Phuc Ngo

Combinatorial structure of rigid transformations in 2D digital images

Rigid transformations are involved in a wide range of digital image processing applications. When applied on discrete images, rigid transformations are usually performed in their associated continuous space, requiring a subsequent digitization of the result. In this article, we propose to study rigid transformations of digital images as fully discrete processes. In particular, we investigate a combinatorial structure modelling the whole space of digital rigid transformations on arbitrary subset of Z^2 of size NxN. We describe this combinatorial structure, which presents a space complexity O(N^9) and we propose an algorithm enabling to construct it in linear time with respect to its space complexity. This algorithm, which handles real (i.e., non-rational) values related to the continuous transformations associated to the discrete ones, is however defined in a fully discrete form, leading to exact computation.

Topology-Preserving Conditions for 2D Digital Images Under Rigid Transformations

In the continuous domain

Combinatorial Properties of 2D Discrete Rigid Transformations under Pixel-Invariance Constraints

\mathbb{R}^{n}

Sufficient conditions for topological invariance of 2d images under rigid transformations

, rigid transformations are topology-preserving operations. Due to digitization, this is not the case when considering digital images, i.e., images defined on

On 2D constrained discrete rigid transformations

\mathbb{Z}^{n}

Well-composed images and rigid transformations

. In this article, we begin to investigate this problem by studying conditions for digital images to preserve their topological properties under all rigid transformations on

Efficient Neighbourhood Computing for Discrete Rigid Transformation Graph Search

\mathbb{Z}^{2}

Geometric preservation of 2D digital objects under rigid motions

. Based on (i) the recently introduced notion of DRT graph, and (ii) the notion of simple point, we propose an algorithm for evaluating digital images topological invariance.

Analysis of Noisy Digital Contours with Adaptive Tangential Cover

Rigid transformations are useful in a wide range of digital image processing applications. In this context, they are generally considered as continuous processes, followed by discretization of the results. In recent works, rigid transformations on ℤ2 have been 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 combinatorial structure of the transformation space and the algorithm leading to its generation.

Convexity invariance of voxel objects under rigid motions

In ℝ2, rigid transformations are topology-preserving operations. However, this property is generally no longer true when considering digital images instead of continuous ones, due to digitization effects. In this article, we investigate this issue by studying discrete rigid transformations (DRTs) on ℤ2. More precisely, we define conditions under which digital images preserve their topological properties under any arbitrary DRTs. Based on the recently introduced notion of DRT graph and the classical notion of simple point, we first identify a family of local patterns that authorize topological invariance under DRTs. These patterns are then involved in a local analysis process that guarantees topological invariance of whole digital images in linear time.