Monotonicity Analysis over Chains and Curves 7 0 0 Dan Kucerovsky and Daniel Lemire 2 n a J Abstract. Chainsare vector-valuedsignals sampling acurve. They 4 areimportanttomotion signalprocessing andtomanyscientificap- 2 plications includinglocation sensors. Weproposeanovelmeasureof smoothness for chains curves by generalizing the scalar-valued con- ] cept of monotonicity. Monotonicity can be defined by the connect- M edness of the inverse image of balls. This definition is coordinate- G invariant and can be computed efficiently over chains. Monotone curves may be discontinuous, but continuous monotone curves are . h differentiable a.e. Over chains, a simple sphere-preserving filter is t shown to never decrease the degree of monotonicity. It outperforms a m movingaveragefiltersoverasyntheticdataset. Applicationsinclude TimeSeriesSegmentation,chainreconstructionfromunordereddata [ points, Optical Character Recognition, and Pattern Matching. 2 v §1. Introduction 1 8 Monotonicityisoneofthesimplestpropertyasignalmayhave. Itoffersa 4 powerfulqualitativedescription(“itgoesup,”“itgoesdown”). Givendata 1 cominginfromeithersensorsorfromanumericalsimulation,monotonicity 0 is independent of the sampling frequency and is robust with respect to 7 0 missing data [8]. Many geometrical objects such as curves are typically / defined in a parametrization-independent way which makes monotonicity h appealing. t a In this paper, we are concernedwith discretely sampled curves(which m we call chains) such as the trajectory of a particle in some vector space. v: This problem has applications in motion capture and tracking [14, 1]. i We expect a “smooth” scalar-valued signal not to change too quickly: X it should be locally constant. Therefore, classical low pass filters such as r the moving average(MA) are often sufficient to help smooth signals. Un- a fortunately, “smooth” chains are not locally constant: consider a loosely ConferenceTitle 1 Editorspp. 1–6. CopyrightOc2005byNashboroPress,Brentwood,TN. ISBN0-0-9728482-x-x Allrightsofreproductioninanyformreserved. 2 D. Kucerovsky and D. Lemire sampledcircle(seeFig.1). Moreover,achainmaylieonasphereorother higher dimensional surface and we may need to preserve this embedding. In Fig. 1, a chain on a circle is filtered using a moving average: we see that the filtered chain can, at best, follow a circle of a smaller radius. A filter is sphere-preserving(resp. circle-preserving)if, when the input data pointsareonasphere(resp. circle),thefiltereddatapointsalsolieonthe same sphere (resp. circle). It is readily shown that no linear filter except the identity can be sphere-preserving (SP) or circle-preserving (CP). In general, an SP filter is CP. We offer a simple SP filter in Section 5. One of the main contribution of this paper is to provide a generaliza- tion of the concept ofmonotonicity which applies to vector-valuedsignals and to curves. This definition is shown to be robust with respect to re- moval of data points and to be efficiently computed. Over curves, we show that monotone curves have many of the same properties as mono- tone functions asfar ascontinuityanddifferentiability areconcerned. We also propose a SP filter which we show to never decrease the degree of monotonicity. Experimentally,weshowthatthedegreeofmonotonicityis inversely correlated with noise and we compare the SP filter with simple MAfilters,provingthenonlinearSPfilterisagoodchoicewhennoiselev- els are low. Applications of this work include chain reconstruction from unordered data points [3] and Optical Character Recognition [15]. §2. Related Work A motionsignalis comprisedoftwocomponents: orientationandtransla- tion. The orientationvector indicates where the object is facing, whereas the translation component determines the object’s location. Recent work has focusedon smoothing the orientationvectors [14, 12], whereasthe re- sults ofthepresentpaperapplyequallywelltoorientationvectors(points on the surface of a unit sphere) as to arbitrary translation signals. In [2, 7, 10], the authors chose to define monotonicity for curves or chains with an arbitrary direction vector: a curve is monotone if its pro- jection on a line is does not backtrack. While this is a sensible choice given the lack of definition elsewhere, we argue that not all applications support an arbitrary direction that can be used to define monotonicity. The definition of monotonicity has been extended to real-valuedfunc- tions [4, 5, 6,11, 16, 17](f :Rn →R)by using contourlines (or surfaces) original filtered Fig. 1. Givensamplesoncircle, asimplemovingaveragedoesnotpreservethe embedding. Monotonicity Analysis 3 but the idea does not immediately generalize to curves and chains. OneapproachtochainsmoothingistouseB-splinesandBeziercurves with the L2 norm [9]. Correspondingly, we could measure the “smooth- ness”ofachainbymeasuringhowcloselyonecanfitittoasmoothcurve. Our approach differs in that we do not use polygonal approximations or curve fitting: we consider chains to be first-class citizens. §3. Monotone Curves Recall that a function f : R → R is said to be monotone increasing if f(x) ≥ f(y) whenever x ≥ y and monotone decreasing if f(x) ≤ f(y) whenever x≥y. A monotone increasingor monotone decreasingfunction is said to be monotone. Recall that B = {x:|x−a|≤R} is called a (closed) ballofradius R centeredarounda: inthe multidimensionalcase, the ball is a generalizationof the (closed) interval. Proposition 1. f :R→Rismonotoneifandonlyiff−1(B)isconnected for all balls B. Anarc-lengthparametrizedcurves:t→s(t)isR-monotoneforR>0 iftheinverseimageofanyclosedballofradiusatmostR,unders,iscon- nected. Straight lines are R-monotone for all R > 0. As motivation the discrete case, we want to compare monotone curves with monotone functions. Monotone functions f :R→R are differentiable almost every- where, and they do not have to be continuous. R-monotone also do not havetobe continuous: the curves(t)=(f(t),f(t)) wheref′(t)=1a.e. is R-monotone for all R > 0. Moreover, they are also differentiable a.e. as the next proposition shows. Proposition 2. Continuous R-monotone curves are differentiable a.e. Proof: Take any point x in the (open) domain of the curve s. Choose another point y so that the arc-length y − x over s is smaller than R. Consider any point z on s between y and x, then z must be contained in all balls of radius R containing both x and y. It follows that s must be differentiable from the left at x. Similarly, s is differentiable from the right at x. If the two derivative from the left and from the right do not match, then it is possible to find y and y′ close to x from the left and the right such that there is a ball of radius R containing both y and y′ but not x, a contradiction. Just like monotone functions, continuous R-monotone curves do not have to be twice differentiable, consider the arc-length parametrized ver- sion of s(t)=(t,|t|t) for t∈(−1,1). Differentiable functions are not necessarily monotone. Likewise differ- entiable curves are not necessarily R-monotone as the next proposition shows. 4 D. Kucerovsky and D. Lemire Proposition 3. Thereis adifferentiable continuousfinite curveswithno cross-over (that is, t → s(t) is one-to-one) which is not R-monotone for any R>0. Proof: Consider a curve following a inward spiral around a fixed point such as s(t)=(2π−t)(cost,sint) for t∈(0,2π]. Functions are monotone or not, and there is no “degree of monotonic- ity.” Similarly, for curves of finite length, it simply matters whether they areR-monotoneforsomefiniteRsinceR-monotonicityisscale-dependent. Proposition 4. GivenaR-monotonecurve,scalingthecurvebyafactor ∞>α>0 makes it αR-monotone. §4. Signal Monotonicity Inthis section,wedefine monotonicityforvector-valuedsignalsorchains as a natural extension of monotonicity for real-valued signals. We show how to compute efficiently the degree of monotonicity. A scalar-valuedsignal(or discrete function) is monotone if and only if the index set of values in any closed interval [a,b] is a set of consecutive integers [j,k]: p ∈ [a,b] ⇔ i ∈ [j,k]. Equivalently, the values of the i signal pi never go down (pi+1 ≥ pi) or never go up (pi+1 ≤ pi). Another equivalent definition is given by the next proposition. Proposition 5. A scalar-valued signal p is monotone if and only if, for i any 3 consecutive samples, pi,pi+1,pi+2, the index set of the values con- tained in any closed interval [a,b] is a set of consecutive integers [j,k]. Equivalently,the index setis aconvexsetunderanappropriatedefinition of convexity. Itiseasytoextendthisdefinitionofmonotonicitytothecaseofvector- valued signals. Unfortunately, a straightforward generalization, based on considering the set of indices of the values contained in any closed ball, would lead us to conclude that the only monotone vector-valued signals are on straight lines and never backtrack. It is not hard to realize no sensible filter could turn any vector-valued signal into a monotone signal. In order to obtain nontrivial results, we need to restrict the class of balls considered, as in the following definition. Definition 1. A vector-valued signal p has a degree of monotonicity R i ifRis thelargestvalue suchthat,consideringonly3consecutivesamples, pi,pi+1,pi+2, the index set of the values contained in any closed ball B of radius at most R is a set of consecutive integers in {i,i+1,i+2}. Monotonicity Analysis 5 5 4 2 6 3 7 1 8 Fig. 2. Given thechain of data points shown, thedegree of monotonicity is at most the size of the radius of the circle given in the picture: it contains points 4 and 6 but not point 5. If the signal values are on a straight line with no backtracking, then the degreeofmonotonicityis∞,andthe degreeofmonotonicityisalways largerthan 0 for finite signals. Fig.2 givesanintuitive view of the degree of monotonicity. This measure of monotonicity is robust in the following sense. Proposition 6. If one point is omitted from a vector-valued signal, the degree of monotonicity cannot decrease. While this discrete definition is similar to the definition given for R- monotone curves, to allow efficient computation, we consider only sets of 3consecutivesamples,thusreplacingaglobalproblembyalocalproblem. Ifwelifttherequirementthatonly3samplesareconsidered,thenasignal is R-monotone if and only if all subchains of length 3 are R-monotone. This suggests that the cost of checking global R-monotonicity grows in a cubicfashionwithrespecttothelengthofthesignalwhichisunacceptable for most applications. In practical applications, maximizing the degree of monotonicity R leads to useful chains. For example, noise tends to reduce R by creating sharpturnsandlocalbacktrackingandahighlymonotonecurve(Rlarge) is more likely to be noise-free. On the other hand, when reconstructing chains from unordered sets of points, as happens in computer vision, we often want to minimize sharp turns and backtracking. Therefore, solving forthechainmaximizingRwhilepassingthroughallavailabledatapoints is a sensible “curve reconstruction” strategy. As a prerequisite to computing the degree of monotonicity, we need a computationally effective way to compute the radius of the circle going through 3 points. Given p1,p2,p3 ∈ Rn, we can compute the radius of thecirclepassingthroughthem(denoted⌢(p1p2p3))byfirstcomputing a=kp1−p2k, b=kp2−p3k, c=kp1−p3k, σ =(a+b+c)/2, and then we have the classical Heron’s formula for the radius of the circle: abc Routcircle = 4 σ(σ−a)(σ−b)(σ−c) p 6 D. Kucerovsky and D. Lemire A C B A B C Fig. 3. Given a chain of 3 data points, we give two cases: (left) the angle ∠ (ABC) > π/2 so we compute the radius of the circle going through ABC, otherwise (right), we compute half the distance between A and C. whenever a,b,c>0. The next theorem gives us a way to compute the (local) degree of monotonicity for any 3 points, to compute the degree of monotonicity of an entire signal simply requires, by definition, to take the minimum of the result for all consecutive 3 points. The theorem essentially says that if ∠(p1p2p3) <π/2, the degree of monotonicity is then half the distance between p1 and p3, andotherwise, it is Routcircle (see Fig. 3). To see that this local form of monotonicity is distinct from the global form suggested earlier, consider a chain in the form of a figure “8.” Theorem 1. The degree of monotonicity for the sequence p1,p2,p3 is 1c if a2+b2 >c2 R:= 2 (cid:26) Routcircle otherwise where a=kp1−p2k, b=kp2−p3k, c=kp1−p3k. Proof: Consider the disk B0 containing p1 and p3, centered at (p1 + p3)/2 and having radius d(p1,p3)/2. The point p2 is outside the disk if and only if cos∠(p1p2p3)= a2+2ba2b−c2 is positive. Thus, p2 is outside the disk ifandonlyifa2+b2−c2 >0. ClearlyR=radius(B0)=d(p1,p3)/2. Next, supposethatp2 isinthediskB0. Wehavethatanyballcontaining p1 and p3 but not p2 must be larger than radius(B0) since B0 is the smallest ballcontainingboth p1 and p3. Now, suppose there is a (closed) ball of minimal radius R containing p1 and p3, but not p2. This implies a non-zero distance, δ > 0, between B and p2. We have that the center of the ball has to be away from the line formed by p1,p3: if not then it must be a ball containing B0. This means we can move the center of the ball slightly closer to p1 and p3 while reducing the radius just enough so that p2 remainsoutside the ball. By repeatingthis process,we showthat δ = 0, a contradiction. Hence, there is no (closed) ball of minimal radius R containing p1 and p3, but not p2. Hence p1,p2,p3 have a degree of outcircle monotonicity R . Monotonicity Analysis 7 §5. Monotonicity and Sphere-Preserving Filters Inthissection,weproposeaSPfilterwhichneverdecreasesthedegree of monotonicity of the signal. Given a signal p , we consider recursive i (IIR) filters of the form p′i =f(p′i−2,p′i−1,pi,pi+1,pi+1). To ease the notation, we write A = p′i−2, B = p′i−1, X = pi, C = pi+1, D = pi+1) so that the equation becomes X′ = f(A,B,X,C,D). Let R(A,B,X,C,D) be the degree of monotonicity of A,B,X,C,D com- puted as min(R(A,B,X),R(B,X,C),R(X,C,D)). The following propo- sition gives us a condition of f to increase the monotonicity of a vector- valued signal. Proposition 7. GivenX′ =f(A,B,X,C,D),iff issuchthatthedegree of monotonicity R(A,B,X′,C,D)≥R(A,B,X,C,D), then the recursive filter p′i =f(p′i−2,p′i−1,pi,pi+1,pi+1) never decreases the degree of monotonicity of a signal. It seems that f should be chosen so that R(A,B,X′,C,D) is as large as possible. To maximizes R(A,B,X′,C,D) with X′ = f(A,B,X,C,D), f should be either B or C. In other words, we improve monotonicity best when we make the sample X “virtually disappear.” Proposition 8. R(A,B,X′,C,D)is minimizedwhenX′ =B orX′ =C and these choices are unique unless ⌢(ABC) =⌢(BCD) in which case any point on the arc of the circle between B and C inclusively qualifies. Fortunately, we can easily define a more interesting SP filter. Given an arc of a circle, denoted α, and a point X, we can project X on α by solving for the point closest X in α. The projection onto a circle can be determined easily using only linear algebra [13]. In the plane, start with equation (x−r1)2+(y−r2)=ρ2 and substitute 3 values of x,y, getting 3 equations. By pairwise subtraction, we can remove the unknown ρ2, and be left with linear system having 2 equations and 2 unknowns (the center of the circle). We apply this by first projecting on the circle and if the projected point does not belong to the given arc we move it to the closest point on the arc (an endpoint of the arc). Let us define X1 to be the projectionofX onthe arcBC ofthe circleABC, anddefineX2 tobe 8 D. Kucerovsky and D. Lemire Fig. 4. Thedegreeofmonotonicityversustheabsoluteinputnoiselevel(MSE) over a synthetic data set generated from points on a unit circle. The SP filter outperforms MA when noise levels are low. the projection of X on the arc BC of the circle BCD. Intuitively, either point X1 or X2 would make a good choice for X′ . To ensure that the degree of monotonicity is never decreased, we set f(A,B,X,C,D)=arg max R(A,B,X,C,D). X′∈{X,X1,X2} This function can be computed quickly and is sphere-preserving. §6. Experimental Results Wegenerateachaininthexyplanebyregularlysamplingaunitcircle3 timesforatotalof30samples. AMAfilterwithwindowwidthk averages each k consecutive data points. We add white noise to every point in the chain and we filter it using simple MA filters with window widths of 3 and 5 samples as well as with the SP filter of the previous section. Each test is repeated 10 times and we keep only the averages. Fig. 4 shows the degree of monotonicity versus the noise level (Mean Square Error) with the three smoothing filters and the unfiltered chain. The noise level ranges from none to over 0.05 (MSE) which corresponds roughly to a 5% noise-to-signal ratio. An example of filtering is given in Fig. 5. §7. Discussion In the unfiltered chain, the degree of monotonicity is inversely corre- lated with the noise level: the Pearson correlation is p = −0.95 (90%). The degreeofmonotonicityseemsagoodindicatorofnoise,whichinpar- ticular suggests that a method for increasing the degree of monotonicity Monotonicity Analysis 9 1.5 unfiltered sphere preserving (width=5) moving average (width=3) 1 0.5 0 -0.5 -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Fig.5. VisualcomparisonoftheSPfilterwiththeMAfilterforlownoiselevels and coarse sampling. would alsofunction as a goodnoise reductiontechnique. As required,the SP filter always increases the degree of monotonicity with respect to the unfiltereddata. 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