Digital Image Processing, Analysis and Computer Vision Using Nonlinear Partial Differential Equations

Partial differential equations (PDEs), which have long been used to express various dynamical phenomena since they describe the continuous change, have become ubiquitous in mathematically oriented scientific domains, like the engineering and physics. Thus, these equations are fundamental for the scientific understanding of the diffusion, heat, electrostatics, electrodynamics, thermo-dynamics, fluid dynamics, elasticity or quantum mechanics.

This chapter addresses the nonlinear partial differential equation (PDE)-based image denoising and restoration domain, detailing our most important contributions in this field. It describes PDE-based filtering models for various types of noise and images. Thus, the first three sections present 2D image restoration techniques related to different noise distributions. The first one introduces several effective nonlinear diffusion-based models, in variational and non-variational form, for additive white Gaussian noise (AWGN) reduction. The nonlinear PDE-based quantum noise filtering solutions are described in the second section, while the variational and anisotropic diffusion schemes dealing successfully with the noise mixtures are discussed in the third section.

Two closely related image processing domains based on nonlinear partial differential equations, namely image inpainting and compression are described in this chapter. The nonlinear PDE-based image reconstruction, or inpainting, field is approached in the first section. Variational and non-variational structural inpainting techniques based on nonlinear PDE models of various orders are described here. Our main contributions in this domain are also surveyed in this section.

The multi-scale image analysis and computer vision techniques based on nonlinear PDE models are discussed in this chapter. The results described here illustrate that many effective multi-scale analysis solutions for various tasks can be achieved by using nonlinear diffusion-based models. Some nonlinear PDE-based scale-space representations are created and used by the high-level feature extraction components of several image analysis techniques proposed by us. Thus, some anisotropic diffusion-based multi-scale texture recognition frameworks are described in the first section. Then, the content-based image recognition using nonlinear PDE-based scale spaces is discussed in the second section. The nonlinear diffusion-based multi-scale image indexing and retrieval solutions are presented in the final section.

This chapter approaches the PDE-based image segmentation domain, describing variational and non-variational PDE-based models for static and video image segmentation. The edge-based segmentation of static images is addressed in the first section of the chapter, where our contribution in this field, representing a nonlinear diffusion-based multi-scale edge detection framework, is discussed. Next, the segmentation solutions using nonlinear PDE-based parametric and geometric active contour models are presented in the second section. Our own contributions, representing some geodesic active contours with level-set functions, are also described here. The video image segmentation field is then addressed in the third section. A temporal video segmentation technique proposed by us, which combines a nonlinear second-order PDE-based multi-scale frame analysis to a deep learning-based high-level feature extraction, is discussed in that section.

The applications of the nonlinear PDE models in the video object detection and tracking field are discussed in this chapter. This computer vision domain has important application areas, including video monitoring, law enforcement, security systems, human–computer interaction, video indexing and retrieval, robotic vision, medical imaging and augmented reality. A detection and tracking task means locating some certain objects in the frames of a video sequence and identifying their trajectories.

The applications of the mathematical models based on nonlinear partial differential equations in the image processing and analysis, computer vision and artificial intelligence domains have been addressed in this book. The work has surveyed the existing research in these fields, but mainly insisted on the author’s most important contributions in them.