Well-posedness of eight problems of multi-modal statistical image-matching

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(1)Well-posedness of eight problems of multi-modal statistical image-matching Olivier Faugeras, Gerardo Hermosillo. To cite this version: Olivier Faugeras, Gerardo Hermosillo. Well-posedness of eight problems of multi-modal statistical image-matching. RR-4235, INRIA. 2001. �inria-00072352�. HAL Id: inria-00072352 https://hal.inria.fr/inria-00072352 Submitted on 23 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Well-posedness of eight problems of multi-modal statistical image-matching Olivier Faugeras — Gerardo Hermosillo. N° 4235. ISSN 0249-6399. ISRN INRIA/RR--4235--FR+ENG. ` THEME 3. apport de recherche.

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(4) Well-posedness of eight problems of multi-modal statistical image-matching Olivier Faugeras , Gerardo Hermosillo Thème 3 — Interaction homme-machine, images, données, connaissances Projet Robotvis Rapport de recherche n° 4235 — — 64 pages. Abstract: Multi-Modal Statistical Image-Matching techniques look for a deformation field that minimizes some error criterion between two images. This is achieved by computing a solution of the parabolic system obtained from the Euler-Lagrange equations of the error criterion. We prove the existence and uniqueness of a classical solution of this parabolic system in eight cases corresponding to the following alternatives. We consider that the images are realizations of spatial random processes that are either stationary or nonstationary. In each case we measure the similarity between the two images either by their mutual information or by their correlation ratio. In each case we regularize the deformation field either by borrowing from the field of Linear elasticity or by using the Nagel-Enkelmann tensor. Our proof uses the Hille-Yosida theorem and the theory of analytical semi-groups. We then briefly describe our numerical scheme and show some experimental results. Key-words: Multi-modal Image Matching, Variational Methods, Registration, Optical Flow, Mutual Information, Correlation Ratio, Euler-Lagrange equations, Initial-value problems, Maximal monotone operators, Strongly continuous semigroups of linear bounded operators, Analytical semigroups of linear bounded operators.. This work was partially supported by NSF grant DMS-9972228, EC grant Mapawamo QLG3-CT-2000-30161, INRIA ARCs IRMf and MC2, and Conacyt.. Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France) Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65.

(5) Résultats sur le caractère bien posé de huit problèmes de mise en correspondance multimodale d’images R´ esum´ e : Les méthodes de mise en correspondance multimodale cherchent un champ de déformation qui minimise un critère d’erreur entre deux images. Ceci est accompli en calculant une solution du système d’EDP paraboliques obtenu a` partir des e´ quations d’Euler Lagrange du critère d’erreur. Nous démontrons existence et unicité de la solution pour ce système parabolique dans huit cas qui correspondent aux alternatives suivantes. Nous considérons que les images sont des réalisations de processus aléatoires spatiaux qui sont soit stationnaires soit non stationnaires. Dans chaque cas on mesure la similarité entre les deux images soit par l’information mutuelle, soit par le rapport de corrélation. Dans chaque cas nous régularisons le champ de déformation soit par un terme d’élasticité linéarisée, soit par de la diffusion anisotrope en utilisant le tenseur de Nagel-Enkelmann. Notre preuve utilise le théorème de Hille-Yosida. Nous décrivons ensuite brièvement la discrétisation des e´ quations et nous montrons quelques résultats expérimentaux. Mots-cl´ es : Mise en Correspondance Multimodale, Méthodes Variationnelles, Flot Optique, In´ formation Mutuelle, Rapport de Corrélation, Equations d’Euler-Lagrange, Problèmes d’évolution, Opérateurs maximaux monotones, Semi-groupes d’opérateurs linéaires bornés fortement continus, Semi-groupes d’opérateurs linéaires bornés analytiques..

(6) Well-posedness of eight problems of multi-modal statistical image-matching. 3. Contents 1 Introduction. 5. 2 The framework 2.1 Functional spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The minimization problem and the associated initial value problem . . . . . . . . . .. 5 5 8. 3 The regularization terms: convexity and coerciveness 9 3.1 The linearized elasticity operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 The Nagel-Enkelmann tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4 The Dissimilarity terms 11 4.1 Definition of the global dissimilarity criteria . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Definition of the local dissimilarity criteria . . . . . . . . . . . . . . . . . . . . . . . 13 5 Existence of minimizers. 15. 6 The Euler-Lagrange equations 18 6.1 The first variation of the regularization terms . . . . . . . . . . . . . . . . . . . . . 18 6.2 The first variation of the dissimilarity terms . . . . . . . . . . . . . . . . . . . . . . 18. . . and 7 Properties of theoperators and are self-adjoint maximal monotone . . . . . . . . . . . . 7.1 7.1.1 The linearized elasticity operator . . . . . . . . . . . . . . . . . 7.1.2 The Nagel-Enkelmann operator . . . . . . . . . . . . . . . . . are the infinitesimal and generators of semigroups 7.1.3 and are the infinitesimal generators 7.2 of analytical semigroups .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 22 22 22 24 25 26. 8 Properties of the functions

(7) and 8.1 Preliminary results . . . . . . . . 8.2 Global criteria . . . . . . . . . . . 8.2.1 Mutual Information . . . . 8.2.2 Correlation ratio . . . . . 8.3 Local criteria . . . . . . . . . . . 8.3.1 Mutual information . . . . 8.3.2 Correlation ratio . . . . . 8.4 Summary . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 28 28 29 29 37 43 44 47 52. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 9 Existence and uniqueness of a solution to the initial value problem (3) 53 9.1 Weak and strong solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 9.2 Classical and regular solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 10 Comments. RR n° 4235. 56.

(8) 4. Olivier Faugeras , Gerardo Hermosillo. 11 Implementation Issues 57 11.1 Density estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 11.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 12 Summary and conclusion. 61. INRIA.

(9) Well-posedness of eight problems of multi-modal statistical image-matching. 5. 1 Introduction Image-matching techniques look for a deformation field that minimizes some error criterion between two images. In the variational framework, the deformation field is modeled as an element of some functional space , and the minimization is achieved by solving the Euler-Lagrange equations of the error criterion, which is a functional of the deformation field. In view of the difficulty of directly solving these equations, a gradient descent strategy is adopted, which in turn may be written as a parabolic system of functional equations. The solution is then taken as the asymptotic state (when goes to infinity) of this system. In [4], we derived a variational approach to the multimodal matching problem by considering that the images are realizations of spatial random processes and taking as error criterion either their mutual information [12] or their correlation ratio [11]. We considered two classes of linear regularization terms corresponding to a linearised elasticity model [5] and to a contour preserving, geometry driven diffusion term based on the Nagel-Enkelmann tensor [9, 1]. The purpose of this paper is twofold. First, we generalize this approach by considering that the mutual information, or the correlation ratio, are functions of the space variable, and take as error criterion the integral of these functions over the image domain. This approach is well suited for dealing with nonstationarities of the statistical relation between intensity pairs. Second, we prove existence and uniqueness of a classical solution of the parabolic system of equations which are obtained in the eight cases which encompass the four statistical criteria and the two classes of linear regularization terms.. 2 The framework. . .

(10) . . We given two and , which we model as functions

(11) images is the closed

(12) are , where and and note ! interval (we limit ourselves to the cases ) + are zero outside the Euclidean norm

(13) in&%&' ). We make the weak assumption that these*functions the ”square” "$# and square-integrable, i.e. they belong to ( . This assumption is completed by another one, i.e. that we observe regularised versions -, and ., of and by a gaussian with standard deviation equal to / . This is realistic in terms of modelling (it is the scale space idea []) and has several mathematical advantages. Let 0 be any open bounded set containing ) + that 0 be regular, i.e. that its boundary 120 be of class ): 3 , and " (in particular we may require ., are in the space 54 0 of the infinitely continuously differentiable real functions on 0 ; they are therefore, as all their derivatives, bounded on 0 ; they are also, as all their derivatives, Lipschitz continuous on 0 . 0 6! , such that the7&9 two )87:9<; problem The matching consists in finding a “regular” mapping + images ., and=. , minimize some criterion which is a functional of the mapping ( is the identity mapping of ).. 2.1 Functional spaces The mapping. RR n° 4235. belongs to the functional space >@?. *) + A ) ) * ) 0 , where ? 0 + ( 0 +B+C. ..

(14) 6. Olivier Faugeras , Gerardo Hermosillo. A) ( 0 + as a real Hilbert space equipped with the +

(15) ) + ) + )

(16) + . We will consider in most parts of this paper Hilbert product ) which induces the norm. ) 6 ? 0 +. . is also considered most of the time as a real Hilbert space equipped with the Hilbert. ) + ) + . product. . .

(17). which induces the product norm . . . .

(18). . . ) + ) + ! "#. . ) + . . ! "#. %$ . ) + " ) + ". . ) +. . " . &. criterion that we minimize with respect to is the sum of two terms, a data term, noted

(19) The , measuring

(20) , which the dissimilarity of the two images and a regularization term, noted ( ) regularization ) A) term is a function ) of the Jacobian ) * ensures that the mapping is regular. This ) +B+C belongs to a subspace of + 0 + 0 ++ of which we choose to be + 0 + () therefore 0 , and ) ) ++

(21) ( ) * ) ) 0 + and 0 + are, in most parts of this paper, considered to be real Hilbert spaces equipped with the Hilbert product ) + -,

(22) ) + ) +. ; ) + ) +. . 0/ / ) + ) + + 0 and + 0 are also considered to be real Hilbert spaces equipped with the Hilbert product ) + 1 ,

(23) ) + ) + ; ) +! ) + . * . * " ! "# " ) " ) ; ) + " ) .+ + + / . / . ) ) + ) +B+ ) +! ) + where * . This Hilbert product induces the product norm * * trace * ) + ; ) +! ) + 1 ,

(24) . 2* * " " " ) ; ) ) ! "3# + + / + & $. / . '. INRIA.

(25) Well-posedness of eight problems of multi-modal statistical image-matching. 7. !. ) + 0 into it will be convenient in Despite the fact that the mappings that we consider .are? from section 7.2 to consider also mappings from 0 into 0 is a complex Hilbert space equipped with the Hilbert product ) + ) + . . where. . ) +. . .

(26). ) + ! "# . . " . ) +" ) +. indicates the complex conjugate and which induces the product norm . . . Similarly, + . ) + 1. ,.

(27). . . . .

(28). ) + 0. . . . ) +. . . ! "#. . . . . $. ) +. . ". . &. ) + and + 0 are complex Hilbert spaces equipped with the Hilbert product ) + ) + ; ) +! ) +B+ ;. . 0* * # ) +

(29) ; ) + ) + . . 0*. (1). *. which induces the product norm . . 1. ,.

(30) . . ) + Re.

(31). ; . ) + Im. ; ! "# .

(32). . . . Re. ) " /. + .

(33). ;. . Re. ? 4. + Im / ) + 1 ,

(34) ;. ). . ". ) + 1 Im.

(35) . ,

(36). ) + technical ) ) reasons, ) + will also have to consider in section 9.2 the functional space For we + + 0 ( 4 0 . ( 4 0 is a real Banach space equipped with the norm. ) + Similarly ? 4 0.

(37) . .

(38). ) + .

(39).

(40). $#&%(' *),+,+,+ ) . We have the useful Lemma 1 We have the following continuous imbedding:. ) ) ? 4 0 +.- ? 0 + . RR n° 4235. 0. "!. is a real Banach space equipped"3# with the norm " product . or equivalently. a.e. in. -. .

(41) .

(42). .

(43) 8. Olivier Faugeras , Gerardo Hermosillo. Proof : If. ) ? 4 0 + we have,) according to the definitions

(44) %*&&2 a.e. + ". . . . Therefore .

(45). . . ! "3# $. . . ) + ". . .

(46). . 0 . &. . .

(47).

(48). !. 2.2 The minimization problem and the associated initial value problem In summary, we look for the minima of the following functional of the mapping :.

(49) ; .

(50) . '. ). . 2). where is a positive weighting parameter, i.e.. . . .

(51) . ,

(52). . ,

(53) . ) ++ . *.

(54) ; '. ) . 0). *. ) +B+ . (2) . The case corresponds to no regularization. The search for minima is done by computing the first

(55) variation (sometimes also called the Euler of the functional

(56) with Lagrange equations or the infinitesimal gradient) / respect to and equating it to 0 to find the extrema. The set of equations. . /. .

(57)

(58). . is known as the Euler-Lagrange equations of the error criterion . They have to be completed with the boundary conditions on 120.

(59). . Rather than solving them directly in the unknown , a task which is usually one

(60) impossible, )

(61) and a + differentiable) function, introduces time also noted from the interval into (we say .+ is an initial field and we solve the following initial value ) such that that problem: . . . . . . ) .+ ) +.

(62) . ) @ + /. . (3). i.e. we start from the initial field and follow the infinitesimal gradient of the functional (the sign is because we are minimising). The question that we answer in this article is that of the existence and uniqueness of a solution to (3). '

(63) In our case / is the sum of two

(64) terms, one corresponding to the dissimilarity functional

(65) and one to the regularization term ( . For this term, the computation of / ( is standard. . where ). . /.

(66) (. is the derivative of ) with respect to *. 9 ). +. . ). .. INRIA.

(67) Well-posedness of eight problems of multi-modal statistical image-matching. 9. 3 The regularization terms: convexity and coerciveness We introduce two regularization functionals and show that they are coercive and convex. We recall that, for a functional ) ).

(68) . (. ). +B+ . *. . coerciveness means. ) + exist constants There

(69) for all. such that. ). . *. ). ). ++ . . trace case we use . .. of . is the usual matrix norm defined on the set. matrices. In our. 3.1 The linearized elasticity operator Our first regularization operator arises from elasticity theory and is defined by the Saint-Venant/Kirchoff model [5] for which. . and. . ). ). *. . + . trace. ). *. + ;. ;. . *. . . trace. ). *. + *. ;. *. (4). are constants called the Lamé coefficients.. Proposition 1 The mapping. . . . trace ) ; + ; . ). . trace. . ) ;. + . is convex.. as a quadratic form of the components "! of , ) + # # #%$ ) ; + and notice that the matrix # has 3 (resp. 2) eigenvalues equal to & 6 (resp. 2) eigenvalues and $ (resp. ). The result follows from the and equal to in the case fact that Proof : We write ). . ! ". . ". ". !. ). ". Proposition 2 The functional (. RR n° 4235. .

(70) . ). . ) *. ) ++ . .

(71) 10. Olivier Faugeras , Gerardo Hermosillo. . . is coercitive, i.e.. such that:. . ). . ). . ) +B+. *. *. . Proof : Clear the the previous proposition if we choose from and . ). . equal to the smallest eigenvalue of. 3.2 The Nagel-Enkelmann tensor Our second regularization operator is defined by functions ) of the form. where , is a . . + % . ). ). *. ). trace. , *. !+ *. (5). symmetric matrix defined at every point of 0 by the following expression:. ) ;. . 7:9 ) / =%* + + ; / / /

(72) = . for. . This matrix was first proposed by Nagel and Enkelmann, [9] for optical flow computation and used more recently by Alvarez et al. [1]. We consider each of its scalar components since in this case this separation is possible. As pointed out in [1],

(73) , has strictly positive eigenvalues. Proposition 3 The mapping. . . . . ). . . is convex. Proof : Clear, since ,. . has strictly positive eigenvalues.. Proposition 4 The functional. is coercitive, i.e.. . . .

(74) ( . ). ) +. *. ). ). ) +B+ *. /. . , /. . . . *. % % %= is the smallest eigenvalue of

(75) .. Proof : We have Where. such that: ). . . . ,. /. 0. ,. INRIA.

(76) Well-posedness of eight problems of multi-modal statistical image-matching. 11. 4 The Dissimilarity terms We analyse two classes of statistical similarity criteria between two images that we call global and local. Both classes are based upon the use of some ) estimates of the joint probability of the grey levels + , is estimated by the Parzen window method in the two images. This joint probability, noted ) 7:9 upon ; + the mapping []. It depends since we estimate the joint probability distribution between the images ., and ., . To be compatible with the scale-space idea and for computational for the convenience, we

(77) choose a Gaussian window with variance Parzen window. We use and note: the notation . . . . % )+ ) + ) + . . . . . . . . %. . % ) +

(78) . ) + . . Notice that and all partial derivatives are bounded and Lipschitz. We will need in the sequel its and the infinite norms 4 . 4 For conciseness, we also use the following notation when making reference to a pair of grey-level 7 ) +

(79) ) + ) ; ) +B+8 intensities at a point :. . ,. ,. ) +. 4.1 Definition of the global dissimilarity criteria. ) +B+ are the values , We note , the random variable whose ) ; samples able whose samples are the values , . The joint probability density distribution (pdf) of

(80) , and

(81) -

(82)

(83)

(84) &%: : global) is defined by the function. % )+ 0. . )87 ) + +. . and. . ). the random vari-. ) (the upper index stands for (6). With the help of the (6) we define the mutual information between the two images

(85) ,estimate

(86) . In order and ., , noted MI

(87) and the correlation ratio, noted CR

(88) to do this we need to ) introduce more random variables besides

(89) , and

(90) . They are summarized in table 1. ) with respect to

(91) , , noted introduced in this table the conditional law of

(92) ) We have +

(93) : ) +. . ,. . ) + )

(94) '

(95) +

(96)

(97) )

(98) of the intensity in the second image , ) 7:9 ; + and the conditional expectation conditionally) to the+ intensity in the first image , . For conciseness, we note the value of this random , indicating that it depends on the intensity value and on the field . Similarly variable the conditional variance intensity the second image conditionally the intensity in the first

(99)

(100) of) the and itsinvalue ) + .toThe

(101) image is noted is abbreviated mean and variance . . . . RR n° 4235. ,. ,.

(102) 12. Olivier Faugeras , Gerardo Hermosillo. ). ) +

(103) ,

(104) . ) Value +.

(105) ,. . Random Variable.

(106) . . ).

(107)

(108) )

(109) , . '. )

(110) +. . . )

(111) + . ) + #. )

(112)

(113) )

(114) , A

(115). '. )

(116) +. PDF ) + ) + ) + . . '. )

(117) + '. )

(118) +. .. + # ) +. ) + . .. . . . Table 1: Random variables: global case. of the second image will also be used. Note that these are not random variables but that they are functions of : ) + # '

(119) ) + (7). . ) + #. . ) ) ) '

(120) + ++ . (8). ) + ) ) + ) + + '

(121) '

(122)

(123)

(124) )

(125) ) +

(126)

(127) % CR

(128) ! . The similarity measures can be written in terms of the quantities defined in table 1.

(129) MI

(130). . . . . ,. Regarding the correlation ratio, and according to table 1:.

(131)

(132)

(133) )

(134) , . ) ) + '

(135) +. . The mutual information and the correlation ratio are positive and should be maximized with respect to the field . Therefore we propose the following. INRIA.

(136) Well-posedness of eight problems of multi-modal statistical image-matching. 13. Definition 1 The two global dissimilarity measures based on the mutual information and the correlation ratio are as follows: '. MI '. CR.

(137) MI

(138)

(139)

(140)

(141)

(142) )

(143) ) +

(144) , . . CR

(145).

(146) .

(147) %.

(148) . ) ) +B+ shows that the mappings Note that this definition are not of the -MI = and . ThereforeCRthe Euler-Lagrange. ( form , for some smooth function ( equations will be slightly more complicated to compute than in this classical case. '. '. 4.2 Definition of the local dissimilarity criteria An interesting generalisation of the ideas developed in the previous section is to make the estimator (6) local. This allows us to take into account nonstationarities in the distributions intensities. centered atof the We weight our estimate (6) with a spatial Gaussian of variance . This means. ) and ) ) (the upper that we for each point in 0 we have two random variables, noted , index stands for local) whose joint pdf is defined by:. . . ) + . . % ) +. ). ) 7 ) + +. . . + (9) where % ) ) + 3. ) + + and ) + ) ) . + 0

(149) + (10) ) point of 0 the local mutual informaWith the help of the estimate (9) we define at every ) + the two images , and . , , noted MI

(150) + , and the local correlation ratio,) noted tion

(151) between CR . In order to do this we need to introduce more random variables besides , and ) ) . We summarise our notations and definitions in the table 2. . The similarity measures can be written in terms of the quantities in table 2 ) defined + ) )

(152) + + ) ) + MI + ' '

(153) ) + %

(154)

(155) ) ) ) + , ) ! CR . . . . . . . . . . As in the global case, the mean and variance of the second image are also used. Note that they are not random variables but that they are functions of and :. ) +#. RR n° 4235. . ) ' + . (11).

(156) 14. Olivier Faugeras , Gerardo Hermosillo. ). ) ) + ) . ,. ). . . ,. ) . . ) )

(157) ) ) ) .

(158) ) ) , ) . . ) + ' ) .+ ) * + ' ) .+ ) + ' . . ). . ) PDF A +. ) Value A+. Random Variable. + # ) +. + # ) +. ) +. . ,. . '. ) +. Table 2: Random variables: local case.. ) ) ) ' * + +B+ We define our similarity measures by aggregating the local ones: ) + ) + ) ) ) +

(159)

(160) MI MI . + ' A + '

(161)

(162)

(163)

(164) ) + % ) ) ) ). ) + #. . . . CR . . . . .. CR . . . . . . . $. . +. . ,. &. . (12). . The mutual information and the correlation ratio are positive and should be maximized with respect to the field . Therefore we propose the following Definition 2 The two local dissimilarity measures based on the mutual information and the correlation ratio are as follows: '.

(165)

(166) CR. '. MI. MI .

(167) . . .

(168)

(169) ) ) ) , ) + . . CR .

(170) 0. INRIA.

(171) Well-posedness of eight problems of multi-modal statistical image-matching. 15. .

(172) . ) ) +B+ shows that the mappings Note just as in the global case, this definition. that,

(173) are MI= and . '. not of the form , for some smooth function ( ( CR Therefore the Euler-Lagrange equations will be slightly more complicated to compute than in this classical case. '. 5 Existence of minimizers. . ) of ) minimizers In this section, we consider the existence is the sum error Our. criterion ++ wherefor) (2).. ) is ) smooth of a of a ”classical” regularization term ) * + + and. dissimilarity term which is only a function of but cannot be written as . Since ( the proof of theorem 5 in section 8.2 of chapter 8 of [7] assumes that this is the case, we have to adapt '

(174) it. Examining this proof we see we need to prove that the dissimilarity term is continuous in . This is proved in theorems 8 and 12 for the correlation ratio in the global and local cases, respectively. Therefore we have the. . . ) + satisfying Theorem 1 There exists at least one function + 0

(175) 1 ,

(176) ) '

(177) ; (

(178) 8+ .

(179) (definition 2) and (

(180) '

(181) '

(182) ' ' CR

(183)

(184) (definition for 1)) or ) ++ CR . (equation (4)) or ( ) * (equation (5)). . . . ). . ) *. ) ++. . Proof : This is a consequence of propositions 2 and 4, theorems 8 and 12 and theorem 5 in section 8.2 of chapter 8 of [7]. In order to prove the same theorem in the case of the mutual information, we need to prove continuity. We have the following. . %*&2 be a sequence of functions of such that almost 2 MI

(185)

(186) . are continuous, )87 ) + += )87 ) + + a.e.) in 0 ) . Since , and ) 7 : ) Because ) Proof . + + + the dominated convergence ) A+ theorem ) A+ implies that += + for all . A similar reasoning shows that '

(187) . Hence, the logarithm '

(188) for all being continuous ) + ) ) + ) ) + ) ) + ) + + + + '

(189) '

(190) '

(191) '

(192) ) + "

(193)

(194) "

(195). Proposition 5 Let @

(196) everywhere in 0 then MI

(197). . We next consider three cases to find an upper bound for . RR n° 4235. ,. . . :.

(198) 16. . Olivier Faugeras , Gerardo Hermosillo. This is the case where. ) ; ) +B+ , ) ) ) ; ) ++B+ + , ) ) ) A+ + ) + +) + '

(199) '

(200) )+ ) + ) A+ '

(201) '

(202) ) + ) ) +) ) + + + '

(203) '

(204) '

(205) . Hence This yields. and. . . ) A+ . . . ) ) + + ) + ) + . % . %. ) + ) ) + + . The function on the righthand side is continuous and integrable in We have , ) ; ) ++ % ) ) ) ; ) +B++ Hence ) + % # + , ) ) ) This yields ) + + ) + +) + ) + + '

(206) '

(207) )+ and ) .+ ) + ) + ) + '

(208) '

(209) and therefore ) + ) ) +) ) .+ ) + ) ) .+ + A+ '

(210) '

(211) '

(212) +

(213) . The function on the righthand side is continuous and integrable in and therefore. This is the case where. . . . ,. ) ;. . ) +B+. . %. INRIA.

(214) Well-posedness of eight problems of multi-modal statistical image-matching. ) + . 17. ) ) ; ) ++B+ ) + % , ) + )+ This yields ) + ) + ) + ) + ) A+ '

(215) '

(216) )+ and ) + ) + ) A+ ) + '

(217) '

(218) and therefore ) + ) ) +) ) + ) + ) ) + + A+ '

(219) '

(220) '

(221) A+ ; . The function on the righthand side is continuous and integrable in Hence. . ) + ) ) + ) . + + '

(222) '

(223) . The dominated convergence theorem implies that.

(224) MI

(225) . . .

(226) MI

(227). We also have the. . Proposition 6 Let @

(228) everywhere in 0 then MI . . %*&2 be a sequence of functions of MI

(229) .. Proof : The proof is similar to that of proposition 5. ) + ) ) + ) + + '

(230) '

(231) . . . . almost. . ) + 0 satisfying

(232) 1 ,

(233) ) '

(234) ; (

(235) 8+ .

(236) (definition 2) and (

(237) '

(238) '

(239) ' ' MI

(240)

(241) (definition for 1)) or ) ++ MI . (equation (4)) or ( ) * (equation (5)). Theorem 2 There exists at least one function. . such that. +. . ). . ) *. ) ++. Proof : This is a consequence of propositions 2 and 4, propositions 5 and 6 and theorem 5 in section 8.2 of chapter 8 of [7].. RR n° 4235.

(242) 18. Olivier Faugeras , Gerardo Hermosillo. 6 The Euler-Lagrange equations In this section we compute the Euler-Lagrange equations of the error criterion (2). As pointed out earlier, this is classical for the regularization terms and slightly more involved for the dissimilarity terms.. 6.1 The first variation of the regularization terms. 9 !). It is straightforward to verify that. ). ). . . +B+ *. ; ) ;. . . + ) /. /. +. We will find it useful to rewrite the righthand side in divergence form:. . . ;@) ;. + ) /. /. . + . div. ). ) ) +B+ 7:9 ; ) + + . trace. . ) + % ) ; + * * (13) # # )" + ) ) %

(243) #&2 1 # # &2 # 1 # # which is equal to

(244) div . +&&2 div + , and div ) + is the th row vector of the matrix 0 . We define the where by linear operator div ) trace ) ) +B+ 7:9 ; ) ++ (14) ) + A ) + is a subspace of ? 0 defined in section 7. The domain of the operator In the case of the Nagel-Enkelman tensor, it is also straightforward to verify that ) + div , / 9 !) ) ++ * ) ) ... + div , / ) *+ by and we define the linear operator ) + div , / (15) ) ... + div , / ) + is a subspace of ? A) 0 + defined in section 7. The domain of the operator where we have noted . . . . . . . .

(245). . . .

(246). . . 6.2 The first variation of the dissimilarity terms.

(247) . In [4], we have given the expressions of the infinitesimal gradients / of the global criteria. We reproduce this result here without proof, the proof being found in [8], since we will need it later.. INRIA.

(248) Well-posedness of eight problems of multi-modal statistical image-matching. 19. Theorem 3 In the global case, the infinitesimal gradient is given by. ) )

(249) + + / )+ where the function (

(250) .

(251) ) + ) (

(252) + ) , ) + , ) ; ) B+ ++ , ) ; ) +B+ '. /. is equal to. ( MI

(253). ). )+. % 1 ) + ' ) + 0 ) + ' ) + . (16). in the case of the mutual information and to. )

(254) ) + ( CR. ) ) ) 0 ) +

(255)

(256) +

(257)

(258) %A+ + . . in the case of the correlation ratio.. This defines two functions . : ) + ) MI

(259) ( MI

(260) ) ) )

(261) ) CR

(262) + ( CR. + ) , , 8) 7:9 ; + ) , , )8:7 9 ;. +B+ /5 , )87&9 ;. (17). +. +B+ /5 , )87&9 ; + ! ) ) :+ The only point that is not contained in [8] is the fact that MI

(263) + (respectively that Proof CR

(264) ). This is a consequence of theorems 7 and 10. ) ; is )convolved with the 2D gaussian A point to be noticed is that the function ( )

(265) ) + ++B+ . The direction of descent at the and the result evaluated at the intensity pair , . , ; ) is+ then obtained by multiplying this value by the gradient of the second image at the point point . and. The case of the local criteria is very similar. We have the following.

(266) ) + ) ) ( +B+ ) , ) + , ) A + where the function ( is equal to ) + % ) + 1 ( ) ) + ) + . Theorem 4 In the local case, the infinitesimal gradient is given by. ) ) + + . ) ;. /. . MI. ) +B+ + ) ; / ,. ) +B+. ) ) A ++ ' '. in the case of the mutual information and to. ) ( C R ) + ) + ) +. . RR n° 4235. .

(267) ) + ) + )

(268) ) + %A+ ) + . (18).

(269) 20. Olivier Faugeras , Gerardo Hermosillo. ). in the case of the correlation ratio.. This defines two functions . ) ) MI + ) ) CR + ) The fact that MI +. and. . . . +. is defined by (10).. :. ( ) + ) , , )87:9 ; MI. + 7:9 + / , ) 7:9 ; + ( ) + ) , , )87&9 ; + + /5 , )87&9 ; + ! CR ) (respectively that CR + ) is a consequence of theorems 11. Proof : and 16. We give the remaining of the proof only in the case of the mutual information, the proof in the case of the ) correlation obtained from this proof and [8]. We compute the Gâteaux

(270) of ' ratio

(271) incanthebedirection ' derivative of :. . . ; .

(272) '

(273) 1 1 ). '. #. . ) + ) ) +) + A . + . 1 ' '

(274) ) ) ; ) ) + + % 1 + + . '

(275) ' 1 . ) ) + ) + ) A + 1' . '

(276) 1 . . ;

(277) 1' 1 . We have. . 1 . . . . . . . . . . . . . .

(278). . . . . . . We first notice that. . . . 1'. ) + 1. '. %. ) A + . ) +

(279) ) . 1 . . ". . . .

(280). 1. The law. . '. ) + . ) + is given by equation (9): % ) + ) 7 ) + + ) + ) + . .

(281). . . !. INRIA.

(282) Well-posedness of eight problems of multi-modal statistical image-matching. ) + 1. Therefore. 1. %. ) + ) . . We now let. ,. )

(283) '. 21. . ) 3( + % ) + ) + 1 . . . . ) ) ) ; + 1 , + ,. . . ) + ; ) B+ + A+ ) ; ) + ; ) ++ / , . ) + . ) ) + ) ; ) B+ + + , , ) ; ) + + ) + / , . . This expression can be rewritten. )

(284) '. . ). % ) + + ) + . ;( ) ) ) ) +B+ + / , ) ; 1 , + ,. . . . . . ) +B+. ) +. . Two convolutions appear, one with respect to the space variable and the other one with respect to the intensity variable . This last convolution commutes with the partial derivative 1 with respect to the second intensity variable : '. )

(285) . Since. . . . ). % + ) +1 . ) ) + ) ; , , ) 1 ( ) + 1 ) . . . ( ) + ) +B+ + ) ; / , ) + + ' ) A ++ ' . ) +B+ . . ) +. * < is convolved with the 2D gaussian The point to be noticed is that the function ( ) ; ) for for the first two variables (intensities) and)B) the) nD+ gaussian (spatial), +B++ the+ remaining .variables and the result is evaluated at the point , of The direction of . , descent at the ; point) + is then obtained by multiplying this value by the gradient of the second image at the point . and defined in section 6.1 are In the remaining of this paper we prove that the operators self-adjoint, maximal monotone, that and are the infinitesimal generators of semigroups of we find the announced Euler-Lagrange equation.. RR n° 4235. .

(286) 22. Olivier Faugeras , Gerardo Hermosillo. contractions of , are also the infinitesimal generators of analytical semigroups of contractions of

(287) , CR

(288) , MI , and CR , defined in section 6.2 are Lipschitz continuous. and that the four functions MI We then use these results to prove the existence and uniqueness of various types of solutions for the initial value problem (3).. % and and defined by our two regularization terms In this section we prove that the operators and generate semigroups. We then are self-adjoint maximal monotone and therefore that. 7 Properties of the operators. show that they also generate analytical semi-groups of operators.. 7.1. . and. . . We recall that a linear operator from a linear subspace maximal monotone if and only if. ). • It is monotone: where. ) & +. ) +. are self-adjoint maximal monotone. . . +. indicates the Hilbert product in .. • It is maximal:. )87:9 ;. Ran. +. )87:9 ;. . of the Hilbert space . ) +. + @ . where Ran denotes the range of the linear operator ; operator is maximal if the equation. . has a solution. . . is said to be. 7:9 ;. . In other words, an. . .. 7.1.1 The linearized elasticity operator. 7:9 and apply the standard variational approach [7, 5]: Proposition 7 The operator defines a bilinear form on the space + which is continuous and coercive (elliptic). Proof : The proof can be found for example in [5]. We reproduce it here schematically. Let + , we consider the bilinear form defined as ) + We begin with the linear operator. , integrate by parts using the Green formula ; " * div " ". $. We use the definition (14) of . INRIA.

(289) Well-posedness of eight problems of multi-modal statistical image-matching. . . . 23. . true for each smooth field " of+ symmetric 0 ( is the) set matrices) and 0 enough . In this vector field formula, trace , is the inside pointing normal " * " * vector to 120 and is the area element of 120 . Using the fact that + , we find. $). + . . where. ) + ) + . ). . ) ) +B+. ) ) B+ + ; . . trace. . ) + . and hence, using Cauchy-Schwarz again, proves continuity. Next we note that. It is ) proved in [5], theorem 6.3-4, that if ). . . 1. ,

(290). and the coerciveness is proved.. 1. ,

(291). ) + . ) + . . . . 1. . . . . 1. . . ,

(292). . . 1. . . +. . . ) +. . . ,

(293). .. , which. . . 1. ,

(294). . such that. . We therefore have the. A) + 8 Proposition 0 . into ?. ) +. . . ) + ) + ) + 0 there exists a constant . . . . . , hence. ) +. ,

(295). . . . By applying several times Cauchy-Schwarz, we find that . +. . ) ) + ) + + ) + ) + ; trace . Hence . . + . ) + ) B+ + . . trace. is a maximal monotone self-adjoint operator from. . ) + +. . . ) + 0. +. *) + 0. . ) follows ) +! Proof : Monotonicity the of proved in) the More + from

(296) + , the proof + proposition..

(297) ) coerciveness previous. ) precisely, since shows that + . 7&9 Regarding ) + that the bilinear form associated to the operator maximality, proposition 7 shows ) apply the Lax-Milgram theorem is continuous and coercive in + 0 . We can therefore in + 0 + to the equation ) + ) + ) + and state the existence and uniqueness of a weak solution for all <? 0 . Since 0 is regular (in particular ), the solution is is in + 0 + 0 and is ) theorem ) + A) + )87:9 a strong solution (see [5], + + 6.3-6). + . In order to prove that Therefore we have 0 + 0 and Ran ) since ) monotone, the operator is self-adjoint, it is sufficient, to prove that is symmetric + it

(298) is maximal ([3], proposition VII.6), i.e. that +

(299) and this is obvious from the proof of proposition 7.. . . . . . RR n° 4235. . . .

(300) 24. Olivier Faugeras , Gerardo Hermosillo. 7.1.2 The Nagel-Enkelmann operator We now treat the case of. :. .

(301). div. 7&9. We prove the analog of proposition 7.. . Proposition 9 The operator continuous and coercive (elliptic).. ). div. ). ,. + /. .. .. . +. . , /. . . defines a bilinear form. on the space +. ) + 0. which is. . Proof : Because of the form , it is sufficient to work on one of the coordinates of) the*+ operator. ( ) 0 + defined and consider the operator by. $. $. $. . . %. div. ). , . +. % /. . and to show that the operator % % % defines a bilinear form which is continuous and coercive. Indeed, we define. )% + . $. . % . . div. ). , . % /. We integrate by parts the divergence term, use the fact that . )% + . . Because the coefficients of . ,. . %. ;. . ) % + . . /. % . , /. from which it follows that for some positive constant . ) + ) + 0 0. . ) + 0 , and obtain. . . are all bounded, we obtain, by applying Cauchy-Schwartz:. . . %. ) % % + ) + . . . % %. % . %.

(302) . -,

(303). are strictly positive, we have . , . . . . -,

(304). %. where is a positive constant, hence continuity. Because the eigenvalues of the symmetric matrix where is a positive constant. This implies that . , / % /. +. on the space . . . ,

(305). . . /. . and hence we have coerciveness.. %. .

(306). ,. . . 7:9. ,. . . ) + apply the Lax-Milgram theorem and state the*) existence We can therefore and uniqueness of a weak A) 0+ is regular (in particular solution in + 0 to the equation ) + for all <? ) 0 + + . Since ), the coefficients of , in 0 , the solution is in + 0 + 0 and is a strong solution (see e.g. [7]). We therefore have the. . . . INRIA.

(307) Well-posedness of eight problems of multi-modal statistical image-matching. . A) + 10 Proposition into ? 0 .. is a maximal monotone self-adjoint operator from. . ) * + . 25. +. ) + 0. +. *) + 0. . . Proof : Monotonicity follows from the coerciveness of proved in the previous proposition. ) ) ) 8 ) : 7 9 Maximality also follows from the proof of proposition 9. According same proposition, we A+ + 0 + + 0 + and Ran + (application oftothetheLax-Milgram have theorem).. is self-adjoint for the same reasons as those indicated in the proof of proposition Finally, 8. 7.1.3. . and . . . are the infinitesimal generators of . semigroups. We recall the definition of a ) semigroup of ; bounded linear operators on a Banach space . Con+ of bounded linear operators from to . This sider a one-parameter family " , family is said to be a semigroup of bounded linear operators if. ) . + " ) , is the identity operator on . ) ; A+ ) +@ 2. " ) " " * + for every . + for every . 3. " The Hille-Yosida) theorem says one-to-one correspondence between semigroup of +

(308) % that there is ) aand contractions ( " maximal monotone operators in a Hilbert space. A for all ) + space maximal monotone operator in a Hilbert . is said to be the infinitesimal generator of the corresponding semigroup noted " ) + The relation between and " is the following. Consider the initial value problem ) +

(309) (19) ) .+ ) + ) + Definition 3. )

(310) ; Because. 1.. . . ) ' ;monotone ) ; for ) +B+ there exists ) + a unique selfadjoint all function

(311) is+ maximal

(312) +

(313) ) + such) that ) . The linear ) + ) + ) + . + , where + is the solution ) is defined by of (19) application " , ) + " . ) + at time . Since " and density to a . it is possible to extend " + by continuity linear continuous operator . This family of operators, also noted " , is the semigroup of contractions corresponding to . . . Thus we have the. Theorem 5 The operators bounded linear operators on . . and for all . . . are the infinitesimal generators of two .. semigroups of. Proof : This is a consequence of the Hille-Yosida theorem and of propositions 8 and 10. A property of the ing. RR n° 4235. . semigroups of bounded operators that we will need later is given in the follow-.

(314) 26. Olivier Faugeras , Gerardo Hermosillo. ) + ) + , " ) + " . Proposition 11 For all. ) + and ) + ) " " + . Proof : The proof can be found for example in theorem 2.4 of chapter 1 of [10].. ) +. We will also need the following two lemmas. Lemma 2 The linear operator ". . is bounded by 1, for. ) theorem: because Proof : It is a consequence of the Hille-Yosida ) +

(315) 8% and 10) we have " + for all operator (propositions . Hence " . . . . . . . . . .. . or. for all . is a maximal-monotone ([3], theorem VII.7).. We will also need the following. or for all . ) has a unique solution for all ? 0 + . Proof : It is sufficient to show that the equation The proofs of propositions 7 and 9 show that ) the+ bilinear forms associated to the operators are and continuous and coercive in + 0 , hence ) the Lax-Milgram theorem tells us that the *) + equation has a unique weak solution in + ) 0 + + for *) + and for all ? 0 . Since 0 is regular the weak solution is in + 0 + 0 and is a strong solution. Lemma 3 The linear operator. . is invertible, for. . . 7.2. . . . . and. . are the infinitesimal generators of analytical semigroups. . . It turns out that, because of the special form of the operators and , the corresponding semigroups can be extended to analytical semigroups. This is necessary in order to obtain the existence, uniqueness and regularity results for the solution of (31). We recall the definition of an analytical semigroup of bounded linear operators. For more details, the interested reader is referred to [10].. ) " $ + . . Definition 4 Let ) bounded linear operator. The family . 1. . ) + " " ) ; " . ) + . . ). * ) . ). . and for . is an analytic semigroup in. . if. , let. ) + . ". be a. . ) + ) + ) + for every . + 3. . ) + " " for A semigroup " will be called analytic if it is analytic in some sector 2.. . . is analytic in . ) and ". . . . . containing the nonnegative ) + We real axis. Clearly, the restriction of an analytic semigroup to the real axis is a ) + semigroup. are interested in the possibility of extending a given semigroup (i.e. "

(316) , and "

(317) ) to an analytic semigroup in some sector around the nonnegative real axis. The reason for this interest ) that+ if is the infinitesimal generator of an analytic semigroup, one can define fractional powers is of which are useful in the study of our semilinear initial value problem (31).. . . INRIA.

(318) Well-posedness of eight problems of multi-modal statistical image-matching. . Theorem 6 The operators groups of operators on .. and. . 27. are the infinitesimal generators of two analytical semi-. ) +is simply an adaptation of the proof of theorem 7.2.7 in [10], Proof : To carry out the proof which ; the Hilbert ) ? we extend product in valued functions as shown in equation (1). Let + . A simple Re + and 0 Imto) complex , computation shows that. . ). . . +. since the operators and propositions 7 and 9 we deduce. ). . +. ). .

(319). ; ). + . .

(320). . +. . ) . +.

(321). ;@). +. . .

(322). are symmetric (propositions 8 and 10). From the proofs of. ) .

(323). . 1. ,

(324). ;. 1 . ,

(325). + . . . 1. ,.

(326). . . 1 ,

(327) + Im ) + ) + )

(328) + 1 ,

(329) % is included It follows that the numerical range " in , the set " , where . . . *. . Choosing such that arg there exists a constant such that * and denoting$

(330) ) ) +B + (20) " for all

(331) ) + " denotes the distance where and the set " .) are in; the + of . This is because the between Next we note that all reals resolvent set resolvent of is the set of s such that is invertible, i.e. the set ;of s such that ; is invertible. But since the complex equation (@ and ) is equivalent to the two real equations and , we can apply ) + to show that each ) + 9 and ) the+ Lax-Milgram the results in the proofs of propositions 7 and theorem ) ) ) that equation has a unique strong solution in + 0 for each therefore + 0 ? has a unique strong solution in + 0 + + 0 + 0 forand 0 +. the equation each ? are in the resolvent set of , this shows that

(332) , which ) contains ) the negative Since ) + of the complement of the closure " + ) of " + + that has a real axis, is contained in a component

(333) and that nonempty intersection with . This implies (theorem 1.3.9 of [10]) that for every in

(334) % ) ) ) + " +B+ ) ; + ) + is the ) operator where . + ) + and defining ) fact ) + , we

(335)

(336) ) + arg Using the that +.

(337) . Moreover, for all

(338) we have infer that and, according to (20), % ) ) ) + " +B+ !. Since Im. ). , we certainly have also. . RR n° 4235. . . . . . . .

(339) 28. Olivier Faugeras , Gerardo Hermosillo. . We can therefore apply theorem 2.5.2 in [10] which allows us to conclude that the generated by can be extended to an analytical; semigroup in a sector where * is defined by * .. . . 8 Properties of the functions. . semigroup arg . . and. In this section we prove that the functions defined by our various criteria (theorems 3 and 4) satisfy some Lipschitz continuity conditions.. 8.1 Preliminary results We remind the reader of the following results on Lipschitz continuous functions. These results will " be used several times in the following. . . Proposition 12 Let be a Banach space and let us denote its norm by be two Lipschitz continuous functions. We have the following:. . . . Let. ; is Lipschitz continuous. If and are bounded then the product. is Lipschitz continuous. and if and are bounded, then the ratio , is Lipschitz continuous. If . % . 1. 2.. 3.. Proof : We prove only 2 and 3. Let. and. . be two vectors of. ) ) ) ) + + +) ) + + * ) +B+ ) + ; ) . ) + . . from which point 2 above follows. Similarly. ) +. ) +. ) + ) + ) + ) + ) +. ) . ) + ) + . If. !. . . , there exists $ ) +. ) + . ) + ) + . . such that. %. . $ . :. ) ) ) ) + + ) ; + ) + + + & ) + ) + . ) + + ) + ) + ; ) + ) + ) + ) + ) +. $ . Hence. ) + . . ) + ) + ;. ) ) + + . ) +. from which point 3 above follows.. In the following, we will need the following definitions and notations. INRIA.

(340) Well-posedness of eight problems of multi-modal statistical image-matching. ;

(341) and)

(342) < ; spaces equipped with * + ; the Banach - and , respectively. . Definition 5) We + note, the norms . 29. . . . . . . . . . ) +. We will use several times the following result. 6. . ) A +. be such that . ) A is+ Lipschitz continuous with a Lemma 4 Let. ) such Lipschitz constant independent of and that is Lipschitz continuous with A+ a Lipschitz constant ( independent of , then is Lipschitz continuous.. . Proof : We have. ) ) A + ) . .

(343) . + ) + + ; ) + ) ) ; +; ) ( ) ; ( + . . ) +. + ; . In section 8.3, we will need a slightly more general version of this lemma.. %. . . . . + . ) +. 0 ) is Lipschitz Lemma 5 Let. be such that ) that + continuous with a Lipschitz constant independent of and and such + is Lipschitz continuous a Lipschitz constant ( independent of , then is Lipschitz

(344) with uniformly on 0 . continuous on. . . Proof : Indeed,. ) ) A + ) * + ). . and the Lipschitz constant. . + ) + ; ) . ; + ; ) ) ( ; ( + ) + ( is independent of 0. . . + .. . ) + ; + <0 . 8.2 Global criteria We discuss the case of the two global criteria. 8.2.1 Mutual Information We first prove that in the mutual information case, there is a neat separation in the definition of the function between its local and global dependency in the field . More precisely we have the following. RR n° 4235.

(345) 30. Olivier Faugeras , Gerardo Hermosillo. Proposition 13 The function. -

(346) defined ) by A+ ) + '' ) +. ) ) + $ A + . satisfies the following equation:. . where the function Proof :. '. . $. ) A + . is defined by. hence. '. The function. $. ) + . ) A + . . . ) + ' %. ) ) ; ,. ) ) ; ,. . 0. . is equal to. ) B+ + A+ . ) B+ + + ) ) ; ,. . ) + + + . ) + + ) ) ; ) +B+ + , ) , ) ; ) +B+ + $. ) ; ) +B+

(347) . and the result follows from the fact that , A simple variation of the previous proof shows the truth of the following -

(348) defined by Proposition 14 The function ) ) + 1 ) + + ) * + % , . )+ ). satisfies the following equation:. . ) +. . . . ) ;. .. In the following, we will use the function (. . !. . (21). + . )

(349) MI ) + -

(350) % ) )+ % ) ) +B+ )

(351) ( MI ) + 0 0

(352) of convolving ( MI

(353) ) with We then consider the result MI. defined as ) + ) +) + ) $ where the function. . . . defined as (see theorem 3). ) + ) +B+ $ . , i.e. the two functions. ) +$ + . . (22).

(354) . . (23). INRIA.

(355) Well-posedness of eight problems of multi-modal statistical image-matching.

(356) defined as ) ) ) + + + . and. ) + ) +. . . .

(357) . We prove a series of propositions.. 31. . ) A. . (24). +. defined by is Lipschitz continuous with Proposition 15 The function a Lipschitz constant

(358) which is independent of . Moreover, it is bounded by .. . . ) + . . . . . : The second part of the proposition follows from the fact that Proof (proposition 13). and In order to prove the first part, we prove that the magnitude of the derivative of the function is bounded independently of . Indeed. % ) + . . . 4 ) + ) + ) + $ 4 . . . 4 ) + . . 4. . The function on the righthand side of the inequality is independent of therefore upperbounded.. ) A. + * ) 0 +

(359) .. Proposition 16 The function ? Lipschitz constant (

(360) which is independent of Proof : We consider. ) A. + ) A A + ) + ) ) + ) +$ According to equation (21), ratio $ is the ) ) ; ) ++ ) , ) A + ,. . . and. We ignore the factor. %. $. *. ) + . ) ) ; ,. and continuous on. is Lipschitz continuous on ?. *) + 0. ) A + ) A ++ ) + $ * of the two functions ; ) +B+ A+ . ) + + A+ .

(361) , with. (25). !. which is irrelevant in the proof. We write. ) ) + A + . . RR n° 4235. . . ) ) ) ) + A * + ) & * *+ ) + * + * + ; * * ) A + * ) * + ) ) * ) + ) * + * * +* +. (26).