# Medical Images Analysis

- Brain Tumor Visualization
- Brain Deformation
- Area-Preserving Intestine Flattening
- Hippocampus Deformation

# 3D Animation

# Brain Tumor Visualization

The video shows a tumor in the human brain and its volume-preserving parameterization on a unit ball.

# Brain Deformation

The deformation of human brain is subtle and not easy to be observed. In order to efficiently detect the deformation, the one-to-one correspondence between two brain surfaces is necessary. With the efficient parameterization algorithms for genus zero closed surfaces, the registration problem between two brains is reduced into the registration problem on the unit sphere, which is much easier since the shape of domain are identical. Then the deformation from one brain to another can be visualized via the homotopy between the identity mapping and the registration mapping between two brains.

# Area-Preserving Intestine Flattening

Polyps are the predecessor of intestinal cancers. However, polyp detection is usually not an easy task, especially when the polyps are hidden in the folds. In order to efficiently detect the polyps on the intestinal surface, an area-preserving mapping can be applied to flatten the surface to a domain of rectangle, so that the polyps can be easily found in the rectangle domain. Then the exact locations of polyps can be obtained by the one-to-one correspondence between the intestinal surface and the rectangle domain.

# Hippocampus Deformation

It is known that Alzheimer's disease can be detected by the deformation of the hippocampus. The deformation can be measured via the one-to-one correspondence between two surfaces of hippocampi. Similar as the brain mappings, the mapping between two hippocampi can be obtained via the registration mapping on the spherical domain of parameterization mappings.

# Surface Morphing

A demo video of surface morphing between four facial expressions.

The aim of surface morphing in 3D is to use less quantity of the mesh frames to construct a smooth 3D video. The registration mappings between each pair of surfaces play the crucial role. With the suitable parameterizations, the registration problems between surfaces are reduced into the registration problems on a domain of a unified simple shape. Then the smooth video in 3D is constructed via the cubic spline homotopy between each pair of registration mappings.

# 3D Video Compression

The original video of the chewing-gum motion (right) and the reconstructed video by the spline homotopy using 1.6% of the original data (left).

The 3D video, captured using the 3D scanner, contains 30 surfaces together with texture images per second. Each surface data is of size roughly 30 megabytes. The goal of the 3D video compression is to use less quantity of the raw data of a 3D video to approximate the original 3D video. Note that each frame of the 3D video is a simply connected open surface, which is conformally equivalent to a unit disk. By using the proposed algorithm, the conformal equivalence can be computed efficiently and accurately. Then the diffeomorphism between each pair of surfaces can be constructed on the disk. Ultimately, the video sequence is reconstructed by the cubic spline homotopy between each pair of diffeomorphisms. As a result, the whole video sequence is reconstructed by merely 1.6% of the original data while the resolution of the video is preserved.

# Texture Mapping

A demo video of texture mapping.

Texture mapping refers to replace the texture of a surface with another one. It has been widely applied to the movie industries, e.g., the famous movie,

*Avatar*. In the movie, actors and actress perform without heavy makeup. Instead, a virtual makeup technique is applied to create a lifelike visual effect. The key issue is to find an appropriate diffeomorphism between the surface and the desired texture image that preserves features. The texture image is regarded as a planar surface in 2D, so that the diffeomorphism between the mesh and the texture image can be computed similarly. With the proposed algorithm, the virtual makeup can be performed efficiently and robustly.

# Virtual Broadcasting

A demo video of automatic Chinese virtual broadcasting.

Due to the fact that each Chinese syllable is composed of 1 to 3 Mandarin phonetic symbols, a virtual broadcasting system can be realized via recording the videos of pronouncing all the 37 phonetic symbols and constructing the morphing sequences.

# Motion Retargeting

A morphing sequence of the chewing-gum motion (left) and the motion retargeting on a target model (right).

Motion retargeting is aimed to use the information of a sequence of 3D video to drive a target surface. In other words, the motion of a target surface model is controlled by the motion of a given 3D video. The application already appears in

*iPhone X*. By applying the proposed algorithm, the conformal equivalence can be efficiently computed. The feature-preserving diffeomorphism between each pair of surfaces is constructed on the disk. Next, a vector field is constructed by the difference of two consecutive frames. Then the embedding of the target surface is controlled by the vector field via the topological equivalence.