Feb.24, 1999

Publications by INRIA on ridges and related topics :

• O. Monga et al. :
  • Thin Nets Extraction Using a Multi-Scale Approach, 1999.
  • Thin Nets & Crest Lines: Application to Satellite Data & Medical Images, 1997.
• J.P. Thirion et al. - EPIDAURE project :
  • The Extremal Mesh & the Understanding of 3D Surfaces, 1996 & 1993.
  • Automatic Retrieval of Anatomical Structures in 3D Medical Images, 1995.
  • New feature points based on geometric invariants for 3D image registration, 1996 & 1993.
  • Image surface extremal points, new feature points for image registration, 1993.
  • The Marching lines algorithm: New results and proofs, 1993.
  • The 3D marching lines algorithm and its application to crest lines extraction, 1996 & 1992.

BibTeX references.

Thin nets are the lines where the grey level function is locally extremum in a given direction. Recently, we have shown that it is possible to characterize the thin nets using differential properties of the image surface. However, the method failed when these structures present different widths. In this paper we show that the extraction process of the thin nets, having different width, requires a multi-scale analysis of the image. To design the fusion process of the multi-scale information, we will study the behavior of the differential properties of the image surface, in particular the curvatures, in scale space. We illustrate the efficiency of the proposed multi-scale approach by extracting roads and blood vessels of different widths in satellite and medical images

In this paper, we describe a new approach for extracting thin nets in gray-level images. The key point of our approach is to model thin nets as crest lines of the image surface. Crest lines are lines where the magnitude maximum curvature is a local maximum in the corresponding principal direction. We define these lines using first, second, and third derivatives of the image. The image derivatives are computed using recursive filters approximating the Gaussian filter and its derivatives. Using an adaptive scale factor, we apply this approach to the extraction of roads in satellite data, blood vessels in medical images, and actual crest lines in depth maps of human faces.

3 stage algorithm:

1. Computation of 1st, 2nd & 3rd order partial derivatives of the image using recursive Gaussian filters.
2. Computation of the principal curvatures, the principal curvature directions, and the Directional derivative of the Maximum Curvature (DMC) along the corresponding principal direction.
3. Extraction of the zero-crossings of the DMC which form the crest lines of the thin nets (TN).

This paper is about a new concept for the description of 3D smooth surfaces: the extremal mesh. In previous morks, we have shown how to extract the extremal lines from 3D images, which are the lines where one of the two principal surface curvatures is locally extremal. We have also shown how to extract the extremal points, which are specific points where the two principal curvatures are both extremal. The extremal mesh is the graph of the surface whose vertices are the extremal points and whose edges are the extremal lines : it is invariant with respect to rigid transforms. The good topological properties of this graph are ensured with a new local geometric invariant of 3D surfaces, that we call the Gaussian extremality and which allows to overcome orientation problems encountered with previous definitions of the extremal lines and points. This paper presents also an algorithm to extract the extremal mesh from 3D images and experiments with synthetic and real 3D medical images showing that this graph can be extremely precise and stable. The extremal mesh and the Gaussian extremality are new insights into the geometrical nature of 3D surfaces with many promising consequences, some of which being lusted at the end of this paper.

This paper describes a method to automatically generate the mapping between a completely labeled reference image and the 3D medical image of a patient. To achieve this, we combined 3 techniques:

1. the extraction of 3D feature lines,
2. their matching using 3D deformable line models,
3. the extension of the deformation to the whole image space using warping techniques.

We present experimental results for the segmentation of structures in Magnetic Resonance images of the brain of different patients; the segmentation of the cortical and ventricle structures. We emphasize the advantages of using crest lines deformable models prior to surface based models. This gives a sparser representation of the data, easier to manipulate, and which makes the convergence of the model much less sensitive to initial positionning. In the future, we hope to use this method to generate anatomical atlases, by the automatic interpretation of large sets of 3D medical images.

Keywords : Deformable Models, Electronic Atlas, Feature Line, Non-Rigid Matching, Warping.

We generalize the definition of the extremal points (EP) to image surfaces and also to hyper-surfaces in any dimensions. These feature points can be used for image analysis and image registration because their relative positions are invariant with respect to rigid transforms. In a previous paper, we have defined the extremal points of the object surface for 3D images, and we have shown how to extract those points and how to use them to perform automatically the accurate registration of 3D medical images. The extremal points of the object surface are also invariant with changes of the image dynamic, because they are intrinsic to the object surface. We show now that another kind of extremal points can be defined in 3D, from the 4D image surface (x, y, z, f(x, y, z)). We explain how to compute and extract those new extremal points and then present registration experiments, comparing the results between the use of the extremal points of the object surface and of the image surface. We conclude by showing that both methods have their own advantages, leading to the extraction of extremely precise feature points and to the reliable registration of 3D images.

This research report is a compilation of two articles describing new results concerning the 3D Marching Lines algorithm. The Marching Lines extracts, with sub-voxel accuracy characteristic 3D lines out of 3D images, such as the crest lines. Those feature lines can then be used to perform higher level image processing tasks, such as 3D image registration, or automatic labeling of anatomical structures, when medical images are processed. The first paper concentrates on the computation of the differential characteri- stics of iso-intensity surfaces and shows how to characterize crest lines points directly from the differentials of the 3D image. The second paper brings the proof of the good topological properties of the reconstructed surfaces and 3D curves obtained with the Marching Lines algorithm. New experiments on real and synthetic data are also presented, showing the high precision and stability of the extracted features lines.

This paper presents a powerful and general purpose tool designed to extract characteristic lines from 3D images. The algorithm, called Marching Lines, is inspired from the Marching Cubes algorithm which is used to extract iso-value surfaces out of 3D images. The Marching Lines extracts with sub-pixel accuracy the 3D lines corresponding to the intersection of two iso-surfaces coming from two different 3D images. We show how to implement this algorithm to ensure that the reconstructed 3D lines have good topological properties mainly that they are continuous and closed. We present also a new method to compute the differential characteristics of iso-surfaces and show an application to the extraction of crest lines in 3D images. We explain that a crest line can be locally defined as the intersection of two surfaces one corresponding to an iso-value in the image and the other one to a crest surface which we define in this paper and whose implicit equation can be directly computed from the voxel values of the 3D image. At last, some experimental results for the 3D image of the skull are presented where crest lines are extracted and used to compute automatically the geometric transform between two 3D scanner images of the same subject taken in two different positions.

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