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Topology-Preserving Rigid Transformation of 2D Digital Images Phuc Ngo, Nicolas Passat, Yukiko Kenmochi, Hugues Talbot

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Topology-Preserving Rigid Transformation of 2D Digital Images Phuc Ngo, Nicolas Passat, Yukiko Kenmochi, Hugues Talbot Topology-preserving rigid transformation of 2D digital images Phuc Ngo, Nicolas Passat, Yukiko Kenmochi, Hugues Talbot To cite this version: Phuc Ngo, Nicolas Passat, Yukiko Kenmochi, Hugues Talbot. Topology-preserving rigid transfor- mation of 2D digital images. IEEE Transactions on Image Processing, Institute of Electrical and Electronics Engineers, 2014, 23 (2), pp.885 - 897. 10.1109/TIP.2013.2295751. hal-00795054v1 HAL Id: hal-00795054 https://hal-upec-upem.archives-ouvertes.fr/hal-00795054v1 Submitted on 3 Jul 2013 (v1), last revised 22 Apr 2014 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. JOURNAL OF LATEX CLASS FILES, VOL. X, NO. X, JANUARY 20XX 1 Topology-preserving rigid transformation of 2D digital images Phuc Ngo, Nicolas Passat, Yukiko Kenmochi, Hugues Talbot Abstract—We provide conditions under which 2D digital im- We then propose, in Sec. IV, some conditions under which ages, considered in the two most common digital topology models a 2D digital image preserves its topological properties under (namely, dual adjacency and well-composedness), preserve their any rigid transformation. Based on these results, we provide topological properties under rigid transformation. This study, that is developed in a discrete framework, leads to the proposal in Sec. V some methodological solutions for analysing and of efficient preprocessing strategies that ensure the topological preprocessing digital images before rigid transformation, in invariance of images under further rigid transformation. These order to preserve their topological properties. This study is results and methods are proved to be valid for various kinds of generic on two sides: (i) the main two digital topology models images (binary, grey-level, label), thus providing a generic set are considered, namely the dual adjacency, and the well- of tools, that can be used in particular in the context of image registration and warping. composedness ones; and (ii) the cases of binary, grey-level and label images are dealt with. Sec. VI concludes the article Index Terms—Digital imaging, rigid transformation, digital by perspective works. For the sake of readability, technical topology, well-composed images, image correction. proofs are reported in Appendix. II. BACKGROUND NOTIONS I. INTRODUCTION A. Notations N digital imaging, the preservation of topological properties A B C is a crucial issue in several application fields, involving 3D The sets are noted , , , etc. Subsets of these sets are I A data (e.g., medical imaging [1]) but also 2D ones (e.g., remote noted A, B, Γ, etc. The power set of a set A is noted 2 . The sensing [2]). In particular, topology preservation –pioneered elements of sets are noted a, b, c, etc., and a, b, c, etc. if the nearly fifty years ago [3], [4]– has been investigated in the set is a cartesian product. By abuse of notation, an element a, that should be noted as a column vector, is noted as a line context of image transformation, both from the viewpoints of a registration [5] and warping [6]. It has to be noticed that efforts vector, e.g., a = (a, b) instead of a = b . have been mainly devoted to handle complex transformations, The functions defined on continuous sets are noted A, B, C, while more simple ones have been globally unconsidered. etc., and the ones defined on discrete sets are noted A, B, C, A B A B A Indeed, the handling of “simple” transformations (e.g., etc. A function F from to is noted F : → . If A ⊆ B −1 translations, rotations) is often assumed to be trivial. This can and B ⊆ , we note F (A) = {F (x) | x ∈ A} and F (B) = be explained by the fact that, in the continuous case (i.e., in {x | F (x) ∈ B}. If F is a bijection, its inverse function is −1 B A A B Rn), most of such transformations are topology-preserving, also noted F : → . The restriction of F : → to A B while this is not necessarily the case for complex ones (e.g., the subset A ⊆ is noted F|A : A → . The composition of A B B C A C those induced by nonrigid registration [7]). Based on this F : → and G : → is noted G ◦ F : → . The “continuous” assertion, it is often thought that simple trans- spaces of functions are noted A, B, C, etc. formations still lead to easy handling of topological properties Adjacency (i.e., binary, irreflexive and symmetric) relations in the digital case (i.e., in Zn). This is a wrong belief. are noted . Equivalence (i.e., binary, reflexive, transitive and In the case of rigid transformations [8], that include the symmetric) relations are noted ∼. We recall that a relation family of rotations, (e.g., (quasi-)shear rotations [9], [10], or (resp. ∼) defined on a set A is actually a subset of A×A, and hinge angle rotations [11], [12], [13]), some topological issues that a b (resp. a ∼ b) means that (a, b) ∈ (resp. ∼). have been identified [14], [15]. These issues are directly or Given a set A, equipped with an equivalence relation ∼, the indirectly induced by the sampling policies that are mandatory equivalence class of a ∈ A with respect to ∼ is noted [a]∼, to guarantee the stability of the transformations inside Zn. and the quotient set of A with respect to ∼ is noted A/∼. In this article, we propose a study devoted to the topological B. Rigid transformations invariance of 2D digital images under rigid transformation. In R2 Sec. II, we first provide background notions required to make 1) Continuous case: In , a rigid transformation is a function the article self-contained. Secs. III–V constitute the core of U : R2 → R2 (1) the article. The main purposes are first detailed in Sec. III. x → R.x + t 2 Phuc Ngo, Yukiko Kenmochi and Hugues Talbot are with the ESIEE- where R is a rotation matrix, and t ∈ R . The function U is Paris and the Universit´e Paris-Est, LIGM UMR CNRS 8049, Paris, France a bijection, and we note T = U −1 its inverse function, which ({p.ngo,y.kenmochi,h.talbot}@esiee.fr). Nicolas Passat is with the Universit´e de Reims Champagne-Ardenne, is also a rigid transformation. We note RigR2 the set of the CReSTIC EA 3804, Reims, France ([email protected]). rigid transformations. JOURNAL OF LATEX CLASS FILES, VOL. X, NO. X, JANUARY 20XX 2 2) Discrete case: These definitions cannot be directly ap- D. Digital topology Z2 plied in the discrete case, i.e., when considering instead 1) Basic notions: Digital topology [16] provides a simple R2 Z2 Z2 of . Indeed, there is no guarantee that U( ) ⊆ . The framework for handling the topology of binary images in Zn. handling of discrete rigid transformations then requires to Beyond its simplicity, it is also a robust framework that has R2 Z2 consider a discretisation operator D : → . In the most been proved to be compliant [17] with other discrete models common cases –and in the present one– D is the standard (e.g., Khalimsky grids [18] and cubical complexes [19]) but rounding function. We can then define the discrete analogues also with continuous notions of topology [20]. Z2 Z2 Z2 Z2 U : → and T : → , of U and T , as Practically, digital topology on Zn mainly relies on two adjacency relations, noted n and n , defined, for any U = D ◦ U Z2 (2) 2 3 −1 | p q Zn −1 , ∈ , by T = D ◦ T|Z2 = D ◦ (U )|Z2 (3) p 2n q ⇐⇒ p − q1 = 1 (4) We note RigZ2 the set of the discrete rigid transformations. p n q ⇐⇒ p − q = 1 (5) 3) Transformation models: Two transformation models can 3 −1 ∞ be considered for discrete (rigid) transformations: the Eulerian In the case of Z2, we retrieve in particular the well-known 4- (or backwards) model, and the Lagragian (or forwards) one. and 8-adjacency relations, namely 4 and 8. The Lagrangian model consists of computing U(Z2), i.e., Let Ω ⊆ Z2, and p, q ∈ Ω. We say that p, q are 4- (resp. 8-) Z2 it determines the image of the “initial” space associated adjacent (in Ω) if p 4 q (resp. p 8 q). From the reflexive- to the rigid transformation. From an imaging viewpoint, this transitive closure of 4 (resp. 8) on Ω, we derive the 4- model is not satisfactory, since U is, in most cases, neither (resp. 8-) connectedness relation ∼4 (resp. ∼8) (on Ω); we injective nor surjective. In other words, if U is applied on a say that p, q are 4- (resp. 8-) connected (in Ω) if p ∼4 q digital image (see Sec. II-C), it may lead to a transformed (resp. p ∼8 q). It is plain that ∼4 (resp. ∼8) is an equivalence image that will present both undefined and conflicted values. relation on Ω; the equivalence classes of Ω with respect to ∼4 By opposition, the Eulerian model consists of computing (resp. ∼8), namely the elements of Ω/∼4 (resp. Ω/∼8) are T (Z2), i.e., it determines the preimage of the “transformed” called the 4- (resp. 8-) connected components of Ω. space Z2 associated to the rigid transformation. From an 2) Dual adjacency and well-composedness models: A finite imaging viewpoint, this model is more satisfactory, since T is set Ω ⊂ Z2 can be modeled as a binary image I ∈ ImB, defined on the whole transformed space Z2, thus guaranteeing defined by I−1({1}) = Ω and I−1({0}) = Ω = Z2\Ω, or vice that any point of a transformed digital image will be unam- versa.
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