● Image Denoising

Image denoising is a fundamental problem in image processing. It aims at eliminating noise from a corrupted image while preserving essential features such as edges and textures. There are three main directions: stochastic modeling, wavelets, and variational (PDE-based) approaches.

Among these approaches, our main research interest lies in the variational approaches. Denoising problem commonly covers additive noise removal and multiplicative noise removal.

1) Let $u_0(x,y) = u(x,y) + n(x,y)$, where $x,y \in \Omega$ and $n(x,y)$ is the additive noise and TV is total variation operator.

$\displaystyle argmin_{u} \int_\Omega (u-u_0)^2 dx+ \lambda TV(u)$

2) Let $u_0(x,y) = u(x,y) * n(x,y)$, where $x,y \in \Omega$ and $n(x,y)$ is the multiplicative noise.

$\displaystyle argmin_{u} \int_\Omega (u+u_0 e^{-u}) dx+ \lambda TV(u)$

Generally, solutions of variational model with TV regularization have many desirable properties like edge preserving. However, the regularization based TV regularization has some drawbacks, most undesirable drawback is so-called staircasing artifact that is the undesired appearance of edges. To overcome the staircasing artifact, many improved models have been suggested. For example, combining first and higher order TV functionals and total generalized vatiation (TGV).

● Image Inpainting

Image inpainting is the process of removing obstacle or noise in the image and inferring the background blocked by the obstacle or noise.

Suppose you take a picture of beautiful scenery or famous artifact, but this is spoiled by something such as unknown person’s head at the corner of photo. Then you can remove it by using inpainting technique and get the intended one.

There are also many other use of image inpainting in various areas as you can imagine. For example, medical imaging, criminal investigation and astrophysical image processing, etc.

In view of image restoration and image interpolation, we use TV(Total Variation) regularization technique for inpainting process. It aims to minimize the $L_2$-norm of difference of original and inpainted image in the unblocked region with TV-norm of inpainted image as a regularizer. The $L_2$-norm of difference is adjusted by spatially-varying parameter.

The recovered region is primarily affected by its boundary. This technique has advantage over other inpainting method such as $H^1$-inpainting in that it fairly recovers sharp edges.

● Image Segmentation

Image segmentation is one of the important tasks in image processing along with de-blurring, de-noising and image registration.

Image segmentation is the process of partitioning images into several segments and the purpose of segmentation is to simplify or transform an image to cluster of data that is more meaningful and easier to analysis.

It is largely applicable to many industrial area. In health and medical field, we can apply this for computer-aided diagnosis by the detection of tumors. We also use this process for detection of obstacles and pedestrians for self-driving car.

In addition to above, there are lots of areas using image segmentation: analysis of satellite images, face recognition, automatic traffic control system, and others. Therefore, image segmentation of indispensable in modern society.

Image segmentation has been developed through various approaches: the statistical models, the variational models and the partial differential equation (PDE)-based models, and the graph cut approaches.

In our study, the primitive approach is the variational and the PDE-based model by active contour.