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Affine invariant pattern recognition using multiscale autoconvolution.pdf 323KB

Afﬁne Invariant Pattern Recognition Using

Multi-Scale Autoconvolution

Esa Rahtu, Mikko Salo and Janne Heikkil

Abstract

This article presents a new afﬁne invariant image transform called Multi-Scale Au-

toconvolution (MSA). The proposed transform is based on a probabilistic interpretation

of the image function. The method is directly applicable to isolated objects and does

not require extraction of boundaries or interest points, and the computational load is

signiﬁcantly reduced using the Fast Fourier Transform. The transform values can be

used as descriptors for afﬁne invariant pattern classiﬁcation, and in this article we

illustrate their performance in various object classiﬁcation tasks. As shown by a com-

parison with other afﬁne invariant techniques, the new method appears to be suitable

for problems where image distortions can be approximated with afﬁne transformations.

Index Terms

Afﬁne invariance, afﬁne invariant features, pattern classiﬁcation, target identiﬁca-

tion, object recognition, image transforms.

E. Rahtu and J.Heikkil¨a are with the Machine Vision Group, Department of Electrical and Information Engineering, University

of Oulu. P.O. Box 4500, 90014 University of Oulu, Finland. E-mail: {erahtu,jth}@ee.oulu.ﬁ

M. Salo is with the Rolf Nevanlinna Institute, Department of Mathemathics and Statistics, University of Helsinki. P.O. Box

68, 00014 University of Helsinki, Finland. E-mail: msa@rni.helsinki.ﬁ

Afﬁne Invariant Pattern Recognition Using

Multi-Scale Autoconvolution

I. INTRODUCTI ON

Every day we confront situations where we have to recognize an object or patterns, like when

seeing the face of a friend, ﬁnding keys from the table, reading a book, smelling food, listening to

a conversation, looking at photos, watching television, etc. These things are often easy for us, but

to build a machine with similar abilities is very difﬁcult. Nevertheless, the many results already

achieved, combined with ever increasing computational power, insure that nowadays it is difﬁcult

to ﬁnd many scientiﬁc or technical disciplines not applying some pattern recognition techniques

in one way or another. Fields like medicine, biology, psychology, marketing, computer vision,

artiﬁcial intelligence, and remote sensing make use of patterns and automated recognition.

One of the most difﬁcult things for a machine to do is to identify an object when seen from

different angles. However, this kind of ability is very much desired in many pattern recognition

applications. There is even the dream of a general pattern recognizer. While it is difﬁcult

to directly achieve such a system, many approximations have been proposed with promising

results. One of these is the idea to model the distortions using some geometric transformation,

like translation, rotation, scaling etc. and then to ﬁnd features which are invariant under these

transformations. These features can be used as a basis for classiﬁcation. For example in models

where only rotation, scaling and translation are involved, well known global methods are Fourier-

Mellin transform [1], Radon transform [2], wavelet and ridgelet based techniques [3], and Haar

integrals [4]. However many times these three degrees of freedom are not enough for adequate

approximation. Restricting ourselves to essentially linear models, which are usually much easier

to handle than strongly non-linear ones, the afﬁne mapping gives the best possible approximation.

It is easy to build afﬁne invariances using segmented information like contours or interest

points. Well known methods in this class include boundary based Fourier descriptors [5], [6],

[7], discrete interest point based methods [8], [9], and some wavelet based descriptors [10],

[11]. Some of these methods are actually even capable of handling the more complex case of

projective mappings [9]. The problem is however that these methods stay very dependent on

the success of the segmentation process. Sometimes one also needs additional techniques, like

geometric hashing [9] or interest point detectors [12], for matching correspondences.

The group of global afﬁne invariant methods is more sparse despite the fact that the ﬁrst

one, afﬁne invariant moments [13], was already presented in 1962. That presentation, however,

contained an error which was later corrected in [14] and [15]. Since then, other methods have

been introduced, e.g. Ben-Arie’s frequency domain technique [16], [17], cross-weighted moments

[18], and trace transform [19]. In this article, we propose a novel approach called Multi-Scale

Autoconvolution (MSA) for global afﬁne invariant feature extraction. Our technique is based on

applying standard point-based invariants in a novel way, combined with probabilistic ideas. The

proposed transform has only moderate computational complexity, and it does not require any

image segmentation steps other than background elimination. This idea was originally introduced

in [20] and further developed in [21]. It is also possible to use MSA for convexity recognition,

and some initial results can be found in [22]. As convexity recognition is not directly related to

afﬁne invariant classiﬁcation we will not consider this here.

In this article, we give a comprehensive treatment of the MSA transform, expanding the

earlier results and complementing them with new experiments. The article is organized as

follows. The upcoming section introduces the Multi-Scale Autoconvolution with its deﬁnition and

properties, whereas Section III considers the actual implementation. In Section IV, we examine

the computational complexity of the proposed method and compare it with some other similar

techniques. Section V illustrates the discriminative power of MSA based features in various

object classiﬁcation tasks, including binary and gray-scale cases also in the presence of noise

and different distortions. Sections VI and VII ﬁnally summarize the results and give directions

for future development. Our goal is to present a novel idea for afﬁne invariant recognition and

to discuss its properties and computational aspects, so that it will be easy to implement for the

use of applications.

II. M

ULTI-SCALE AUTOCONVOLU TION

In this section, we deﬁne an afﬁne invariant image transform called Multi-Scale Autoconvo-

lution (MSA). We start with an introduction to the ideas behind MSA and then go further into

the mathematical details and the properties of this transform. But before continuing, let us ﬁrst

deﬁne the spatial afﬁne transformation and afﬁne invariance as follows:

Deﬁnition 1: Deﬁne the afﬁne transformation A = A{T,t} by



= A(x)=Tx + t,

where t, x ∈

and T is a 2 × 2 nonsingular matrix whose elements belong to

. Any such

transformation is invertible with inverse A

−1

(x)=T

−1

x − T

−1

Deﬁnition 2: Suppose f(x):

→

, f ≥ 0, is an image intensity function corresponding to

a gray-scale image in

. We may apply an afﬁne transformation A to this image, which gives

a new gray-scale image in

with image function f



, where



(x)=f ◦A

−1

(x)=f(T

−1

x − T

−1

t).

We call f



the A transformed version of f.

Deﬁnition 3: A feature I, extracted from the function f, is said to be afﬁne invariant if it

produces the same value for f and the A transformed version of f, for any afﬁne transformation

A. The deﬁnition of MSA

Let f(x):

→

with f ≥ 0 be an image intensity function in L

(

) ∩ L

(

) and

let p(x)=

f 

f(x) be the normalized version of f, so that



p(x)dx =1. Then p(x) is

a probability density function, and we may take X

, X

, and X

to be independent random

variables with values in

so that p

)=p(x

) (we write p

for the probability density

function of a random variable X). Consider three samples (x

, x

) of these random variables

as a basis for the following transformation

u = α(x

− x

)+β(x

− x

)+x

(1)

where (α, β) are the coordinates for u in the space spanned by the vectors x

− x

and x

− x

and with the origin at x

Now let A{T,t} be an afﬁne transformation as in Deﬁnition 1. If one takes the A transformed

versions of the sample points x

, x

and x

, i.e. x



= Tx

+ t, x



= Tx

+ t, and x



= Tx

+ t,

and deﬁnes another transformation by



= α(x



− x



)+β(x



− x



)+x



, (2)

it can be rewritten as



= α(Tx

− Tx

)+β(Tx

− Tx

)+Tx

+ t

= Tu + t. (3)

One can see that also the points u and u



are connected by A if they have the same coordinate

values (α, β) in the corresponding spaces. This well known phenomenon is illustrated in Figure

1. As known, afﬁne transformations are bijective, so every u has exactly one corresponding u



Now recall that the points x

, x

, and x

were samples of the random variables X

, X

, and

, and deﬁne a new random variable

α,β

= X

+ α(X

− X

)+β(X

− X

) (4)

which has u as the corresponding sample.

Similarly deﬁne



α,β

= X



+ α(X



− X



)+β(X



− X



where X



= TX

+ t, X



= TX

+ t, and X



= TX

+ t and hence also U



α,β

= TU

α,β

+ t.

It follows (see Lemma 2.1 below) that p



)=p



), where p



is the normalized version of



(x)=f ◦A

−1

(x)=f(T

−1

x − T

−1

t).

Substituting x = U



α,β

we get



α,β

)=f(T

−1



α,β

− T

−1

= f(T

−1

(TU

α,β

+ t) − T

−1

t)=f (U

α,β

x

x=Tx+t

x

Fig. 1. Illustration of how a point triplet behaves under afﬁne transformation.

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