Exploiting Eigenvalues of the Hessian Matrix
for Volume Decimation

Abstract

In recent years the Hessian matrix and its eigenvalues became important in pattern recognition. Several algorithms based on the information they provide have been introduced. We recall the relationship between the eigenvalues of Hessian matrix and the 2nd order edge detection filter, show the usefulness of treating them separately and exploit these facts to design a combined threshold operation to generate sparse data sets.

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	talk given at WSCG 2001

Selected Figures

Figure 1: Property of 2D Eigenvalues images.

Images shown:

a) Intensity

b) gradient magnitude

c) Laplacian

d) the bigger eigenvalue

e) the smaller eigenvalue The Laplacian (c) is a sum of the Hessian matrix eigenvalues (d) + (e). Eigenvalue images, however, can be thresholded leaving only salient features in the original image.

Another 2D example, for a CT head

The same as above demonstrated for a CT image.

Figure 2: Property of 3D eigenvalue images.

Part of the interface and demonstration of a typical situation for 3D volumes: Image corresponding to the middle eigenvalue lack a good contrast. This eigenvalue is therefore excluded from the decimation process.

Table 1: Overview of Results.

Results for some typical volume data sets. The volume can be, for visualization purposes, represented by approximately 10 % of its voxels.

Figure 3: Comparison of engine block DVRs.

The decimation introduced is based on the definition of a variant of 2nd order edge filter which is important for bio perception. This is mostly noticeable comparing the DVRs of this data set. In the decimated volume, areas corresponding to edges are emphasized and provide the observer with a better topological information.

Figure 4: Comparison of CT head DVRs.

The decimated volume (left) is represented by just 10 % of its original (right).

Figure 5: Comparison of MRI head DVRs.

The decimated volume (left) is represented by just 12.5 % of its original (right).

Acknowledgements

This work has been funded by the V^isM^ed project. V^isM^ed is supported by Tiani Medgraph, Vienna and the Forschungsförderungsfonds für die gewerbliche Wirtschaft, Austria.

Project Duration

summer - October 2000

This page was last updated by Jiri Hladuvka on February 14, 2001.
If you have any comments, please send a message to jiri@cg.tuwien.ac.at.