No doubt that mathematics research area can be expand to computer vision area such as image processing. Indeed, this area can be a new perspective in Mathematics and Computer Science. One of the popular model in image processing can be using partial differential equation (PDE) to describe the evolution of an image. Therefore, some numerical methods are needed in order to solve PDE in image processing. Let us observe the following partial differential equation in image processing, assume that is a pixel value of 2D image, then PDE for image is given as follows,

where is a Lipschitz function and is a Possion operator. This equation is known as Perona-Malik equation.

Example: denoising image

For instance, if is equal to a constant 1, then the equation exactly becomes a diffusion equation. Thus, if this equation is implemented to image processing then this can be called a bluring or denoising technique. Please see the following images as the illustration of bluring.

As it shown in Fig. 1, the bluring equation or diffusion equation is able to press the high density into low density. Logically, this high density can be used to describe the high value of pixel in an image. For instance, a noise in image can be represented as an outlier pixel value in image.

Then if this equation is applied in image the results can be found in the following results. Figs. 2 and 3 are obtained from the finite difference approach for bluring/denoising equation.

From the above results (Figs. 2-3), the bluring equation is able to reduce the noise, however the quality of the image can not maintained. Next, if the Lipschitz function is not in a constant, then the better results will be obtained.

For more detail about this research, please read my following research works

• Gunawan, P. H., & Gumilar, A. F. (2019, April). Mac-Cormack’s Scheme for Shock Filtering Equation in Image Enhancement. In Computer Science On-line Conference (pp. 83-89). Springer, Cham.
• Rabbani, H., & Gunawan, P. H. (2018). Kinerja OpenMP pada Pengolahan Citra dengan Model Curvature Motion. Indonesian Journal on Computing (Indo-JC), 3(1), 97-102.
• Iryanto, I., Fristella, F., & Gunawan, P. H. (2016). Pendekatan Numerik pada model Isotropic dan Anisotropic diffusion dalam Pengolahan Citra. Indonesian Journal on Computing (Indo-JC), 1(2), 83-96.