Deterministic Sampling on the Circle using Projected Cumulative Distributions
DDeterministic Sampling on the Circleusing Projected Cumulative Distributions
Daniel Frisch and
Uwe D. Hanebeck
Intelligent Sensor-Actuator-Systems Laboratory (ISAS)Institute for Anthropomatics and RoboticsKarlsruhe Institute of Technology (KIT), Germanye-mail: [email protected], [email protected]
Abstract
We propose a method for deterministic sampling of arbitrary continuous angular density functions.With deterministic sampling, good estimation results can typically be achieved with much smallernumbers of samples compared to the commonly used random sampling. While the Unscented KalmanFilter uses deterministic sampling as well, it only takes the absolute minimum number of samples.Our method can draw arbitrary numbers of deterministic samples and therefore improve the qualityof state estimation. Conformity between the continuous density function (reference) and the Diracmixture density, i.e., sample locations (approximation) is established by minimizing the difference ofthe cumulatives of many univariate projections. In other words, we compare cumulatives of probabilitydensities in the Radon space.
1. Introduction
Context.
State estimation or control techniques for nonlinear systems often use samples (or particles)to represent the occurring densities.Obtaining discrete samples (on continuous domains) from continuous probability density functions(PDFs) is therefore an important module in many state estimators and controllers. The “brute force”approach, often used to obtain ground truth for reference, is Monte Carlo Sampling with large numbersof random samples. There are universal but rather slow random sampling methods [1] and fastermethods specialized for certain densities like the von Mises-Fisher distribution [2].In embedded systems subject to real-time constraints and limited memory, the number of samplesshould be rather small. With deterministic samples (instead of stochastic samples), comparable resultscan be achieved with much fewer samples.Applications for deterministic sampling and filtering particularly in directional statistics includepredictive control [3], heart phase estimation [4], wavefront orientation estimation [5], and visualSLAM [6].
Considered Problem.
In this work we consider the problem of deterministic sampling of arbitrarycontinuous densities on the circular domain with an arbitrary number of samples.
State-of-the-art.
The minimalistic and popular deterministic sampling method of normal densitiesin the Euclidean domain is the basis of the Unscented Kalman Filter (UKF) [7, 8]. The efficientconcept of the UKF has successfully been transferred to the circular domain [9, 10], however inheritingequivalent limitations (only three samples, specific types of densities). Using higher order moments,the number of samples can be increased to five [11] and multiples of five with superposition techniques[12]. For specific densities, sampling based on the cumulative density function (CDF) has beenproposed [12] but is not invariant w.r.t. interval choice.Weighted samples in an equidistant grid are very well suited for the circular domain [13], the sphere[14], and the torus [15] but expensive to extend to a high number of dimensions. UKF-like sampling a r X i v : . [ ee ss . S Y ] F e b igure 1: Wrapped Laplace Distribution (blue) on the circular domain (black), with proposeddeterministic sampling result for 35 samples (red).methods by contrast are applied to higher-dimensional directional estimation for orientations on thehypersphere [16, 17], for multivariate circular estimation on the torus [18], and for dual quaternions onspecial Euclidean groups [19, 20], general Lie groups [21], or arbitrary Riemannian manifolds [22, 23] –all without exponential increase of computational cost.How can we make deterministic sampling more flexible, i.e., provide more samples than UKF-likeschemes, but avoid Cartesian products? One way to achieve this is based on the Localized CumulativeDistribution (LCD) and a modified Cramér-von Mises distance. The LCD transforms any density(either continuous or Dirac mixture (DM)) to a continuous representation via kernel convolution. Themodified Cramér-von Mises distance is basically an L norm of the difference of densities [24] butadditionally averages over all kernel widths. LCD and modified Cramér-von Mises distance togetheryield a distance measure between continuous and DM densities in any combination [25], which hasbeen successfully applied in the Euclidean domain [26], especially for Gaussian densities [27, 28].Early adaptions to directional estimation applied the LCD in the Euclidean tangent space of thedensity’s mean, placing samples on the coordinate axes only [29] or distributing them in the entiretangent space [30]. Direct application of the LCD on non-Euclidean manifolds has been performedfor sample reduction (DM to DM comparison) on the sphere [31] and for dual quaternion samplereduction in the special Euclidean group SE(2) [32]. Unfortunately, this method cannot easily beapplied to arbitrary density functions and manifolds, because the involved integrals often do not existin closed form.For the special case of the von Mises-Fisher density there is also a very efficient deterministicsampling method that places samples on an arbitrary number of “beams” in a star-like arrangement[33]. It is very fast and more flexible than UKF-like, but the star-like arrangement doesn’t alwayscover the state space homogeneously and purely according to the density function. Contribution.
In this paper, we present a method to optimally approximate a continuous angulardensity function f C ( x ) on the circular domain with a Dirac mixture density (DMD) f DM ( x ) with anarbitrary number of samples. 2 . Overview Key Idea.
We propose to extend the projected cumulative distribution (PCD) from the Euclideanspace R d [34] to the circular domain S and use it for deterministic sampling. By projecting toone-dimensional marginal distributions, we reduce multivariate problems to a set of univariate ones. Inthe univariate setting, cumulative distributions are uniquely defined and can easily be approximatedeven for arbitrary density functions.In other words, we match a continuous density with a DMD in the Radon domain. To optimallycapture and transfer all of the density’s details, it is important to include many different projections,which we implement in an iterative manner. Problem Formulation. f C ( x ) , x ∈ S is an arbitrary continuous density function on the circle,considered as reference density here. The goal is to obtain a DMD f DM ( x ) = 1 L L X i =1 δ ( x − ˆ x i )with sample locations ˆ x i ∈ S , i ∈ { , , . . . , L } . This DMD should optimally approximate the givencontinuous reference density, limited in accuracy only by the allowed number of samples L . Requiredinputs areI1 the number L of wanted samples,I2 a numerical function handle of a continuous angular reference density function f C ( x ) , x ∈ S .Obtained outputs are the sample locations ˆ x i ∈ S .
3. Projection of the Circular Domain
Projection along a certain direction u ∈ S allows to compare one-dimensional PDFs f ( r | u ) ata time. CDFs F ( r | u ) are uniquely defined in one dimension and can also be easily calculated fromthe PDFs via the trapezoidal rule with proposal samples (if no closed-form solution is available).Furthermore it is easy to compare two one-dimensional CDFs.The following two types of projections f ( r | u ) of circular densities f ( x ), exponential map andorthographic projection, appear to be equally convenient for our purpose. Consider the circular domain as a real interval of length 2 π by cutting the unit circle open at anarbitrary position u ∈ S f ( r | u ) = f " cos( r − ∠ u )sin( r − ∠ u ) , ≤ r ≤ π , , otherwise , (1)where ∠ u = atan2( u (2) , u (1) ) is an angular representation of u . Consider the Euclidean embedding of the circular manifold S in R . We then perform a linearprojection using the direction vector u rrr = u > xxx , rrr . In terms of densities, we calculate the marginal distributionalong u f ( r | u ) = Z S f ( x ) δ ( r − u > x ) d x | α = ∠ x = π Z α =0 f " cos( α )sin( α ) δ r − " cos( ∠ u )sin( ∠ u ) > " cos( α )sin( α ) d α = π Z α =0 f " cos( α )sin( α ) δ ( r − cos( α − ∠ u )) d α = X i =1 f " cos( α i + ∠ u )sin( α i + ∠ u ) | sin( α i ) | , | r | ≤ , , | r | > , (2)with α i = ( arccos( r ) , i = 1 , π − arccos( r ) , i = 2 . See Fig. 2 for a visualization of two orthographic projections.
4. Implementation
With a suitable projection at hand, we can now start approximating the continuous density. It iswell known that samples of any one-dimensional density, like our projected PDF, can easily be drawnwhen the inverse of the CDF is available. Therefore, we seek to obtain the following intermediateresults one by one in the course of this section: 4igure 3: Procedure for deterministic sampling of a projected von Mises-Fisher density, usingorthographic projection (2). Upper part: we evaluate f ( r ) (blue) at the fixed evaluation points t h j (black) as well as previous sample locations (red). Lower part: Trapezoidal integration on saidevaluation points is performed (blue). Compare the ground truth obtained with a numerical ODEsolver (yellow). Then, one-dimensional deterministic sampling is performed (black), yielding anapproximating DM distribution function (red). See also Alg. 1 for a more detailed description.• reference PDF f C ( x ) (is given),• projected PDF f C ( r | u ),• projected CDF F C ( r | u ),• inverse CDF F C − ( p | u ),• sample locations r i ,• sample updates ∆ x i .The procedure will then be repeated iteratively for different projections u . The projected PDF f C ( r | u ) is available in closed form by inserting the given f C ( x ) into (1) or (2).Since we are permitting arbitrary density functions, a closed-form representation of the accordingCDF F C ( r | u ) = Z rt = −∞ f ( t | u ) d t lgorithm 1: Calculate sample steps that make a DMD approximate a continuous density bymatching the cumulatives, in the univariate (projected) setting.
Function { ∆ r i } Li =1 ← sample1D( f rrr ( · ) , { r i } Li =1 ) Input: f rrr ( · ): continuous reference density in one dimension, { r i } Li =1 : current sample approximation Output: { ∆ r i } Li =1 : proposed step for each sample, to improve similarity to f rrr ( · ) n t h i o L h i =1 // Fixed evaluation points { t j } L e = L h + Lj =1 ← n t h i o L h i =1 ∪ { r i } Li =1 { F j } L e j =1 ← cumtrapz( { t j } L e j =1 , { f rrr ( t j ) } L e j =1 ) { F j } L e j =1 ← n F j + − F L e o L e j =1 // Centering for i ← to L do F det ← i − L // Deterministic sampling (cid:16) j L , j R (cid:17) ← adjacent( F det , { F j } L e j =1 )// Quadratic interpolation m ← f j R − f j L t j R − t j L ( a, b, c ) ← F j L + R xt j L m · (cid:16) x − t j L (cid:17) d x ! = F det ( x quad1 , x quad2 ) ← roots( a, b, c )// Linear interpolation x lin ← F det − F j L m + t j L // Updated sample location r e i ← select_best( x quad1 , x quad2 , x lin ) end // Assign r i and r e i (cid:16)(cid:8) r sort i (cid:9) Li =1 , { j i } Li =1 (cid:17) ← sort( { r i } Li =1 ) for i ← to L do ∆ r j i ← r e i − r sort i // Sample step end is not possible in general. However, we know that the integrand f ( r | u ) has limited support, i.e., r ∈ [0 , π ] for the exponential map projection (1), and r ∈ [ − ,
1] for the orthographic projection (2).To obtain an approximation of F ( r | u ), we apply the composite trapezoidal rule with an adaptive setof function evaluation points t j .A fixed set of homogeneous function evaluation points t h j inside the support interval is alwaysused to ensure a good general approximation of the CDF’s global shape. Additionally, in order tomaintain proper accuracy of the numerical integral even in the case of very localized PDFs with smallextent, the projected samples r p i in the currently assumed approximating density f DM ( r | u ) are alwaysincluded into the set of function evaluation points.Summarizing, after composite trapezoidal integration of f C ( r | u ) with said evaluation points, wenow have a piecewise linear representation of the projected reference CDF, F C ( r | u ). We draw deterministic samples p i that are uniformly distributed in [0 ,
1] , p i = 2 i − L , i ∈ { , , . . . , L } , lgorithm 2: PCD-based deterministic sampling of conditional circular densities.
Function { ˆ x i } Li =1 ← sampleS1( f xxx ( · ) , L ) Input: f xxx ( · ): continuous circular density, x ∈ S , L : number of wanted samples Output: { ˆ x i } Li =1 : deterministic samples on the circle that approximate f xxx ( · ) N ← // Projections per iteration// High quality for visualization M ← // Number of iterations λ ← . // Update step decrease factor// Initialization λ ← { ˆ x i } Li =1 ← rand( L, S ) for m ← to M do ϕ ← rand( , S ) { ∆ˆ x i } Li =1 ← for n ← to N do // Symmetric projections ϕ ← π · ( n − /N + ϕ u ← " cos( ϕ )sin( ϕ ) // Project the samples ˆ x i → r i { r i } Li =1 ← n u > ˆ x i o Li =1 // Project the density f xxx ( · ) → f rrr ( ·| u ) // according to Sec. 3 f rrr ( · ) ← project( f xxx ( · ) , u )// Get projected sample updates// using Alg. 1 { ∆ r i } Li =1 ← sample1D( f rrr ( · ) , { r i } Li =1 )// Get sample updates in R { ∆ x i } Li =1 ← { backproject( ∆ r i ) } Li =1 { ∆ˆ x i } Li =1 ← { ∆ˆ x i + ∆ x i } Li =1 end λ ← λ · λ for i ← to L do // Perform sample update ˆ x i ← ˆ x i + λ ∆ˆ x i /N // Restrict to S ϕ i ← atan2( ˆ x (2) i , ˆ x (1) i ) ˆ x i ← " cos( ϕ i )sin( ϕ i ) endend and propagate them through the inverse CDF to obtain deterministic samples r i of f C ( r | u ) r i = F C − ( p i | u ) , i ∈ { , , . . . , L } . Under the assumptions that have been made with the trapezoidal rule, our representation of f C ( r | u )is piecewise linear, and thus F C ( r | u ) is a piecewise quadratic. Therefore, evaluation of F C − ( p i | u )7 a) von Mises (b) wrapped Cauchy (c) wrapped normal (d) wrapped exponential(e) von Mises mixture (f) custom distribution, sinu-soidal (g) piecewise constant (h) uniform Figure 4: Illustration of various circular distributions and deterministic samples obtained with theproposed method. Continuous probability density function (blue) on the angular domain S (black),with sampling results (red). For better visualization, the length of the red lines representing theunweighted samples has been set to the maximum density function value (mode) instead of the sampleweight 1 /L .for any p i to obtain r i involves two things. First, a search for the relevant interval, i.e., an adjacentpair ( t L , t R ) from the trapezoidal function evaluation points t i such that F C ( t L | u ) ≤ p i < F C ( t R | u ).Second, the quadratic (or sometimes linear) function that represents the CDF in this segment has tobe inverted, what is easily done in closed form.Of course, if a closed-form representation of the projected CDF or its inverse is available, wecan use that directly for sampling, with no need for trapezoidal integration. For example, a fastapproximation of the von Mises-Fisher density’s cumulative (in conjunction with the exponential map)is available in closed form [35].At this point we have the deterministic sample locations r i in the projected space that is definedby the projection direction u .Compare Fig. 3 for a visualization of CDF-based sampling in the projected space. The projected sample locations r i now have to be backprojected to the original domain S . Wetypically use updates from several symmetrically arranged projections simultaneously.The projected samples generated as described in Sec. 4.2 are not naturally associated with theexisting samples from previous iterations. Thus, we have to find an appropriate association first.Projection also helps us here: in the one-dimensional case, the association that minimizes the globaldistance of associated point pairs can simply be obtained by element-wise comparison of the sortedsets. The according global distance is also called Wasserstein distance.Refer to Alg. 1 for a pseudocode representation of the procedure described in Sec. 4 up to here. To equally consider all dimensions, we propose to use a symmetric set of N projections in eachiteration step. For N = 2 projections per iteration, we choose projections that are orthogonal (90°8 a) Fixed (b) Adaptive Figure 5: Deterministic circular sampling using (a) only a fixed set of 30 evaluation points t h j versus(b) the 30 fixed points plus the previous samples, for better numerical integration. The difference forthis quite “narrow” von Mises-Fisher distribution ( κ = 500) is notable.between them) but with random orientation, see Fig. 2 for an example. The individual sample updatesfrom each projection are averaged, thus yielding the total update ∆ˆ x i of the current iteration step. The procedure is repeated until the arrangement of the samples obtains an acceptable quality.In order to asymptotically reach a stationary state, we propose to multiply sample updates with anexponentially decreasing factor λ . This accounts for the fact that more and more information (frommore projections) is already present in the sample locations, and the amount of extra informationprovided by every additional iteration decreases.Refer to Alg. 2 for a more detailed presentation regarding the iterative sample update scheme.
5. Evaluation
The flexibility of the proposed method is demonstrated by showing obtained deterministic samplesform various different density functions, see Fig. 4.Our adaptive choice of evaluation points for numerical integration allows for an accurate approxi-mation even for “narrow” densities, where fixed evaluation points alone would not be sufficient. SeeFig. 5 for an example.
6. Conclusions
We present a method to generate any number of deterministic samples for any continuous densityfunction on the circle.It does not require gradient-based numerical optimization like LCD-based methods. Instead, weuse the trapezoidal rule with adaptive support points on a given interval in an iterative method.Furthermore, the distance measure is simple and undisputable: Matching the cumulatives is alwaysan adequate solution for univariate densities. No parameters or weighting functions have to be chosen.With the help of the PCD, we can apply the same elementary method (matching one-dimensionalcumulatives) to higher dimensions.In the future, we will extend this method to higher-dimensional geometries such as the hypersphereand the torus. While calculating the projected density was easy on the circle, it will be more difficultin higher dimensions. We will look for closed-form solutions that work for specific types of densities.Furthermore, numerical integration techniques with an adaptive choice of evaluation points will be9ursued and also pure sample reduction techniques, where no integration is necessary. Presumably,orthographic projection is a good choice for hyperspherical higher-dimensional extensions of the circle,and the exponential map for the Cartesian product of circles, i.e., toroidal manifolds.
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