Using Channel Representations in Regularization Terms: A Case Study on Image Diffusion
Authors: Christian Heinemann, Freddie Åström, George Baravdish, Kai Krajsek, Michael Felsberg, Hanno Scharr
Organizations: IBG-2: Plant Sciences, Forschungszentrum Jülich, 52425 Jülich, Germany · Department of Electrical Engineering, Linköping University, SE-581 83 Linköping, Sweden · Department of Science and Technology, Linköping University, SE-601 74 Norrköping, Sweden
In this work we propose a novel non-linear diffusion filtering approach for images based on their channel representation. To derive the diffusion update scheme we formulate a novel energy functional using a soft-histogram representation of image pixel neighborhoods obtained from the channel encoding. The resulting Euler-Lagrange equation yields a non-linear robust diffusion scheme with additional weighting terms stemming from the channel representation which steer the diffusion process. We apply this novel energy formulation to image reconstruction problems, showing good performance in the presence of mixtures of Gaussian and impulse-like noise, e.g. missing data. In denoising experiments of common scalar-valued images our approach performs competitive compared to other diffusion schemes as well as state-of-the-art denoising methods for the considered noise types.
Figures & tables
Figure 1: Overview of proposed method. Contrast is increased in ”Filter & Update” for a better visualization.
Figure 2: Example for equidistantly distributed B-Spline basis functions. Five channels are used here.
Figure 3: a) Typical structure of S(u) for a certain spatial position. The size of the matrix is equal to the number of channels. In this example 11 channels are used and non-zero entries are centered around channel 9. b) Structure of ctS(u)c for the cameraman image and a Gaussian kernel w with standard deviation 3. Black indicates low and white high values close to 1.
Figure 4: Quantitative denoising results of the images Skyline (69007), Hawk (42040), Bird (197027), Rollercoaster (235098), Boat &House (140088) and Lighthouse (228076) from the Berkeley image database [ 12 ] and standard test images (Cameraman, House, Lena). The numeric name is the same as in the database. Used methods are linear diffusion, NLD [ 13 ] , AD [ 20 ] , CS [ 6 ] , the novel LCD, the novel NLCD, MF [ 8 ] , BM3D [ 5 ] and NLM [ 2 ] . SSIM versus different Gaussian noise levels is plotted. For details see text. Best viewed in color
Figure 5: Visualization of certain parts of different images (Cameraman, Hawk, Boat&House) of noise level σ=30 . A visual comparison is done between the introduced methods.
The assessment of image denoising results depends on the respective application area, i.e. image compression, still-image acquisition, and medical images require entirely different behavior of the applied denoising method. In this paper we propose a novel, nonlinear diffusion scheme that is derived from a linear diffusion process in a value space determined by the application. We show that application-driven linear diffusion in the transformed space compares favorably with existing nonlinear diffusion techniques.
Freddie Åström, Michael Felsberg, George Baravdish +1
Computer Vision Laboratory, Linköping University, Sweden · Center for Medical Image Science and Visualization, Linköping University, Sweden · Department of Science and Technology, Linköping University, Sweden +1
In this work, we introduce a novel tensor-based functional for targeted image enhancement and denoising. Via explicit regularization, our formulation incorporates application dependent and contextual information using first principles. Few works in literature treat variational models that describe both application dependent information and contextual knowledge of the denoising problem. We prove the existence of a minimizer and present results on tensor symmetry constraints, convexity, and geometric interpretation of the proposed functional. We show that our framework excels in applications where nonlinear functions are present such as in gamma correction and targeted value range filtering. We also study general denoising performance where we show comparable results to dedicated PDE-based state of the art methods.
Freddie Åström, Michael Felsberg, George Baravdish
Heidelberg University Heidelberg Collaboratory for Image Processing (HCI) Im Neuenheimer Feld 205 69 120 Heidelberg, Germany · Linköping University Department of Electrical Engineering 581 83 Linköping, Sweden · Linköping University Department of Science and Technology 581 83 Linköping, Sweden
This paper deals with the case of using nonlinear diffusion filters to obtain piecewise constant images as a previous process for segmentation techniques. We first show an intrinsic formulation for the nonlinear diffusion equation to provide some design conditions on the diffusion filters. According to this theoretical framework, we propose a new family of diffusivities; they are obtained from nonlinear diffusion techniques and are related with backward diffusion. Their goal is to split the image in closed contours with a homogenized grey intensity inside and with no blurred edges. We also prove that our filters satisfy the well-posedness semi-discrete and full discrete scale-space requirements. This shows that by using semi-implicit schemes, a forward nonlinear diffusion equation is solved, instead of a backward nonlinear diffusion equation, connecting with an edge-preserving process. Under the conditions established for the diffusivity and using a stopping criterion for the diffusion time, we get piecewise constant images with a low computational effort. Finally, we test our filter with real images and we illustrate the effects of our diffusivity function as a method to get piecewise constant images. The code is available at https://github.com/cplatero/NonlinearDiffusion.
Javier Sanguino, Carlos Platero, Olga Velasco
Health Science Technology Group, Technical University of Madrid, Ronda de Valencia 3, 28012, Madrid, Spain.