Adel I. El-Fallah

The evolution of mean curvature in image filtering

A new formulation for inhomogeneous image diffusion is presented in which the image is regarded as a surface in 3-space. The evolution of this surface under diffusion is analyzed by classical methods of differential geometry. A nonlinear filtering theory is introduced in which only the divergence of the direction of the surface gradient is averaged. This averaging preserves edges and lines, as their direction is non-divergent, while noise is averaged since it does not have non-divergent consistency. Our approach achieves this objective by evolving the surface at a speed proportional to mean curvature leading to the minimization of the surface area and the imposition of regularity everywhere. Furthermore, we introduce a new filter that renders corners, as well as edges, invariant to the diffusion process. Experiments demonstrating the adequacy of this new theory are presented.<<ETX>>

Nonlinear adaptive image filtering based on inhomogeneous diffusion and differential geometry

The inadequacy of the classic linear approach to edge detection and scale space filtering lies in the spatial averaging of the Laplacian. The Laplacian is the divergence of the gradient and thus is the divergence of both magnitude and direction. The divergence in magnitude characterizes edges and this divergence must not be averaged if the image structure is to be preserved. We introduce a new nonlinear filtering theory that only averages the divergence of direction. This averaging keeps edges and lines intact as their direction is nondivergent. Noise does not have this nondivergent consistency and its divergent direction is averaged. Higher order structures such as corners are singular points or inflection points in the divergence of direction and also are averaged. Corners are intersection points of edges of nondivergent direction (or smooth curves of small divergence in direction) and their averaging is limited. This approach provides a better compromise between noise removal and preservation of image structure. Experiments that verify and demonstrate the adequacy of this new theory are presented.

On mean curvature diffusion in nonlinear image filtering

Mean curvature diffusion is shown to be a position vector diffusion, tending to scalar diffusion as a flat image region is approached, and providing noise removal by steepest descent surface minimization. At edges, it switches to a nondiffusion state due to two factors: the Laplacian of position vanishes and the magnitude of the surface normal attains a local maximum.

Invariance of edges and corners under mean-curvature diffusions of images

We have recently proposed the use of geometry in image processing by representing an image as a surface in 3-space. The linear variations in intensity (edges) were shown to have a nondivergent surface normal. Exploiting this feature we introduced a nonlinear adaptive filter that only averages the divergence in the direction of the surface normal. This led to an inhomogeneous diffusion (ID) that averages the mean curvature of the surface, rendering edges invariant while removing noise. This mean curvature diffusion (MCD) when applied to an isolated edge imbedded in additive Gaussian noise results in complete noise removal and edge enhancement with the edge location left intact. In this paper we introduce a new filter that will render corners (two intersecting edges), as well as edges, invariant to the diffusion process. Because many edges in images are not isolated the corner model better represents the image than the edge model. For this reason, this new filtering technique, while encompassing MCD, also outperforms it when applied to images. Many applications will benefit from this geometrical interpretation of image processing, and those discussed in this paper include image noise removal, edge and/or corner detection and enhancement, and perceptually transparent coding.

Mean curvature evolution and surface area scaling in image filtering

Proposes a geometrical surface representation for an image, and introduces a nonlinear adaptive filter that diffuses the surface in time at a speed proportional to the mean curvature. Linear variations in intensity (edges) are inclined planes of vanishing mean curvature, and are thus invariant. Noise is characterized by high mean curvature and will be diffused. The authors show that this diffusion resolves the conflict of removing noise while preserving edges. A novel nonlinear scale space filtering relating surface area to the diffusion speed is introduced resulting in very efficient algorithms. Experiments demonstrating excellent performance and efficiency are presented.<<ETX>>

Preprocessing for improved performance in image and video coding

In previous work, we reported on the benefits of noise reduction prior to coding of very high quality images. Perceptual transparency can be achieved with a significant improvement in compression as compared to error free codes. In this paper, we examine the benefits of preprocessing when the quality requirements are not very high, and perceptible distortion results. The use of data dependent anisotropic diffusion that maintains image structure, edges, and transitions in luminance or color is beneficial in controlling the spatial distribution of errors introduced by coding. Thus, the merit of preprocessing is for the control of coding errors. In this preliminary study, we only consider preprocessing prior to the use of the standard JPEG and MPEG coding techniques.

Structure preserving inhomogeneous diffusion in image filtering

We introduce a new theory relating the magnitude of the image surface normal to an inhomogeneous diffusion that solely diffuses (averages) the mean curvature of the image surface. We discuss the remarkable properties of this diffusion stressing the regularity it imposes on regions and boundaries while preserving the locality of edges and lines. Experiments demonstrating the excellent performance of the algorithms in the areas of noise removal and enhancement are presented.<<ETX>>

Progressive perceptually transparent coder for very high quality images

In the perceptually transparent coding of images, we use representation and quantization strategies that exploit properties of human perception to obtain an approximate digital image indistinguishable from the original. This image is then encoded in an error free manner. The resulting coders have better performance than error free coding for a comparable quality. Further, by considering changes to images that do not produce perceptible distortion, we identify image characteristics onerous for the encoder, but perceptually unimportant. Once such characteristic is the typical noise level, often imperceptible, encountered in still images. Thus, we consider adaptive noise removal to improve coder performance, without perceptible degradation of quality. In this paper, several elements contribute to coding efficiency while preserving image quality: adaptive noise removal, additive decomposition of the image with a high activity remainder, coarse quantization of the remainder, progressive representation of the remainder, using bilinear or directional interpolation methods, and efficient encoding of the sparse remainder. The overall coding performance improvement due to noise removal and the use of a progressive code is about 18%, as compared to our previous results for perceptually transparent coders. The compression ratio for a set of nine test images is 3.72 for no perceptible loss of quality.

Image filtering by gradient inverse inhomogeneous diffusion

A method is developed for the synthesis of a nonlinear adaptive filter based on solutions to the inhomogeneous diffusion equation. The approach is based on the specification of the first derivative of the signal in time (scale). A general solution is derived and is then specialized to the scale invariance case, in which the diffusion coefficient is shown to be the gradient inverse. A novel discrete realization of the inhomogeneous diffusion equation is developed for the noise removal problem, and experimental results are shown. The proposed algorithm not only removes noise but simultaneously enhances and localizes edges. It is extremely simple and parallel, and does not require the detection of any of the many possible line and edge configurations. Since the algorithm is sensitive to the local context, it satisfies human vision requirements more than conventional methods which rely on minimizing the mean square error.<<ETX>>

Progressive analysis-based perceptually transparent coder for still images

The encoding of images at high quality is important in a number of applications. We have developed an approach to coding that produces no visible degradation and that we denote as perceptually transparent. Such a technique achieves a modest compression, but still significantly higher than error free codes. Maintaining image quality is not important in the early stages of a progressive scheme, when only a reduced resolution preview is needed. In this paper, we describe a new method for the progressive transmission of high quality still images, that efficiently uses the lower resolution images in the encoding process. Analysis based interpolation is used to estimate the higher resolution image, and reduces the incremental information transmitted at each step. This methodology for high quality image compression is also aimed at obtaining a compressed image of higher perceived quality than the original.