## Access Type

Open Access Dissertation

## Date of Award

January 2019

## Degree Type

Dissertation

## Degree Name

Ph.D.

## Department

Computer Science

## First Advisor

Jing Hua

## Second Advisor

Farshad Fotouhi

## Abstract

Morphometric analysis of 3D surface objects are very important in many biomedical applications and clinical diagnoses. Its critical step lies in shape comparison and registration. Considering that the deformations of most organs such as heart or brain structures are non-isometric, it is very difﬁcult to ﬁnd the correspondence between the shapes before and after deformation, and therefore, very challenging for diagnosis purposes.

To solve these challenges, we propose two spectral based methods. The ﬁrst method employs the variation of the eigenvalues of the Laplace-Beltrami operator of the shape and optimize a quadratic equation in order to minimize the distance between two shapes’ eigenvalues. This method can determine multi-scale, non-isometric deformations through the variation of Laplace-Beltrami spectrum of two shapes. Given two triangle meshes, the spectra can be varied from one to another with a scale function deﬁned on each vertex.

The variation is expressed as a linear interpolation of eigenvalues of the two shapes. In each iteration step, a quadratic programming problem is constructed, based on our derived spectrum variation theorem and smoothness energy constraint, to compute the spectrum

variation. The derivation of the scale function is the solution of such a problem. Therefore, the ﬁnal scale function can be solved by integral of the derivation from each step, which, in turn, quantitatively describes non-isometric deformations between two shapes. However, this method can not ﬁnd the point to point correspondence between two shapes.

Our second method, extends the ﬁrst method and uses some feature points generated from the eigenvectors of two shapes to minimize the difference between two eigenvectors of the shapes in addition to their eigenvalues. In order to register two surfaces, we map both eigenvalues and eigenvectors of the Laplace-Beltrami of the shapes by optimizing an energy function. The function is deﬁned by the integration of a smooth term to align the eigenvalues and a distance term between the eigenvectors at feature points to align the eigenvectors. The feature points are generated using the static points of certain eigenvectors of the surfaces. By using both the eigenvalues and the eigenvectors on these feature points, the computational efﬁciency is improved considerably without losing the accuracy in comparison to the approaches that use the eigenvectors for all vertices. The variation of the shape is expressed using a scale function deﬁned at each vertex. Consequently, the total energy function to align the two given surfaces can be deﬁned using the linear interpolation of the scale function derivatives. Through the optimization the energy function, the scale function can be solved and the alignment is achieved. After the alignment, the eigenvectors can be employed to calculate the point to point correspondence of the surfaces. Therefore, the proposed method can accurately deﬁne the displacement of the vertices. For both methods, we evaluate them by conducting some experiments on synthetic and real data using hippocampus and heart data. These experiments demonstrate the advantages and accuracy of our methods.

We then integrate our methods to a workﬂow system named DataView. Using this workﬂow system, users can design, save, run, and share their workﬂow using their web-browsers without the need of installing any software and regardless of the power of their

computers. We have also integrated Grid to this system therefore the same task can be executed on up to 64 different cases which will increase the performance of the system enormously.

## Recommended Citation

Hamidian, Hajar, "3d Surface Registration Using Geometric Spectrum Of Shapes" (2019). *Wayne State University Dissertations*. 2220.

https://digitalcommons.wayne.edu/oa_dissertations/2220