Sufficient Conditions for Topological Invariance of 2D Images under Rigid Transformations

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

doi:10.1007/978-3-642-37067-0_14

Cited by 5 publications

(7 citation statements)

References 14 publications

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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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Section: Rigid Transformations and Preservation Of Well-composednessmentioning

confidence: 99%

A Tutorial on Well-Composedness

2017

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“…In addition, the DRT graph explicitly models the "topological links" between such digital transformations, and thus allows the incremental construction of discrete rigid transformations via elementary image modifications pixel by pixel. The DRT graph can be used in a local fashion, e.g., in pattern-based strategies, as proposed in [19] for analysing the topological invariance of digital images under arbitrary rigid transformations. Beyond the theoretical aspects of the DRT graph, its high-order polynomial complexity makes it difficult to generate the whole graph for large images, and to use it directly in imaging applications such as registration or warping [1,9,22,29].…”

Section: Introductionmentioning

confidence: 99%

On 2D constrained discrete rigid transformations

Ngo

Kenmochi

Passat

et al. 2014

Ann Math Artif Intell

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Rigid transformations are involved in a wide range of digital image processing applications. In such a context, they are generally considered as continuous processes, followed by a digitization of the results. Recently, rigid transformations on Z 2 have been alternatively formulated as a fully discrete process. Following this paradigm, we investigate – from a combinatorial point of view – the effects of pixel-invariance constraints on such transformations. In particular we describe the impact of these constraints on both the combi-natorial structure of the transformation space and the algorithm leading to its generation

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“…In the case of discrete rigid transformations [6] (i.e. translations and rotations, followed by a digitisation), some topological issues were identified in [7]. These are directly induced by the sampling, which is mandatory to guarantee the stability of the transformations within Z n .…”

Section: Introductionmentioning

confidence: 99%