International Journal of Computer Vision 28(3), 213–221 (1998)c© 1998 Kluwer Academic Publishers. Manufactured in The Netherlands.

Diffeomorphisms Groups and Pattern Matching in Image Analysis

ALAIN TROUV ELAGA, Institut Galilee, Universite Paris13, Av J-B Clement, 93430 Villetaneuse, France

trouve@math.univ-paris13.fr

Received October 4, 1995; Accepted February 6, 1996

Abstract. In a previous paper, it was proposed to see the deformations of a common pattern as the action ofan infinite dimensional group. We show in this paper that this approach can be applied numerically for patternmatching in image analysis of digital images. Using Lie group ideas, we construct a distance between deformationsdefined through a metric given the cost of infinitesimal deformations. Then we propose a numerical scheme to solvea variational problem involving this distance and leading to a sub-optimal gradient pattern matching. Its links withfluid models are established.

Keywords: pattern matching, diffeomorphisms group, Lie group, deformations, physically based models,geodesic distance

1. Introduction

Deformable template matching has received consider-able attention during the last decade. In this paper, werestrict to high dimensional representation templatesdefined as functionf : M → R whereM is a smoothmanifold. Since our main attention is on grey levelimage matching,M will stand for [0, 1]2 or preferablyfor the two-dimensional torusM = R2/Z2. The tem-plate is a model for a family sharing common features(for instance, the family of radiographic images of theright hand). Assuming thatf is a new observed im-age belonging to this family, the problem of patternmatching in this context is to find an invertible map-ping φ : M → M such that f ∼ f ◦ φ. A commonframework to solve this problem is to define the optimaldisplacement fieldu as the solution of

u = argmin1

2

∫M

| f (x)− f (x + u(x))|2 dx

+ 1

2〈u, u〉1/2

e , (1)

where u belongs to an adequate Hilbert spaceHequipped with a scalar product denoted〈 , 〉e. Thisframework, as well as landmarks based matching, has

been quite extensively studied (Besl, 1989; Booksteinand Green, 1993; Ruprecht and M¨uller, 1992; Szeliskiand Coughlan, 1994) with successful applications tobrain mapping (Miller et al., 1993) and an algorithmbased on stochastic relaxation is proposed in (Amitet al., 1991) for the computation ofu where〈u, u〉e =∫

M(12u1)

2 + (12u2)2 dx (12 is the iterated Lapla-

cian). However, this approach is not entirely satisfyingsince we have no guaranty thatx → x +u(x) is invert-ible and this problem is more and more accurate whenlarge deformations are involved. The problem is thatlarge deformations can be computed if we relax the con-straints on the metric, however, there is often no goodsolution allowing simultaneously a good and regularone-to-one matching (see Christensen et al. (1996) foran interesting discussion on this matter). A more con-venient framework could be to consider that the “good”variable is not the displacement fieldu but the deforma-tion mappingφu defined byφu(x) = x +u(x) living inthe group of continuously invertible mapping Aut(M).Then as a first approximation, the norm|u|e = 〈u, u〉1/2

e

can be interpreted as a distance between the identityI d andφu. Obviously,φu is invertible only for suffi-ciently smallu and|u|e is only an approximation up toa first-order error of a true distance. Hence, for a large

214 Trouve

deformationφ, one can considerφ as the concatenationof small deformationsφui . More precisely, ifφ = 8n

where the8k are recursively defined by80 = I d andφk+1 = φuk+1 ◦ 8k, the family8 = (8k)0≤k≤n de-fines a polygonal line in Aut(M) whose lengthl (8)can be approximated byl (φ) = ∑n

k=1 |uk|e. At a naivelevel, one could define the distanced(I d, φ) as the in-fimum of l (8n) for all family 8 such that8n =φ.For a more rigorous setting, one should considerthe time continuous curve8= (8t )0≤t≤1 in Aut(M)solutions of

∂8

∂t= Ut ◦8t , 80 = I d, (2)

for a continuous time familyUt of vector fields inHand one should define

d(I d, φ) = inf81=φ

l (8) with l (8)def=∫ 1

0|Ut |e dt.

This is the program defined by Azencott (1994). Afully rigorous construction of this distance, in the con-text of infinite dimensional Lie groups, is performedin (Trouve, 1995a, 1995b). We show that the subsetof φ ∈ Aut(M) for which d(I d, φ) can be defined isa subgroupA of invertible mappings. We extend thisdistance between two arbitrary mappingsφ andψ byright invariance:d(φ, ψ) = d(I d, ψ ◦ φ−1). Hence,in this extended framework, the new problem shouldbe to find inA

φ = argmin1

2

∫M

| f − f ◦φ|2 dx+ 1

2d(I d, φ)2. (3)

As a simple example, consider the one-dimensionalcaseM = [0, 1] and defineH as the Sobolev space

H = {u ∈ L2([0, 1],R) | u′ ∈ L2([0, 1],R),

u(0) = u(1) = 0}.

The problem (3) can be interpreted in the context ofspeed recognition as time warping under an elastic con-straint. The associated groupA appears to be the ab-solutely continuous invertible mappingφ : [0, 1] →[0, 1] and the associated distance is given byd(φ, ψ) =arccos(

∫ 10

√φ′

sψ′s ds). This example is presented in

(Piccioni et al., to appear) where it is also proved thatthe problem (3) has a solution which can be computedwith dynamical programming (one can refer to Younes(1995) for an important extension of this 1D example).In several dimensions, the construction can be achieved

if H is a separable Hilbert space with an admissiblemetric, that is essentially, if the supremum norms ofuand∇u satisfy

supx∈M

|u(x)| + supx∈M

|∇u(x)| ≤ K |u|e, (4)

for a uniform constantK , whereM is assumed to bea compact Riemannian manifold without boundary. Inthis general case, the distanced has not generally anexplicit expression but we show in (Trouv´e, 1995b)that the optimal mappingφ solution of (3) always ex-ists. Moreover,A is complete for the distanced andcan be equipped with a weak structure of infinite di-mensional manifold. The Hilbert spaceH is then thetangent space at identity, i.e., the infinitesimal defor-mation aroundI d. This construction is quite generaland valuable for any metric〈 , 〉e satisfying (4). Thefreedom on the choice of〈 , 〉e lets us design the met-ric according to our precise application. Recalling that〈 , 〉e gives the cost of infinitesimal deformations, themetric should be chosen according to the plausibility ofsome deformations. The optimal choice of this metricis still an open problem and the reader could refer to(Lanitis et al., 1995 and references herein) for majorsteps in this direction.

2. Gradient Descent versus PDE

Let us say thatφ cannot be computed numericallywithout using an algorithm based on simulated an-nealing in our infinite dimensional setting due to thenumber of local minima of our functional definedin (3). However, in (Trouv´e, 1995b), we propose asub-optimal algorithm based on a gradient descent onE(φ) = 1

2

∫M( f − f ◦ φ)2 dx. More precisely,A can

be equipped with a right invariant weak structure ofRiemannian manifold (see Trouv´e (1995b) for details)where the tangent spaceTφA atφ is given by

TφA = { u ◦ φ | u ∈ H},

with the scalar product given by

〈u ◦ φ, v ◦ φ〉φ = 〈u, v〉e.

We have proved in (Trouv´e, 1995b) thatE is differen-tiable everywhere and that

dφE(v) = −∫

M( f − f ◦ φ)(∇ f ◦ φ, v)ds,

Pattern Matching in Image Analysis 215

where(z, z′) denotes the usual scalar product onR2.Hence, one can define∇φE for anyφ ∈ A with theusual property

dφE(v) = 〈∇φE, v〉φ.

The idea of our sub-optimal algorithm is to follow thegradient∇E with a control on the distanced fromidentity. More precisely, we have proved in (Trouv´e,1995b) that there exists a solution8 ∈ C(R+,A) sat-isfying

d8

dt= −∇8t E. (5)

Now, consideringq : R+ → R+ defined by

q(t) = E(8t )+ 1

2

(∫ 1

0

∣∣∣∣d8ds

∣∣∣∣8s

ds

)2

,

we can prove thatq reaches its minimum. Hence, defin-ing t = inf{t ≥ 0 | q(t) = inf q}, we denoteφ = 8t

called the sub-optimal solution of the matching prob-lem. The sub-optimal solution can be computed nu-merically as far as∇φE can be easily computed foranyφ ∈ A.

Our gradient algorithm is interesting in itself sinceit can be rewritten in term of PDE. Indeed, assumingthat

〈u, v〉e = 〈Lu, v〉2,

where〈 , 〉2 denotes the usualL2 scalar product andL is a self-adjoint positive differential operator, thenstraightforward computations show that (5) is equiva-lent to the fluid model PDE{

∂8∂t (t, x) = v(8(t, x), t),

Lv + b8t = 0,(6)

whereb8t is a field of restoring forces given by

bφ(y) = ( f ◦ φ−1 − f )(y)∇ f (y)|Jφ−1|(y).

This new formulation shows that starting from any ad-missible self adjoint positive differential operator (inthe sense of (4)), our gradient algorithm inA is a canon-ically associated PDE driving nonlinear kinematics. Asa meaningful example, starting with a linear elasticity

operatorL, (6) is nothing else than a viscous fluid equa-tion as introduced in (Christensen et al., 1996). How-ever, we think that our geometrical approach in thecontext of infinite dimensional Lie group with a rightinvariant Riemannian metric sheds a new light on non-linear template registration procedures. An importantpoint is that the associated distance can be an intrinsicmeasure of the deformations. Moreover, we see thatwe have a considerable freedom in the choice of themetric allowing to escape outside the classical phys-ically based models (not so natural in many context)towards learned metric in a set of examples.

3. Numerical Scheme

In practical situations, sinceM is the two-dimensionaltorus, we could define the metric〈u, v〉e in terms ofthe Fourier coefficients(un)n∈Z2, (vn)n∈Z2 of u andv.Assume the following form for〈 , 〉e

〈u, v〉e =∑n∈Z2

∑i =1,2

anuinv

in, (7)

where(an)n∈Z2 is a positive real-valued sequence suchthat an = a−n for any n ∈ Z2. Then, using Sobolevimbeddings, we can ensure that〈 , 〉e satisfies the con-dition of admissibility if there exists 5/2 ≤ γ1 ≤ γ2,K1 > 0 andK2 > 0 such that

K1(n2

1 + n22

)γ1 ≤ an ≤ K2(n2

1 + n22

)γ2.

Now, if we assume that

an = K ((ξn1)2 + (ξn2)

2 + 1)γ , (8)

for γ ≥ 5/2 andξ > 0, then the Hilbert spaceH be-comes the Sobolev spaceH γ (M) so that the regularityof the elements ofH increases with the value ofγ .The variableξ should be interpreted in this context asa scaling factor.

Remark. Let us note here that our metric is not phys-ically based in the sense of our discussion in Section 2.However, our two parametersξ andγ describe a suffi-ciently rich family of metrics in many contexts.

The computation of∇φE is particularly simple underthe assumption (7). Indeed, using Parseval’s identity,

216 Trouve

we have for anyu ∈ B

dφE(u ◦ φ) =∫

M( f − f ◦ φ−1)|Jφ−1|(∇ f, u) dx

=∑n∈Z2

2∑i =1

Finui

n

=∑n∈Z2

2∑i =1

an

(Fi

n

an

)un,

where Fin =F(( f − f ◦ φ−1)|Jφ−1| ∂ f

∂xi) and F is

the discrete Fourier transform. SincedφE(u ◦ φ)=〈∇φE, u ◦ φ〉φ = 〈∇φE ◦ φ−1, u〉e, we deduce that∇φE = (∇φE1,∇φE2) is defined by

∇φEi =[F−13F

(( f − f ◦ φ−1)|Jφ−1| ∂ f

∂xi

)]◦φ,

(9)

where3 is the multiplication of the Fourier coeffi-cients by 1/an. The case3 = I d corresponds to themetricL2 defined by〈u, v〉2 = ∫

M(u, v)dx. Then, fora general metric satisfying (7), the equality (9) saysthat∇φEi is given as a linear filtering of the usualL2

gradient. Unfortunately, we need to computeφ−1 toget∇φE. However, the evolution equation for8−1

t ismore tractable. Indeed, sinceφ−1 ◦ φ(x) = x for allx ∈ M , we get

d8−1

dt(x) = dx8

−1

(d8

dt◦8−1

),

so that

d8−1

dt(x)

= dx8−1

[F−13F

(( f ◦φ−1 − f )|Jφ−1| ∂ f

∂xi

)].

(10)

Now, the evolution equation of8−1t does not need the

computation of8t so that we will prefer to use (10).Assume thatf and f are known on a grid of size

2r × 2r . Let Mr = 12r R2/Z2 be the corresponding

discretization of the two-dimensional torusM . To dis-cretize our algorithm, we need the following operators:

Restriction operator: This operatorRr : Mr → M isdefined as the canonical injection.

Gradient operator: For anyg : Mr → R, we define

∇1g(x) = 2r

[g

(x +

(1

2r, 0

))− g(x)

],

∇2r (x) = 2r

[g

(x +

(0,

1

2r

))− g(x)

],

and

∇r g = (∇1g,∇2g).

Differential operator: For anyφ : Mr → M we de-fine Dφ : Mr → L(R2,R2) by

Dxφ(u(x)) =(

∇1φ1(x) ∇1φ2(x)

∇2φ1(x) ∇2φ2(x)

)(u1(x)

u2(x)

)

whereφ = (φ1, φ2) andu : Mr → R2.Composition operator: Let g : Mr → R and φ :

Mr → M . We define by linear interpolation the“composition” ofg andφ denotedg◦φ.

With the previous operators, we can now define ournumerical scheme. Letfr = f ◦ Rr and fr = f ◦ Rr .Let8 : R+ → A be the solution of the gradient des-cent for E and let9 : R+ → A be defined by9t =8−1

t . We will compute an approximation of9 on Mr

by the following explicit scheme. Starting from theidentity, we apply the iterative scheme

1. F(n + 1) = ( fr ◦9(n)− fr ) det(D9(n))∇r fr2. F(n + 1) = ifft2 ◦3r ◦ fft2(F(n + 1))3. V(n + 1) = D9(n)(F(n + 1))4. 9(n + 1) = 9(n)+ δV(n + 1)

whereδ is a parameter controlling the time step, fft2(resp. ifft2) is the fast (resp. inverse fast) Fouriertransform and3r the multiplication of the Fourier co-efficient by 1/an.

The use of the FFT in step 2 on a grid whose size is apower of 2 allows a fast computation of∇E. The com-putation ofd = ∫ t

0 | d8sds |8sds can be easily computed

if we add a 5th step. In order to speed up the algorithm,one can start with a lower value ofr and increases thisvalue during the algorithm. When the size doubles, thevalues of8 on the new points can be easily computedby linear approximation.

Pattern Matching in Image Analysis 217

Figure 1. ξ = 0.2, γ = 6.

4. Numerical Results

In this section, we present some numerical results on aset of pairs (template, observation) in order to validatenumerically our approach.

We start with two synthetic black and white imagesof rabbits.

The Fig. 1 should be read in the following way. In thetop-left corner, we have the template and in the bottom-left corner the observation. The top-right corner givesthe image of f ◦ φ−1 and the bottom-left corner pro-vides a mesh representation ofφ−1. Hence, we havea visual evaluation of the quality of the matching by acomparison of the two top images.

Note that all the figures will follow the above format.The pictures have size 64× 64 but we start with a sizeof 32 × 32 and we refine the grid after several itera-tions. On the template rabbit, we have singled out par-ticularly interesting points (with a black cross) as wellas their corresponding points throughout the matchingon both remaining pictures. Note that the matchingis quite accurate. In particular, note the differences inthe shapes of the arms in the template and the obser-vation and the accuracy of the matching. Note alsothe difference of size between the two right ears aswell as the different positions of the white spot on thebody. For the metric〈 , 〉e, we have chosenξ = 0.2 andγ = 3 (see (8) for the definition ofξ andγ ). In thisexample, the choice ofξ is more important than thechoice ofγ . One have to choose a fairly small valueof ξ to allow important local deformations of the leftear.

We present in Fig. 2 the “C”-experiment borrowedfrom (Christensen et al., 1996). In this example, one

Figure 2. “C”-experiment.γ = 6, ξ = 0.5. The top-right picturerepresents heref ◦ φ.

can see as a (not so) small patch of material can deforminto a longer curved patch. The important point hereis that even if the regularity requirement of the de-formation is high (γ = 6, the initial metric is quiterigid), the gradient algorithm succeeds. This regular-ity of the transformation removed the instability prob-lem due to the quasi-singularities which appear withusual viscous fluid models (corresponding approxima-tively to γ = 1). As a consequence, we do not needto use the concatenation procedure consisting in prop-agating the template when the Jacobian of the transfor-mation drops below a certain value and starting withthe new propagated template. Our result is obtained inone shot.

Note that the transformation makes appear somehighly compressed regions but stays nonsingular andfairly regular (note also that the light grey frame bor-der is not affected by the transformation). We show inFig. 3 the values of the Jacobian of the transformation.

Figure 3. Jacobian of the transformationφ−1. Light-grey corre-sponds to low value of the Jacobian.

218 Trouve

Figure 4. “C”-experiment. γ = 6, ξ = 0.5. Top-right: f ◦ φ;Bottom-left: singular region ofφ (zoomed).

We propose on this difficult case a comparison withthe linear approach given by (1). We use the metric〈 , 〉e and we follow the lines of (Amit et al., 1991).The result obtained is presented in Fig. 4

Remark. In fact, even for the linearized approach of(Amit et al., 1991) or (Miller et al., 1993), our metricwhich is diagonalizable in the Fourier basis is of partic-ular interest. Indeed, since the metric is admissible, wecan prove thatu → E(φu) (whereφu(x) = x + u(x))is differentiable inH , and one can easily compute thegradientfor the metric on Hof E. In particular, usingthe FFT, the complexity of the gradient computation isO(n2 log(n)) for an image of sizen × n. As a con-sequence, for an image of size 256× 256, the exactcomputation of the gradient needs no more time thanthe computation of its projection on approximatively 8eigenvectors for usual elastic models.

However, Fig. 4 is our best result, for an experimen-tally tuned Lagrangian multiplier put in front of〈u, u〉e.As discussed in Section 1, we cannot prevent the trans-formation to be noninvertible as shown in the zoomedarea ofφ in Fig. 4.

We turn now to real images. We start with picturesof leaves. In Fig. 5, the template and the observationare identical up to a rotation of 20 degrees.

On a realistic implementation of such a matchingalgorithm, one should first subtract this rigid deforma-tion with one of the standard algorithms. However, weassume that such a rough algorithm can let a residualrotation less than 20 degrees so that we should be able

Figure 5. γ = 4, ξ = 0.5,α = 0.1.

to correct this remaining rotation. It has been men-tioned in the linear approach (Amit et al., 1991), thatparadoxically, if nonrigid deformations are pretty wellmatched as far as the deformation is small, the rotationsremain a problem due to the boundary conditions if wework on the square or on the two-dimensional torus.However, we will show that rotations can be handledquite satisfactorily in our approach if we modify theinitial metric〈 , 〉e in an adequate way. More precisely,we define a pseudo infinitesimal rotation as the vectorfield u0 given by

u0(x) = k(|x|)Rπ/2(

x −(

1

2,

1

2

))+(

1

2,

1

2

),

Rπ/2 =(

0 11 0

).

(Rπ/2 is the rotation ofπ/2) andk(ρ) is a smoothed stepfunction given byk(ρ) = arctan(a(1/2 − ρ)), wherea > 0. We show in Fig. 6 the vector fieldu0 on a gridon size 32× 32 fora = 40.

Figure 6. Vector fieldu0.

Pattern Matching in Image Analysis 219

Assume thatu0 is normalized so that|u0|e = 1.Now, let p0 be the orthogonal projection onRu0 for thescalar product〈 , 〉e. Letq0 = I d − p0 be the projectiononRu⊥

0 . Then, for anyα ∈ R such that 0< α ≤ 1, wedefine the new scalar product onH , denoted〈 , 〉e,α, by

〈u, v〉e,α = α2〈p0(u), p0(v)〉e + 〈q0(u),q0(v)〉e.

Under this new metric,〈u0, u0〉1/2e,α = α so that the cost

ofu0 decreases withα. Therefore, the distance betweenId and the pseudo rotations exp(tu0) decreases withα. The computation of∇E for this new metric has thesame complexity than for the original one, but as wesee in Fig. 5, this new metric is well appropriate tohandle rotations.

Remark. These changes of metric is a powerful tooland should not be restricted to the problem of rota-tions. One can proceed similarly with other specialvector fields which appear to be interested in a pre-cise application. The selection of these vectors fieldsshould be certainly done according to a learning set ofdeformations along the lines of (Lanitis et al., 1995).However, this work is still to be done, and needs furthertheoretical developments. Our feeling is that this exam-ple clearly shows that the metric should not be chosenaccording to an often arbitrary “physical” model, butaccording to the observed global deformations.

In Fig. 7, the template and the observation are twodifferent leaves with a rotation of 20 degrees of thetemplate (note: in particular, the differences betweenboth lower-half parts).

Figure 7. γ = 4, ξ = 0.3,α = 0.05.

The matching here is precise up to an error less thantwo pixels for the size 128× 128. The major part ofthe matching process is done on images of size 32×32and we use the new metric defined above. On the meshgrid representation ofφ−1 we see clearly the action ofthe component coming fromu0. However, we see thatthe obtained matching is much more complex than asimple rotation. Note that without any change on themetric, the obtained matching is not visually satisfyingas shown in Fig. 8.

In particular, the lower parts of the images are mis-matched and the error is about 20 pixels in this area forthe size 128× 128.

In Fig. 9, we have extracted two images of a sequenceof a pumping heart, where the first one is consideredas the template and the second one as the observation.

Figure 8. γ = 4, ξ = 0.3,α = 1.

Figure 9. γ = 4, ξ = 0.5,α = 1.

220 Trouve

Figure 10. γ = 4, ξ = 0.3,α = 0.1.

Since, no rotation is involved, we have used the metric〈 , 〉e (i.e.,α = 1). As far as we can judge visually, theobtained matching seems correct. Our method seemsparticularly well suited for deformations of soft biolog-ical organ in medical imaging where the deformationsare highly nonrigid with no salient features.

We end the presentation of our numerical results byan image of a human face in two positions (Fig. 10).

The matching is particularly difficult since, unlikethe previous examples, the background is not uniform.Moreover, we have a movement of the head on afixedbackground. Obviously, the topological constraints aretoo strong to have a natural matching. However, wewould like to show that even in this delicate situation,our algorithm gives satisfying results. We use the met-ric 〈 , 〉e,α with α = 0.1 since the main displacement isa pseudo rotation, but note that the matching does notreduce at all to this rotation. In Fig. 11 we present thefirst iterations of the matching.

5. Conclusion

In this paper, we have shown that our theoreticalapproach leads to an effective algorithm of pattern

Figure 11. Observation and iteration 5, 10, 15, 25 and 35.

matching. This algorithm appears as a generalizationof the linearized approach of (Amit et al., 1991). Al-though for small deformations the two algorithms givesimilar results, our approach works for large deforma-tions. Moreover, we think that the mathematical frame-work is well adapted to handle much more generalsituations. For instance, we study in (Trouv´e, 1995b)the case of a general compact Riemannian manifoldMwithout boundary so that we are not limited to flat met-rics as in the case of the two-dimensional torus, and thecase whenf has values in a finite dimensional manifoldX. Hence, this framework is well adapted for signalprocessing as done in (Piccioni et al.) or for 3D patternmatching. Moreover, we can use it for classificationtasks or for data restoration since one can consider moregeneral deformations of the template asθ(x)∗ f (φ(x))whereθ(x) has values in a Lie groupG acting of themanifold X as proposed in (Trouv´e, 1995b).

Acknowledgments

The author would like to thank professor RobertAzencott for many helpful discussions. His beautifulideas on the application of the Lie group theory in pat-tern recognition are one of the main sources of thepresent paper.

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