# A method for recognizing the shape of a Gaussian mixture from a sparse sample set.

**ABSTRACT** The motivating application for this research is the problem of recognizing a planar object consisting of points from a noisy observation of that object. Given is a planar Gaussian mixture model rhoT (x) representing an object along with a noise model for the observation process (the template). Also given are points representing the observation of the object (the query). We propose a method to determine if these points were drawn from a Gaussian mixture rhoQ(x) with the same shape as the template. The method consists in comparing samples from the distribution of distances of rhoT (x) and rhoQ(x), respectively. The distribution of distances is a faithful representation of the shape of generic Gaussian mixtures. Since it is invariant under rotations and translations of the Gaussian mixture, it provides a workaround to the problem of aligning objects before recognizing their shape without sacrificing accuracy. Experiments using synthetic data show a robust performance against type I errors, and few type II errors when the given template Gaussian mixtures are well distinguished.

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**ABSTRACT:**One way to characterize configurations of points up to congruence is by considering the distribution of all mutual distances between points. This paper deals with the question if point configurations are uniquely determined by this distribution. After giving some counterexamples, we prove that this is the case for the vast majority of configurations.In the second part of the paper, the distribution of areas of sub-triangles is used for characterizing point configurations. Again it turns out that most configurations are reconstructible from the distribution of areas, though there are counterexamples.Advances in Applied Mathematics 01/2004; · 0.88 Impact Factor -
##### Article: Use of the Kolmogorov-Smirnov, Cramer-von Mises and related statistics without extensive tables

Journal of the Royal Statistical Society. Series B. - 01/1951;

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Purdue University

Purdue e-Pubs

ECE Faculty Publications Electrical and Computer Engineering

2010

A method for recognizing the shape of a Gaussian

mixture from a sparse sample set

H. J. Santos-Villalobos

M. Boutin

This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for

additional information.

Santos-Villalobos, H. J. and Boutin, M., "A method for recognizing the shape of a Gaussian mixture from a sparse sample set" (2010).

ECE Faculty Publications.Paper 47.

http://docs.lib.purdue.edu/ecepubs/47

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A method for recognizing the shape of a Gaussian mixture

from a sparse sample set

Hector J. Santos-Villalobos and Mireille Boutin

School of Electrical and Computer Engineering, Purdue University

465 Northwestern Ave. West Lafayette, IN 47907 USA;

ABSTRACT

The motivating application for this research is the problem of recognizing a planar object consisting of points

from a noisy observation of that object. Given is a planar Gaussian mixture model ρT(x) representing an object

along with a noise model for the observation process (the template). Also given are points representing the

observation of the object (the query). We propose a method to determine if these points were drawn from

a Gaussian mixture ρQ(x) with the same shape as the template. The method consists in comparing samples

from the distribution of distances of ρT(x) and ρQ(x), respectively. The distribution of distances is a faithful

representation of the shape of generic Gaussian mixtures. Since it is invariant under rotations and translations

of the Gaussian mixture, it provides a workaround to the problem of aligning objects before recognizing their

shape without sacrificing accuracy. Experiments using synthetic data show a robust performance against type I

errors, and few type II errors when the given template Gaussian mixtures are well distinguished.

Keywords: Bag of distances, Gaussian mixtures, Fingerprints, Information retrieval, Kolmogorov-Smirnov,

Shape matching, Shape similarity

1. INTRODUCTION

The problem of searching for information is ubiquitous. In cyberspace, for example, many different engines

allow users to find relevant webpages through a text query consisting of a few words. However, the information

available in electronic format goes beyond text. Recognizing the relevant information when the query and/or

the data being searched cannot be summarized as text is challenging. One of the main difficulties is the richness

and complexity of the data, which often hinders our ability to focus on what is relevant for the search. Part

of the problem is that there are often many different ways for a query to appear in the data but no effective

way to remove this ambiguity from the data itself. Thus, a lot of research today is concerned with developing

recognition method for non-text data (e.g. images, sounds, or videos), which match the speed and accuracy of

text-based methods.

One problem of interest is shape recognition, that is to say the recognition of an object, such as a curve,

surface, or volume, up to a rotation and translation of that object. As the ambiguity of representing a shape

is well understood as a group transformation, shape recognition is a good starting point before attacking other

recognition problems where the ambiguity cannot be parameterized explicitly.

Further author information: (Send correspondence to H.J.S.V.)

H.J.S.V: E-mail: hsantosv@purdue.edu, Telephone: 1 787 466 4460

M.B.: E-mail: mboutin@purdue.edu, Telephone: 1 765 494 3538

Computational Imaging VIII, edited by Charles A. Bouman, Ilya Pollak, Patrick J. Wolfe,

Proc. of SPIE-IS&T Electronic Imaging, SPIE Vol. 7533, 753305 · © 2010 SPIE-IS&T

CCC code: 0277-786X/10/$18 · doi: 10.1117/12.848604

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The shape recognition paradigm we follow involves two key steps. 1) A representation for the shape of an ob-

ject is obtained (e.g. a feature vector or even the object itself). 2) A comparison between the two representations

is performed. In this paradigm, there is a tradeoff between complexity and faithfulness at the representation

level, which translates into a tradeoff between speed and accuracy at the comparison level. However, some recent

work1has shown that, in some cases, the tradeoff is nil for the vast majority of possible objects. For exam-

ple, it was shown that the (unlabeled) pairwise distances of a point-set (called a bag of distances) is a faithful

representation for all, but a set of measure zero of point-sets.

For translating this paradigm into practice, one must first understand how noise in the measurements affects

the object, and subsequently its representation. One can then, either modify the comparison method, so it can

deal with the noisy observed data, or find a way to estimate the shape representation from the data. For the

case of an object represented by a point-set, a Gaussian mixture (GM) model can be used to represent the

measurement noise. Previous work2,3has shown that the distribution r(Δ) of the Euclidean distance between

two points drawn independently at random according to this GM generalizes the bag of distances concept while

providing a faithful shape representation for the shape of generic GMs. In this paper, we propose a comparison

method for the case where one GM is a known template and the other GM consists of an observed sparse set of

points (one point per Gaussian). Such a setup occurs, for example, in the problem of fingerprint identification

using minutiae, where an affine analysis can provide the parameters of the GM, and the fingerprint query is

performed inline using observed minutiae. The comparison method we proposed is fast enough to be executed

in real time, and as our experiments indicate, its accuracy is very good.

The rest of this paper is divided into three sections. The following section summarizes the theoretical results

on which our approach is based. Section 3 contains our proposed comparison method. Then, Section 4 presents

a numerical evaluation of the method’s accuracy. We conclude in Section 5.

2. THEORETICAL BACKGROUND

The fact that distributions of invariants can be used to faithfully represent objects modulo some group actions

was proved by Boutin and Kemper,1for the case of point-sets. In particular, the set of pairwise distances of

a point-set, what we call the Bag of Distances (BoD), was shown to be a lossless representation for the vast

majority of point-sets. More specifically, the following theorem was proved. See [2] for a simple proof.

Theorem 1. There exists a polynomial f in 2n variables such that if the points p1,p2,...,pn ∈ R2satisfy

f (p1,p2,...,pn) ?= 0, then for any other point-set ¯ p1, ¯ p2,..., ¯ pn having the same bag of distances as that of

p1,p2,...,pn, there exists an orthogonal matrix M ∈ R2×2, a translation vector T ∈ R2and a permutation

π ∈ Snsuch that

¯ pi= Mpπ(i)+ T, for all i = 1,...,n.

The point-sets that do not lie on the zero-set of the aforementioned polynomial f are called generic point-sets.2

Some recent work aims to generalize Thm. 1 to the case of non-deterministic point-sets. In particular, the

problem of representing the shape of a GM ρ(x) was considered. In 3, the probability density function r(Δ) of

the distance Δ between two points x1and x2drawn independently from ρ(x) was proposed as a representation.

In 2, r(Δ) was shown to be a faithful representation of the shape of the vast majority of planar GM. More

specifically, the following theorem was proved.

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Theorem 2.3Suppose that two Gaussian Mixtures ρ(x), ¯ ρ(x) are such that their respective means forms a

generic point-set. Then ρ(x) and ¯ ρ(x) have the same distribution of distances, r(Δ) = ¯ r(Δ), if and only if they

have the same shape, i.e. if and only if there exists an orthogonal matrix M ∈ R2×2and a translation vector

T ∈ R2such that

ρ(x) = ¯ ρ(Mx + T).

Theorem 2 states that if r(Δ) = ¯ r(Δ), then ρ(x) ≡ ¯ ρ(x). Therefore, we can avoid the difficult task of aligning

the GMs by comparing the distribution of distances r(Δ) and ¯ r(Δ).

3. THE PROPOSED METHOD

Distance

samples

Distance

samples

Drawn from

same

distribution?

Samples

˜ ρQ(x)

ρT(x)

Figure 1. Flowchart of the tasks performed by the proposed method.

Given is a sparse set of points p1,p2,...,pn∈ R2drawn from a Gaussian mixture ρQ(x) with n Gaussian

components, each with standard deviation matrix σ2·I2×2. The parameters n and σ are known, but the remaining

parameters of ρQ(x) are unknown. We assume that the samples were drawn from distinct Gaussian components

of ρQ(x). Also, given is a template Gaussian mixture ρT(x) with n Gaussian components, each with standard

deviation matrix σ2· I2×2.

Our goal is to determine if ρT(x) and ρQ(x) have the same shape, that is to say if ρT(x) = ρQ(Mx + T),

for some rotation matrix M ∈ R2×2and translation vector T ∈ R2. The method we propose contains five steps,

namely.

Step 1: Use p1,p2,...,pnto obtain an approximation ˜ ρQ(x) of ρQ(x).

Step 2: Draw N independent samples d1,d2,...,dN from r(Δ), the distribution of distances of ρT(x),

and draw N independent samples˜d1,˜d2,...,˜dN from ˜ r(Δ), the distribution of distances of ˜ ρT(x).

Step 3: Measure the likelihood if the samples d1,d2,...,dN and the samples˜d1,˜d2,...,˜dN were drawn

from the same distribution.

Step 4: Repeat step 2 and step 3 a total of K times.

Step 5: Final decision.

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Below we describe each step in details.

Step 1: Approximation of ρQ(x).

Since the set of sample points p1,p2,...,pn is sparse, and the points were drawn from distinct Gaussian

mixture components, we set

˜ μi= pi,i = 1,2,...,n

Then, we set ˜ σ = λσ . As we set the standard deviation of the components Σi= λ2σ2· I2×2, all the parameters

of ˜ ρQ(x) are then defined. In our numerical experiments, (See Section 4) we found that λ ∈ [0.66,...,1.33].

Sparse samples

ρQ(x)

˜ ρQ(x)

Figure 2. Approximation of ρQ(x) from the given point samples.

As roughly 95% of the samples drawn from a Gaussian distribution will fall within a distance 2σ of its mean,

the regions of high density of each of the components of ˜ ρQ(x) are likely to overlap with those of ρQ(x), and

these ˜ ρQ(x) and ρQ(x) are likely to be similar (See Fig. 2)

Step 2: Sampling of the two distributions of distances.

To obtain independent samples from a distribution of distances of a Gaussian mixture (either ˜ ρQ(x) or ρT(x)),

we draw two independent samples x1 and x2 from the given Gaussian mixture and measure their Euclidean

distance |x1− x2|L2. We do this a total of N times in order to obtain N independent distance samples. The

distance samples obtained from ρT(x) are labeled d1,d2,...,dNand those from ˜ ρQ(x) are labeled˜d1,˜d2,...,˜dN.

Step 3: Evaluation of the p-value of the hypothesis.

In order to decide if the distance samples d1,d2,...,dN and the distance samples˜d1,˜d2,...,˜dN were drawn

from the same distribution, we use the Kolmogorov-Smirnov (KS) test. The KS test4,5is a statistical test that

quantifies the dissimilarities between two sample sets. More precisely, the KS test measures the distance

D∗= max

Δ∈R(|RT(Δ) − RQ(Δ)|),

where RT(Δ) and RQ(Δ) are the cumulative distributions of the sample sets {d1,d2,...,dN} and {˜d1,˜d2,...,˜dN},

respectively. The quantity D∗is then used to estimate the likelihood (p-value) that the null hypothesis is true

at a 5% significance level; the null hypothesis being that the distance samples were drawn from the same distri-

bution. Type I errors (i.e. rejecting the null hypothesis when it is true) occur when the method incorrectly finds

that the shape of the query ρQ(x) matches that of the template ρT(x). Conversely, type II errors (i.e. accepting

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