 6, 6, 111 obtained via the blotching transformation { /2, 4 1 / 5 , 7 / } replacing the segments [7, 5, 71 and [4, 6, 41 with constant semiregressions. That the result is a regression follows since it is a blotched version at the shortest distance: (5.6569)2.

by Hi is denoted Hi (U). Lemmas 4 and 5 show that, for p E (0, oo), all lomo regressions are blotched versions of the signal being regressed; fo rp = 03, some, but not necessarily all regressions are blotched versions of the signal.

Lemma 4: Let p E [ 1, 03) and w be a lomo p-regression of U. Then w is a blotched version of v .

Forp < 1, as the following lemma shows, the constant segments in a lomo regression may not be constant regressions, but they are constant semiregressions. The example after the lemma shows that constant semiregressions must be considered when p < 1.

Lemma 5: Let p E (0, l) , let w be a p-projection of a signal v on A and let w = w' 1 w2( * 1 wm be the segmentation into constant signals of w ; let v = v11v21 (vm be the segmentation of v corresponding (i.e., having some coordinates) to that of w . Then, for each i ~ / 1 , m l wi is a constant semiregression of U'.

C. %e Blotching Algorithm

ing algorithm. As a result of the discussion above, we have the follow-

Algorithm 1 (Blotching Algorithm) input: A signal v of length n > 2, a desired degree CY L n of local monotonicity and a value of p in (0, 031 output: A set of lomo-a regressions of the signal

( forp < 03 it gives the entire set of lomo regressions 1) for each i in/ 1, 2" - ' 1 compute all blotched

2) for the blotched versions that are lomo-a,

3 ) choose those lomo-cr versions at the smallest

versions in of 0.

compute their distance to v

distance from U.



For p = 1, the collection of blotched signals may be infinite; to have a practical algorithm, the tube algorithm is used together with the blotching algorithm. Since there are 2“ - blotching operations, the complexity of the algorithm is exponential. Fo rp 2 l, not all blotching transformations must be performed: some produce signals at a larger distance than others; Lemma 6 shows that the error does not increase if a shorter segment of the signal is regressed.

Lemma 6: Let v be a segment of w, let v’ and w’ be their corresponding constant p-regressions. Then, for p E

Thus for p z 1, blotching transformations may be tried in a particular order such that not all blotched versions need be computed (for p < 1, blotching transformations may replace a segment with a constant semiregression that is not a regression, and Lemma 6 does not guarantee an improvement of the algorithm). This partial order << on the collection of blotching transformations Hi is the re- verse of set inclusion for the corresponding spaces in the collection S. Transformations Hi and Hj are related Hi << Hi if the transformation Hi replaces (at least) all segments that the transformation Hi replaces; that is, if each integer interval in Hi is a subset of an integer interval in Hi (or if Sj is a subset of Si). From Lemma 6, for p 2 1 Hi << Hj implies dp[v , H i ( v ) ] I dp[v , H j ( v ) ] ; this also follows from the fact that the projecting set Sj of Hj is a subset of the projecting set Si of Hi. The order is made explicit by representing the transformations as nodes of a graph (see Fig. 7) arranged by levels: the first level contains the minimal (do-nothing) transformation, the second level contains transformations that replace only one segment of length 2; in general, at each level different from the first one, the transformations are immediate successors (according to the order defined) of the transformations in the previous level; the nth level contains only the transformation/ 1, n/which replaces the whole signal with its constant regression(s) .

Algorithm 2 (Rej-ined Blotching Algorithm) input: a signal v of length n > 2, a desired degree

(0, 031, dp(v , v’) 5 d p ( w , w’).

01 1 n of local monotonicity and a value of P E (1, 001

output: a lomo-a regression w of v Begin

Label each blotching transformation on each level as MAY-TRY For i: = 1 to n do {consider each of the n levels}

For each blotching transformation on level i that is MAY-TRY do Begin

Apply the transformation to the signal v. If the resulting blotched version is

lomo-a, Then, label all successors of this blotching operation as MAY-NOT-TRY

Fig. 7 . Graph of the partial order defined on the set of blotching transformations for signals of length 5 .

End From the blotched versions that are lomo-a, let w be one at smallest distance from U.

End

Example: For length-5 signals, the 16 blotching transformations {Hi} are arranged on 5 levels as follows. The corresponding graph is shown in Fig. 7.

first level: Ho = {+}

secondlevel: H I = { /1 ,2 / } H,= { /2 ,3 / )

H3 = { / 3 > 4 / } H4 = { /4 ,5 / } thirdlevel: H5= { / 1 , 3 / } H 6 = { /1 ,21 /3 ,4 / }

H7= { /1 ,2L/4 ,5 / } H8= { /2 ,4 / }

H9= { /2?3L/4 ,5 /1 H1o= {/3,5/1

H12= {/1,3L/4,5/1

HI4 = { /2 ,5 /1

fourthlevel: HI, = { /1 ,4 / }

H13 = {/1,2//33 5 / }

fifth level: H15 = { /1 ,5 / }

Implementing algorithm 2 requires that the graph be stored in memory or implemented in hardware. The speed is increased at the cost of the memory used.

D. n e Tube Algorithm - - X E, where each of

its m factors E; is a nonempty closed interval of RI, is called a tube. A signal v belongs to the tube El X * - *

x E, if for each i ~ / 1 , mL vi E Ei. Fig. 8 shows a tube, each of its factors is indicated with a thick vertical line segment.

In general, a tube contains signals of different lomoton- icities (the lomotonicity of a signal is the highest degree of local monotonicity that it possesses); the tube algorithm finds a signal in the tube with maximal lomotonicity. In turn, the tube algorithm may be used to compute lomo regressions under the square and product metrics.

A Cartesian product set El x



1 2 3 4 5 6 1 8

Fig. 8 . A tube (indicated with thick vertical lines) and three signals in it. Fig. 9. A collection of segments is obtained from the tube shown

The tube algorithm finds a collection of transitions that all signals in the tube have; based on this, it finds a finite subset of the tube that still contains a signal of largest lomotonicity and searches this finite subset for a signal of largest lomotonicity .

If a tube does not contain a constant signal, the signals in it have at least one forced transition. We denote the fact that all signals in a tube have an increasing (decreasing) transition at some coordinate value in/i , j /as [ + 1, /i, j / ] ( [ - l , / i , j / ] ) and call it a forced transition. For example, the signals in the overlapping tube [0, 31 X [ 2 , 51 X [4, 71 X [ 2 , 5 ] X [0, 31 have forced transitions [ + 1, /1, 2/] and [-1, /3 , 4fl (note the similarity in notation for constant segments and forced transitions).

In order to find a signal with largest lomotonicity in a tube, a sufficient and necessary collection of forced transitions is found; the collection is sufficient in the sense that a signal with largest lomotonicity in the tube does not need to have more transitions than those in the collection; it is necessary in the sense that a signal that fails to obey one of the forced transitions is not a signal of the tube. This permits the derivation of a finite subset of the tube containing a representative collection of signals. A search for a signal with largest lomotonicity is done on this finite subset.

Algorithm 3 (Tube Algorithm) input: a tube T output: a signal in the tube with largest lomotonicity

1) find a sufficient and necessary collection U of

2) find a representative collection L of signals

3) compute the lomotonicity of the signals in [i 4) choose a signal with largest lomotonicity

forced transitions of the signals in T

having such transitions

A procedure for finding the collection U of necessary and sufficient transitions, as required in Step 1 of the tube algorithm, is given. Let T = El x . - . x E, be the given tube; for each i i n / l , m/, let

x = max (Zi)

y = min (Zi)

p = min { j: x E z k for each k in/j , i / }

q = max { j: x E zk for each k in/i, j / }

r = min { j: y E zk for each k in/j , i / }

s = max { j: y E Zk for each k in/i, j / }

For each i (omitting subscripts i for simplicity), the constant signals ( / p , q / , x) and ( / r , s/, y) are the longest constant segments of signals in T whose widthslp, q/and / r , s/contain i and whose levels x, y are the extrema of Zi, if x = y, only one constant segment is defined. Fig. 9 shows the collection of such constant segments obtained from a tube.

The collection of constant segments that results as i ranges on /1, m / is reduced to a subcollection in two steps. First, each segment that has a width that is a subset of the width of another segment is subtracted from the collection; if there are two segments with equal widths, one of them is subtracted regardless of their levels. The collection of remaining segments is arranged { ( / a j , bj/, xi ) } according to starting coordinates (if r < s then a, C a,). Next, segments that are part of a subcollection of three or more segments whose levels either increase or decrease (but not both) with i and such that the first and last widths intersect, are subtracted, except for the first and last segments. The resulting smaller collection 0 is denoted {(/ai, b,k xi)}. Fig. 10 shows such a reduced collection of segments, obtained from the collection in Fig. 9.

The collection 0 of constant segments determines a collection of forced transitions; each pair of consecutive segments ( / a i , bi/, xi), ( / a i + I , bi+ I / , x i + J in 0 determines one forced transition [ + 1 , /ki, 4 / ] , or [ - 1, /ki, & / I , where/ki, &/=/ai, b i /n /a i + - 1 , bi + , /; the transition is increasing if xi < xi + and decreasing if xi > xi + (see Fig. 11). It is not hard to show that the collection of forced transitions determined by each pair of consecutive segments is necessary and sufficient.

When three or more consecutive forced transitions are of the same sign, they can be localized further, without harming the necessity or sufficiency of the collection: if

spectively, [ - l , / k i , /;/], [ -1 , /ki+l , l ; + l / I - * * [-1, / k j , l j / ] ) are consecutive increasing (respectively, decreasing) forced transitions, they are replaced by more localized forced transitions; the first and last transitions are replaced, [+ l , / k , , li/] by [ + l , / l i , li/] and [+l , /kj , l j / l by [ + 1 , / k j , k j / ] , and the intermediate transitions [ + 1, / k , , &/I , r E l i + 1 , j - 1/, are replaced each by [+1, / h , h / ] where, for each r , h is a number in/kr, &/(see Fig. 12 where the transitions in Fig. 12(a) become local-

[+ l , / k i , l i / I , [ + l , / k i + l , l i+ l / I [ + l , / k j , lj/I (re-



Fig. 10. A subcollection of segments from the collection in Fig. 9 is found.

1 2 3 4 5 6 1 8 9

Fig. 1 1 . The segments in Fig. 10 determines the forced transitions [ - 1 , /3, 4fl and [ + 1 , / 5 , 6 f l .

(a)

(b)

Fig. 12. The forced transitions shown in (a) are localized, as indicated in (b).

ized in Fig. 12 (b)); analogously if the transitions are all decreasing. Call the resulting collection of transitions T.

In step 2 of the tube algorithm, given the resulting col- lections 0 of constant segments and T of forced transitions, a collection L of signals having those transitions only is obtained as follows:

Algorithm 4 input: a collection of forced transitions U and the collec-

tion of constant segments 0 it is obtained from output: a collection L of signals having the indicated transitions

1) choose one coordinate from each forced transition

2 ) form a collection d of constant segments with ending coordinates given by those from Step 1 and with levels given by the corresponding levels of the constant segments in the collection 0

3) concatenate the segments in J, this gives one of the required signals

4) repeat steps 1-3 until all combinations of coordinates from forced transitions have been chosen. This gives the collection L.

The number of signals in L is the product of the widths of the forced transitions, so the complexity of the tube algorithm is combinatorial; a bound on this number is found in Appendix B.

E. AlRorithms for the Square and Product Metrics When using the blotching algorithm for p = 1 , each

blotching transformation dictates the replacement of a collection of segments from the original signal with constant regressions; if all such segments have odd lengths, each regression is unique and the blotching transformation produces one signal only. Otherwise, one or more of the segments to be replaced have even lengths, and a tube results; the factors of the tube are singleton sets and closed intervals [ a , b ] , where a and b are the central ranked values of the components of each segment of even length being replaced. After the tube algorithm finds a signal with largest lomotonicity , the blotching algorithm proceeds. For example, (under the square metric) the blotching transformation (/ 1 , 2 A 4, 5,') on the signal [4, 5, 1 , 3, 21 produces the tube [4, 51 x [4, 51 x [3, 31 x [2, 31 x [2, 31 containing the signal [4, 4, 3, 3, 31.

The blotching algorithm can be used to compute lomo regressions under the product metric, by blotching with the midrange. Also, a lomo oo-regression of a signal is a lomo signal on the boundary of an oo-ball (which is a tube) centered at the given signal; a tube that is centered at the given signal is grown until it contains a signal with a lomotonicity larger than or equal to the specified degree of local monotonicity. The tube does not have to be grown continuously, the radius of the tube (ball) is increased stepwise, each time increasing it to the minimum value that increases the width of at least one segment in the col- lections of segments; only a finite number of tubes is considered and, with a radius no greater than that of a ball that contains the constant regression of the whole signal, a lomo oo-regression is found. For the product metric, the algorithm based on the tube algorithm runs much faster than the blotching algorithm; the reasons being that the product of the widths of the forced transitions is less than the number of blotching transformations necessary to find a lomo regression and that there is no need to compute approximation errors (Appendix B).

Summarizing, the tube algorithm may be used to compute lomo regressions forp = 03; forp = 1 a combination



- 4 .

0 2 4 6 8 10

(a)

I I I I

0 2 4 6 0 10

(b)

0 2 4 6

(C) 0 10

0 2 4 6 8 10

Fig. 13. Original signal (white squares). (a) Three-point median filtered version (black diamonds). (b)-(d): LOmO-3 regressions (black squares) under the (b) Euclidean metric; (c) square metric; (d) product metric.

( 4



Fig. 14. A signal U and a lomo-5 regression metric.

v :- n

w . under the Euclidean

Fig. 15. A signal v and a lomo-3 regression w , under the Euclidean metric.

of both algorithms must be used; the blotching algorithm may be used alone for each value of p different from 1.

F. Examples First we compare the performance of locally monotonic

regression with that of the median filter. Since we consider finite length signals only, we have a choice regard- ing the behavior of the median filter at the extremes of the signal. If a padding median filter (one that appends elements of value equal to the value of the first element and last elements of the signal at its ends, e.g., Fig. 1) is used, a signal of the same length is obtained but the first and last elements are overemphasized. If no elements are ap- pended, a median filter of window size 2k + 1 that filters a signal of length m produces a signal of length m - 2k. We use a nonpadding median filter so that all elements of the signal to be filtered are considered equally important.

The (rough) signal [ l , -1, 2 , - 2 , 3, -3, 4 , -4, 51 of length 9 depicted by white squares in Fig. 13 is filtered with a (nonpadding) median filter of window size 3; the filtered signal (of length 7) is shown in Fig. 13(a) as black diamonds. Note the ineffectiveness of the filter at reduc- ing the roughness of the signal. The same signal was lomo regressed, under the Euclidean, square, and product metrics. The lomo-3 regressions are shown in black diamonds in Figs. 13(c)-(e), the smoothing under the criterion of local monotonicity being very effective.

Certainly the median filter is a useful device, particu- larly for the smoothing of images, because of its (relative) computational simplicity. However, local monotonicity is not a characteristic that the median filter necessarily pre- serves or attains.

In Fig. 14, both a rough signal U of length 16 and a lomo-5 regression w under the Euclidean metric are

1 2 3 4 5 6 7 8 9

Fig. 16. An interval (indicated by thick vertical line segments) of dimension 9, and 3 signals in it.

shown. In spite of the fact that the regression cannot be very similar in appearance to the original signal U , it does show an underlying trend present in U . Fig. 15 shows a less rough signal U and a lomo-3 regression w under the Euclidean metric.

V. CONCLUSION A theoretical framework for a new type of finite-length

discrete signal processing was presented. Numerous criteria of smoothness have been proposed (e.g., see [27]); many are defined in the frequency domain and give rise to linear smoothers. Under the criterion of smoothness of local monotonicity, locally monotonic regression provides a nonlinear tool for the optimal smoothing of discrete signals. Locally monotonic regression provides a further understanding on the concepts of local monotonicity and of the smoothing of discrete signals.

Locally monotonic regressions show an underlying pattern that may not be easy to grasp in the original signal. Many physical processes give rise to locally monotonic signals, or signals that may be approximated as locally monotonic. For example, the sampled height as a function of time of an airplane trajectory may be locally monotonic; if noisy versions of such signals are observed, lomo regression provides an approach for estimation, for example, in [13], the definition of regression is such that algorithms that compute p-regressions are maximum likelihood estimators [ 141, [20] of signals embedded in white additive noise, for a wide family of noise densities.

The algorithms presented currently require large amounts of computational resources to smooth long signals; the problem remains complex. The algorithms do provide a way of smoothing signals and the algorithm that computes lomo a-regressions using the tube algorithm is the least expensive, computationally. If the requirements on the optimality of the approximations are relaxed, it is possible to design algorithms that compute suboptimal lomo approximations, fast and inexpensively [ 161 ; they are the subject of current research.

The application of the technique of projection for the processing of signals using shape descriptors different from local monotonicity provides additional tools for the shaping of signals, and is being currently explored [28]. In particular, local convex/concavity can be used as a smoothness criterion with the advantage that a sinusoidal


2808 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 41. NO. 9, SEPTEMBER 1993

signal is locally convex/concave of some degree; this is not the case under the criterion of smoothness of local monotonicity: a continuous sinusoidal signal, regardless of its frequency, is not locally monotonic.

APPENDIX A Proof of Lemma I : Let p E (1, CO), then the ball

Bp(8, 1) is strictly convex; let A be a nonempty, proper, convex, closed subset of R“. Let x be a point not in A and p = D p ( x , A); then Bd[B,(x, p)] n A, which is the set of regressions of x with respect to A , is nonempty. If the cardinality of RA@) is larger than one, in the relative topology for the line spanned by two of its elements, the interior of the line segment between them is in A and in the interior of Bp (x, p ) . Points in the interior of Bp (x, p ) are at a distance less than p from x; therefore, there is a ball of radius smaller than p and centered at x that inter- =

Proof of Lemma 2: Let a, the degree of local monotonicity, be given. The set A of locally monotonic signals is closed because its complement is open: let U E (R“ - A); since U is not lomo, it has components U, and U’ with U, > u,+l, U’ < u,+l where Ii - j l I cx - 2. Let p = min{lu, - U , + I ~ , Iu, - ~ , + ~ 1 } / 2 . L e t Z b e t h e o p e n b a l l &,(U, p ) (Z is an n-dimensional interval whose edges have equal length 2 p ; it is the Cartesian product of n open intervals: Z I I = (U, - p , U, + p ) . From the definition of p , it follows that each signal in Z has a skeleton with the same ith and jth components, then Z is an open set containing U and points of R” - A only. This shows that R” - A is open (see Fig. 12 where all signals in the interval have the fifth and sixth components of their sign skeleton

Proof of Lemma 3: A fl r is a subset of Bd(A): if a signal has two consecutive components with the same value, then by slightly perturbing one of them, a nonlomo-3 signal arbitrarily close to A can be obtained. On the other hand, using arguments similar to the proof of the previous lemma, it can be shown that the set A fl I?‘ of strictly monotonic signals is open: thus, A n I“ is a subset of the interior of A. Since A contains its boundary, it follows that Bd(A) = A f l I‘ and Int(A) = A n r‘. Thus the boundary of A equals the set of locally monotonic signals that have at least two consecutive equal components. Also, it can be shown that the interior of A is equal to the interior of M (the set of monotonic signals)

Proof of Lemma 4: Let p , v and w be as in the hypothesis. Suppose that w11w21 * Iwm is the segmentation of w into constant signals. Then the pth power of the p-distance between U and w is

sects A, contradicting the definition of p .

equal to - 1 and + l ) .

consisting of strictly monotonic signals only.

[dp(v , w)]P = [dp (u l , w’)y + * * + [dp(vm, W“)]P

where, for each i E / 1, m / , vi is the signal with same coordinates as wi. Suppose that for some i ~ / 1 , m/ wi is not a constant regression of the corresponding segment v1. Then, since the error Id,(v’. wi)lP is a convex function of

the components of w‘, by appropriately perturbing any of these components, the distance between U and w may be decreased while the sign skeleton of w remains the same, contradicting the assumption that w is a locally monotonic

F o r p = 00 the condition of the lemma is sufficient but not necessary: there exist lomo regressions that are blotched versions of the regressed signal but there also may exist regressions that are not blotched versions. Con- sider the distance d,(v, w) = max {d,(vl, w’), * * - , &(Um, w“)} between a nonlomo signal U and a lomo regression w of v ; as long as the segment(s) w J with the largest distance &(U’, w’) are replaced with midranges, the remaining constant segments of w need not be constant regressions: the maximum is not changed. This suggests flexibility in the design of an algorithm that computes lomo regressions under the product metric; the tube algorithm yields regressions that are not always blotched versions of the signal being regressed.

Proof of Lemma 5: Again let p, v and w be as in the hypothesis; if for s o m e j ~ / 1 , m/ , w’ is not a constant semiregression of U’, then by perturbing w’, dp(v’, w’) can be made closer to a local minimum so that a new signal that is at a smaller p-distance from U and has the same sign skeleton as w is obtained, contradicting the mini-

m Proof of Lemma 6: The lemma is proved for the case

where the length of w is the length of U plus one; (if v = w, the lemma is clearly true) the general case then follows. Let v = [ v l , - * , vn] be a signal with constant

v, + be the signal obtained by appending one additional component to v (without loss of generality) at the end, and having constant regression w’ = [wi, , w;, wA+ let w ” = [w;, , w;].

Letp E (0, 03) and suppose that dp(w, w’) < d p ( v , U’). Either U’ # w” or U’ = w’’. If U’ # w” then wrr is a constant signal and is at a smaller distance from v than U ’

is

regression of U .

mality of dp (U, w).

regression U’ = [ v i , * , v;] and let w = [ v I , - - 9 Vn,

[dp(v, wrr)Ip = [dp(w, w’)Ip - lvn+l - wA+IIp

5 [dp(w, W’)IP < tdp(v, v’>IP

and then U’ is not a constant regression of U , which is a contradiction. If v’ = w’’ then

[dp(w, w’>Ip = [dp(v , v’>Ip + Jwn+l - wA+1JP

1 [dp(v , v’>IP

which is also a contradiction. Next, consider the case p = 00. In general, for each

constant signal c , the distance d,(c, x) between c and a signal x is I C - m( + r / 2 , where m is the midrange of x, r is its range, and c is the value of the components of c ; if c is the constant regression of x under the product metric then c = m and the distance is equal to half the range of the signal being regressed. w, + I is either in the interval Iv,,,, v,,,l, bounded by the smallest and largest - y . , I - - ...... ...



components of U , or it is not; in the first case the distance does not change and in the second it increases, which shows the lemma.

APPENDIX B 1. Complexity of Lomo Regression

Let c(n) be the number of elements in the partition 62' of A into convex cones, for a signal of length n . We pro- pose this number as a measure of the complexity of the problem of finding the set of lomo-a! regressions of a signal, and derive an expression for c(n) for the case Q! = 3.

A positive component of the sign skeleton of a lomo-3 signal may be followed by a positive or null component only; similarly, a negative component may be followed by a negative or null component only. A null component may be followed by any type of component. The number of sign skeletons that a lomo-3 signal may have is equal to the sum of the elements of a power of a certain 3 X 3 matrix m obtained as follows. Let g:/ - 1, 1 /-+/ 1, 3/be the function g(r) = 2 - r that maps the first and last values of a sign skeleton to matrix coordinates; the element of m n P 2 at row g(sl) and column g(s, - is the number of sign skeletons with first component s1 and last component s, - that a lomo-3 signal may have. For n = 3, the maximal value of the elements of m is 1 (e.g., the skeleton [0, -13 corresponds to element 2, 3 of m). A signal of length 3 has a sign skeleton with two components and its skeleton is one of nine (= 32) possible skeletons, among them, only seven correspond to lomo-3 signals:

m = 1 1 1 i' O I Lo 1 1 J

Likewise, it may be checked that the sum of the elements of

m 2 = i' 2 3 2 '1 L1 2 2 1

gives the number of sign skeletons that a lomo-3 signal of length 4 may have. Thus c(3) = 7 and c(4) = 17. In each case, the element (i, j ) is the number of skeletons with first and last components g-'(i) and g - ' ( j ) , respectively, that a lomo-3 signal may have.

In general, for a! = 3, c (n ) is the sum of the components of ridp2. This can be proven using the following induction step. Let n 1 4 and suppose that each element mi, in m" - gives the number of sign skeletons with first and last components g - ' ( i ) and g-I( j ) that a lomo-3 signal of length n - 1 may have. Let U = [ u l , * * * , U,] be

v,- and [ v , - ~ , v " - ~ , U,] must be lomo-3 as well; the skeleton possibilities for the first ( n - 1)-segment are given by m n P 3 and the skeleton possibilities for the last

a lomo-3 signal of length n; the segments [U', . * > v n - 2 ,

3-segment are given by m. Their product m" - 2 gives the skeleton possibilities for U : the product of the ith row of om" - and the jth column of m gives the number of sign skeletons that a lomo-3 signal of length n, with first skeleton component s, = g-I(i) and last skeleton component s, + I = g- ' ( j ) , may have.

Since m is symmetric, it may be expressed m = pTdp where d is a diagonal matrix with the eigenvalues of m on the diagonal and p is a unitary matrix: p-' = pT. The eigenvalues of m are x1 = 1 + Jz, x2 = 1, h3 = 1 - h, and

Since mk = p,Tcdkp~,

c(n) = (3/2)(X;-2 + A!-') + &(Ayp2 - A;-2).

F o r n = 3, 4, 5, 6, 7, and 8, c (n ) is given by 7 , 17, 41, 99, 239, and 577, respectively. The complexity depends on n and grows exponentially; for n large, c(n) = 2.9

. For a > 3 the complexity is somewhat smaller. x n - 2

2. Complexities of Blotching and Tube Algorithms The complexity of the blotching algorithm is effectively

expressed by the number of blotching transformations performed on a signal of length n ; this number is 2" - I .

In addition (in Step 2 of the blotching algorithm) for each lomo blotched signal, the distance to the original must be computed.

We bound the complexity of the tube algorithm with the maximal number of signals in the set L that may be obtained, given a tube of n factors. A signal of length n may have at most n - 1 transitions. The number of signals in L is the product of the widths of the forced transitions. The sum of the widths of the forced transitions is at most n - 1 . Thus, the bound is the maximum value of the product nj mi given that C j mi = n - 1, where each mi is a natural number. This maximum value is easily shown to be 3" * 2' I 3', where a = c - b, b = p(r) ; c , r are the quotient and remainder of (n + 1)/3, respectively, andp:/O, 2 / + / 0 , 2/is the functionp(r) = 2 - r . The relatively small exponents a , b indicate that the complexity of the tube algorithm is much lower than that of the blotching algorithm (compare 3"/3 to 2").

REFERENCES

S. G . Tyan, "Median filtering: Deterministic properties," in Two- Dimensional Digital Signal Processing I I : Transforms and Median Filters, T. S . Huang, Ed. Berlin: Springer-Verlag, 1981, pp. 197- 217. N. C. Gallagher and G . L. Wise, "A theoretical analysis of the properties of median filters," IEEE Trans. Acoust., Speech, Signal Pro- cessing, vol. ASSP-29, pp. 1136-1 141, 1981. J . Brandt. "Invariant signals for median-filters,'' Uril. Math., vol.

D . Eberly, H . G . Longbotham, and J . Aragon, "Complete classifi- cation of roots of one-dimensional median and rank-order filters," IEEE Trans. Signal Processing, vol. 39, pp. 197-200, 1991.

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 41, NO. 9, SEPTEMBER 1993

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New York: Wiley. 1981.

Alfredo Restrepo (S’84-M’90) was born in Bo- gota, Colombia, on November 28, 1959. He received the Ingeniero Electronic0 degree from the Pontificia Universidad Javeriana at Bogota in 1983, and the M.S. and Ph.D. degrees from the University of Texas at Austin in 1986 and 1990, respectively.

He worked as a Laboratory Engineer at Texins de Colombia and a Lecturer at the Universidad Javeriana. Later he was a Teaching Assistant and a Research Assistant in the Laboratory for Vision

Systems at the University of Texas. Currently, he is a Research Professor at the Universidad de 10s Andes in Bogota. His research interests are nonlinear signal processing, statistical signal processing, and computer vision.

Alan C. Bovik (S’80-M’80-SM’89) was born in Kirkwood, MO, on June 25, 1958. He received the B.S. degree in computer engineering in 1980, and the M.S. and Ph.D. degrees in electrical and computer engineering in 1982 and 1984, respectively, all from the University of Illinois, Urbana- Champaign.

He is currently the Hartwig Endowed Fellow and Professor in the Department of Electrical and Computer Engineering, the Department of Com- puter Sciences, and the Biomedical Engineering

Program at the University of Texas at Austin, where he is also the Director of the Laboratory for Vision Systems, During the spring of 1992, he held a visiting position in the Division of Applied Sciences, Harvard University, Cambridge, MA. His current research interests include image processing, computer vision, three-dimensional microscopy, and computational aspects of biological visual perception. He has published over 160 technical articles in these areas.

Dr. Bovik received the University of Texas Engineering Foundation Hal- liburton Faculty Excellence Award, was an Honorable Mention winner of the International Pattern Recognition Society Award for Outstanding Con- tribution, and was a National Finalist for the 1990 Eta Kappa Nu Outstand- ing Young Electrical Engineer Award. He has been involved in numerous professional society activities. He has been an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING since 1989 and Associate Ed- itor for the international journal Paftern Recognition, 1988. He has been a member of the Steering Committee of the IEEE TRANSACTIONS ON IMAGE PROCESSING, since 1991; General Chairman of the First IEEE International Conference on Image Processing, to be held in Austin, TX, in November 1994; Local Arrangements Chairman of the IEEE Computer Society Work- shop on the Interpretation of 3-D Scenes, October 1989; Program Chair- man, SPIE/SPSE Symposium on Electronic Imaging, February 1990; and Conference Chairman, SPIE Conference on Biomedical Image Processing, 1990 and 1991. He is a registered Professional Engineer in the State of Texas and is a frequent consultant to local industrial and academic insti- tutions.


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2796 ON 41, Locally Monotonic Regression...2796 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 41, NO. 9, SEPTEMBER 1993 Locally Monotonic Regression Alfred0 Restrepo, Member, IEEE,