Combinatorial structure of rigid transformations in 2D digital images

Ngo, Phuc; Kenmochi, Yukiko; Passat, Nicolas; Talbot, Hugues

doi:10.1016/j.cviu.2012.08.014

Cited by 18 publications

(91 citation statements)

References 27 publications

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“…The DRT graph has already allowed us to contribute to the state of the art on rigid transformations from a combinatorial point of view, by establishing the complexity of "free" [12] and "constrained" [16] discrete rigid transformations. From an algorithmic point of view, it has been used to characterise topological defects in transformed images [17].…”

Section: Discrete Rotations and Discrete Rigid Transformationsmentioning

confidence: 99%

“…In other words, by considering a brute-force search, a solution, i.e., a global optimum, can be determined. However, the DRT graph G of an image of size N × N, has a high space complexity O(N 9 ) [12] that induces the same time complexity both for its construction and exhaustive search.…”

Section: Discrete Rigid Transformation Graph and Image Registrationmentioning

confidence: 99%

“…3.1). Then, we develop a construction strategy that relies on a sweeping plane technique introduced in [12] (Sec. 3.2).…”

Section: K-neighbourhood Construction Algorithmmentioning

confidence: 99%

“…This paradigm relies on a combinatorial structure, called discrete rigid transformation graph (DRT graph, for short) [12]. This structure describes the quantification of the parameter space of rigid transformations, in the framework of hinge angles, pioneered in [13][14][15].…”

Section: Discrete Rotations and Discrete Rigid Transformationsmentioning

confidence: 99%

See 3 more Smart Citations

Efficient Neighbourhood Computing for Discrete Rigid Transformation Graph Search

Kenmochi¹,

Ngo

Talbot³

et al. 2014

Advanced Information Systems Engineering

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International audienceRigid transformations are involved in a wide variety of image processing applications, including image registration. In this context, we recently proposed to deal with the associated optimization problem from a purely discrete point of view, using the notion of discrete rigid transformation (DRT) graph. In particular, a local search scheme within the DRT graph to compute a locally optimal solution without any numerical approximation was formerly proposed. In this article, we extend this study, with the purpose to reduce the algorithmic complexity of the proposed optimization scheme. To this end, we propose a novel algorithmic framework for just-in-time computation of sub-graphs of interest within the DRT graph. Experimental results illustrate the potential usefulness of our approach for image registration

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Section: Discrete Rotations and Discrete Rigid Transformationsmentioning

confidence: 99%

Section: Discrete Rigid Transformation Graph and Image Registrationmentioning

confidence: 99%

“…3.1). Then, we develop a construction strategy that relies on a sweeping plane technique introduced in [12] (Sec. 3.2).…”

Section: K-neighbourhood Construction Algorithmmentioning

confidence: 99%

Section: Discrete Rotations and Discrete Rigid Transformationsmentioning

confidence: 99%

See 2 more Smart Citations

Efficient Neighbourhood Computing for Discrete Rigid Transformation Graph Search

Kenmochi¹,

Ngo

Talbot³

et al. 2014

Advanced Information Systems Engineering

Self Cite

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“…They are much used in remote sensing [166], medical imaging [141,155], image registration [10], and image warping [57]. This is not anymore the case in the discrete world [134,135]: starting from a binary image defined on a square grid, it is often mandatory to discretize the result of a continuous rigid transformation of this image since its domain must belong to Z 2 (see Figure 48). …”

Section: Rigid Transformations and Preservation Of Well-composednessmentioning

confidence: 99%

A Tutorial on Well-Composedness

2017

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Due to digitization, usual discrete signals generally present topological paradoxes, such as the connectivity paradoxes of Rosenfeld. To get rid of those paradoxes, and to restore some topological properties to the objects contained in the image, like manifoldness, Latecki proposed a new class of images, called wellcomposed images, with no topological issues. Furthermore, well-composed images have some other interesting properties: for example, the Euler number is locally computable, boundaries of objects separate background from foreground, the tree of shapes is well-defined, and so on. Last, but not the least, some recent works in mathematical morphology have shown that very nice practical results can be obtained thanks to well-composed images. Believing in its prime importance in digital topology, we then propose this state-of-the-art of well-composedness, summarizing its different flavours, the different methods existing to produce well-composed signals, and the various topics that are related to well-composedness.

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Homotopic Digital Rigid Motion: An Optimization Approach on Cellular Complexes

Passat

Ngo

Kenmochi

2021

Lecture Notes in Computer Science

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Topology preservation is a property of rigid motions in R 2 , but not in Z 2 . In this article, given a binary object X ⊂ Z 2 and a rational rigid motion R, we propose a method for building a binary object X R ⊂ Z 2 resulting from the application of R on a binary object X. Our purpose is to preserve the homotopy-type between X and X R . To this end, we formulate the construction of X R from X as an optimization problem in the space of cellular complexes with the notion of collapse on complexes. More precisely, we define a cellular space H by superimposition of two cubical spaces F and G corresponding to the canonical Cartesian grid of Z 2 where X is defined, and the Cartesian grid induced by the rigid motion R, respectively. The object X R is then computed by building a homotopic transformation within the space H, starting from the cubical complex in G resulting from the rigid motion of X with respect to R and ending at a complex fitting X R in F that can be embedded back into Z 2 .

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Combinatorial structure of rigid transformations in 2D digital images

Cited by 18 publications

References 27 publications

Efficient Neighbourhood Computing for Discrete Rigid Transformation Graph Search

Efficient Neighbourhood Computing for Discrete Rigid Transformation Graph Search

A Tutorial on Well-Composedness

Homotopic Digital Rigid Motion: An Optimization Approach on Cellular Complexes

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