Anisotropic Geometric Level Set Method for Geometric 3D Image Processing

Description

Processing three dimensional images is a task of growing interest in various applications. Especially in medical imaging different image generation hardware such as CT or MRI devices, and more recently also 3D ultrasound devices deliver large image data at high resolution for further post processing and analysis. These images and especially 3D ultrasound images are characterized by high frequent noise typically due to measurement errors. Often, one is interested in the extraction of certain level surfaces from the data, which bound volumes or separate regions of interest. Frequently the actual intensity value is of minor importance and dependent on the modality in the image generation process. Methods which behave invariant under transformations of the intensity or gray scale are called morphological. They only effect the morphology of the image, which coincides with the geometry of the level sets. 

We present a novel anisotropic level set method for the denoising of large, digital 3D images (cf. parametric surface approach). The peculiarity of the method is, that it is able to preserve edges and corners on level sets while still allowing tangential smoothing along the edges. Furthermore it is characterized by a rich class of invariant shapes. Indeed ellipsoids given as level sets of a quadratic polynomial are unaffected by the corresponding evolution. The new anisotropic geometric diffusion approach is characterized by substantial tangential smoothing, especially compared to the original Perona Malik scheme, and conserves sharp edges and details in the example much better than the anisotropic Perona Malik diffusion.

The core of the method is an evolution driven by anisotropic geometric diffusion of level surfaces. For a given gray scale image we consider the following level set equation

with Neumann boundary conditions, and where the anisotropic diffusion tensor (here given in coordinates of principal directions e₁, e₂ of curvature and the normal N on a level set)

depends on a presmoothed shape operator . Therefore it depends on presmoothed principal curvatures and principal directions of curvature and thus is sensitive to the identification of the important surface features. By using the well known Perona Malik Function G(s) = 1/(1+s²)it decreases the diffusivity in certain directions in close vicinity to edges or corners. Here we make us of the observation that edges on surfaces are indicated by one dominant and one sub-dominant principal curvature. The curvature direction corresponding to the dominant curvature can be considered as the tangential direction along the edge. 

Therefore mainly two parameters are at the disposal of the user to influence the performance of the method:

The difference between the actual and the presmoothed shape operator plays an essential role in the control of the evolution problem. Furthermore, the built-in prefiltering is essential to make the proposed method robust and mathematically well-posed. Indeed, it turns out that in the equivalent formulation of a force steering the evolution in normal direction 

the force is proportional to the difference between presmoothed shape operator and the Shape operator on the original level set. In more detail, we have:

.

We conclude from this observation that the evolution is influenced by the chosen regularization type. We choose regularization via local L² - projection of the image onto a polynomial space.