POINT CONFIGURATION INVARIANTS UNDER ... - Science Direct

Pattern Recognition, Vol. 27, No. 11, pp. 1523 1532, 1994 Elsevier Science Ltd Copyright © 1994 Pattern Recognition Society Printed in Great Britain. All rights reserved 0031-3203/94 $7.00 + .00

Pergamon

0031-3203(94)00065-4

POINT C O N F I G U R A T I O N INVARIANTS U N D E R SIMULTANEOUS PROJECTIVE A N D PERMUTATION TRANSFORMATIONS t REINER LENZ3~and PETER MEER§ ~lmage Processing Laboratory, Department of Electrical Engineering, Link6ping University, S-58183 Link6ping, Sweden §Department of Electrical and Computer Engineering, Rutgers University, Piscataway, NJ 08855-0909, U.S.A. (Received 23 June 1993; in revised form 28 April 1994; received for publication 20 May 1994)

Abstract--The projective invariants used in computer vision today are permutation-sensitive since their value depends on the order in which the features were considered in the computation. We derive, using tools from representation theory, the projective and permutation (p2) invariants of the four collinear and the five coplanar points configurations. The p2-invariants are insensitive to both projective transformations and changes in the labeling of the points. When tised as model database indexing functions in object recognition systems, the p2-invariants yield a significantspeedup. Permutation invariants for affinetransformations are also discussed. Projective invariants

Permutation invariants

Feature indexing

l. INTRODUCTION Invariants of simple feature sets (e.g. configurations of a few points or lines) became a frequently used tool in computer vision. Given the set of features s and an operator g belonging to a transformation group, the invariant I [ . ] must satisfy l[s] = I[gs] for any g in the group. If g is a projective transformation, projective invariants are obtained. An extensive survey of the applications of projective invariants in computer vision can be found in references (1, 2). Typical applications of invariants in computer vision are in object recognition problems and recovery of three-dimensionalstructure from uncalibrated cameras. A simple example can illustrate the usefulness of invariants. Assume that several planar (or 'shallow') objects are present in the visual field. Two images of the scene are taken from two different positions. (Similarly we can have only one image and models of the objects available for a generic pose.) Making abstraction of occlusions, the two images are connected by a projective transformation g. Let sA be a set of features extracted from the first image, and sa the set of features extracted from the second. A projective invariant I [. ] associated with SAsatisfies I [SA] = I [gsA] for any change in the relative location of the two cameras. Thus, the problem of matching the features SA in the second image is reduced to investigating the "i"Part of this work was done while the authors were at the Cognitive Processes Department, Auditory and Visual Perception Laboratory, Advanced Telecommunications Research Institute, 2-2 Hikaridai, Seika-cho Soraku-gun, Kyoto 61902, Japan.

Object recognition

relations among the invariants I[SA] and l[sa] independent of the transformation between the two images. The objects of interest can be identified by the presence of invariants associated with their features. Most projective invariants are permutation-sensitive. That is, the value of a projective invariant depends on the order in which the features were considered in its computation. A different ordering of the set, namely, associating indices with the elements in a different way, usually yields a different value for the projective invariant. In the above example, the permutation sensitivity of an invariant yields huge computational burden. To identify the feature set sn in the second image, the features must be considered in the same order in both images when computing the invariants. That is, they must be in correspondence before the computation of the invariants. However, this information is not available. Therefore all possible combinations of pairings must be checked, which also increases the probability of erroneous decisions. Interchanging the indices of the elements is equivalent with a permutation group acting on the set. The permutation group has its own perm= utation invariants, expressions whose value is unchanged by the reordering of the elements in the set. Let the projective transformation T and the permutation n act on the set s. In this paper we derive invariants of s under both the group of projective transformations and the group of permutations. A projective invariant Q [. ] satisfies the condition Q Is] = Q[7~-] for any (say) planar projective transformation. The permutation invariant P[.] satisfies the condition P[s] = P[ns] for all possible permutations of the indices labeling the features in the set. A projective and perm-

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R. LENZand P. MEER

utation (p2) invariant J [ . ] of the set s must satisfy J[s] = J [ n s ] = J [ T s ] = J[rtTs] = J[Trrs] for any T in the transformation group and any n in the permutation group. The value of J [ ' ] remains the same when the set undergoes a projective transformation. It is also independent of the way the indices are associated with the features in either the original or the transformed set. Invariants provide a fast indexing method into databases of models and are often used in object recognition systems [e.g. references (3-5)]. However, the employed invariants are permutation-sensitive and therefore an invariant has to be stored for any of the possible orderings of the set, When two sets are matched by similar values of a p2-invariant the order of elements within the sets becomes irrelevant. The sets are matched independent of their ordering and storage of one pZ. invariant per set suffices. The p2-invariants can be used to design fast and robust feature correspondence algorithms. ~6) In this paper we give the expressions of the p2-invariants ofprojectively transformed collinear and coplanar point configurations. The p2-invariants are obtained exploiting the properties of the fundamental projective invariant, the cross-ratio. Invariants of transformation groups were also used to derive optimal filters for pattern recognition tasks. _j;i,j= 1...6, three are equal to 1. The remaining 18 products span an 18 dimensional vector space V 2. Following the procedure described in the previous section we obtain an 18 dimensional representation of The space V2 can be decomposed into six onedimensional and six two-dimensional irrducible subspaces. Two of the one-dimensional subspaces are connected to the alternating representation of Y4 and lead to the invariant 1212] (11). Projection of the 18-dimensional vector of V 2 into the other four onedimensional irreducible subspaces yields the following four p2-invariants: 2 6 --

~

I 3 [ 2 _ ~2 ] = 1 4 [

] =--1412]

(14)

223 -- 322 -- 32 + 2 1212] = 2(2-

which transform into linear combinations of each other. For example:

...........

-6 -4 -2 -i~ 1 2 ~ 6

325 + 324 - 23 + 322 -- 32 + 1

J , [2] -

2 2 ( 2 - 1)2 226 - 625 + 924 - 823 + 922 - 62 + 2

J2[22 -

2 2 ( 2 - 1)2

J312] = 3 A [ 2 ] = - 3.

(15)

These p2-invariants can be regarded as being of second-order since they were generated using products of the cross-ratio expressions. Only the first two are nontrivial. It can be shown that all second-order p2-invariants of four collinear points are linear combinations of the four invariants, equation (15). Some of these linear combinations are already known.

(~) Invariant

25 2O

(2 2 -- 2 + 1) 3 J11 [2] -

2 2 ( 2 - 1) 2

Jl [2] + J3 [2].

(16)

1

2

4 6 Cross-ratio

(b) Fig. 1. (a) The invariant -1212]. (b) Its absolute value.

The p2-invariant Jl~[2] is often mentioned in the mathematical literature [e.g. reference (16), p. 317; reference (17), p. 127], and was used in computer vision by M a y b a n k (~s) to investigate nonplanar conics. The linear combination

6

6

J12[2] = ~

~

i=l

j=l,j#i

2~2j=JxE2]+J412-]

(17)

Point configuration invariants

Invariant

II III

S/ -6

-4

/

/

\',

-2

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