^{1}

^{*}

^{2}

^{*}

^{3}

^{*}

^{4}

^{*}

Ahmad et al. in their paper [1] for the first time proposed to apply sharp function for classification of images. In continuation of their work, in this paper we investigate the use of sharp function as an edge detector through well known diffusion models. Further, we discuss the formulation of weak solution of nonlinear diffusion equation and prove uniqueness of weak solution of nonlinear problem. The anisotropic generalization of sharp operator based diffusion has also been implemented and tested on various types of images.

Nonlinear diffusion filtering is a well-established tool for image denoising and simplification. Starting with the pioneering work by Perona and Malik [

The sharp function, on the other hand, is a well-known functional analytic concept to measure the oscillatory behaviour of functions. It goes back to the maximal function which was introduced by Hardy and Littlewood [

The idea of applying the sharp operator to measure the oscillation and classification of images was first proposed by Ahmad and Siddiqi [

In this paper, we propose an alternative way to steer nonlinear diffusion filters via the sharp operator without using derivatives to measure edges. We show that the results of these diffusion filters are comparable to classical versions while the underlying sharp operator has a rich theoretical background. Motivated by the available diffusion processes in image processing, we propose an extension of the sharp operator for measuring aniso- tropic structures. To use this to steer anisotropic diffusion processes, we show how a fast variant of it can be implemented and used in practice.

The paper is organized as follows. Section 2 gives a review of classical nonlinear diffusion filters for image processing. In Section 3, we shortly describe the aspects of the theory for the maximal function, bounded mean oscillation functions, and the sharp function, which are necessary for this paper. The main idea of this paper, namely, the use of the sharp operator in nonlinear diffusion filters and its generalization to the anisotropic setting, is presented in Section 4. To evaluate the methods in practice, Section 6 describes some computational experiments. A summary and an outlook conclude the paper in Section 7.

Diffusion is interesting as image processing tool since it is a physical process that equilibrates concentration without creating or destroying mass. The idea behind the use of the diffusion equation in image processing arose from the use of Gaussian filter in multiscale image analysis. It can be founded by a system of several axioms like linearity, translational and rotational invariance, and average grey value preservation, that marks the begin- ning of the scale-space concept [

with standard deviation

where the given image f is used as initial condition

Isotropic nonlinear diffusion. The major drawback of linear diffusion is the delocalisation and blurring of image edges. To circumvent this problem, Perona and Malik [

The diffusivity g is chosen as a decreasing function of the edge detector

with

Anisotropic nonlinear diffusion. Nonlinear isotropic diffusion often shows problems to remove noise close to image edges. It can be helpful to use an anisotropic diffusion filter

in such cases as proposed by Weickert [

In this section, we give a short introduction to the sharp operator and its background. There is a rich theory behind it, and we point out the main results connected to it.

The Hardy-Littlewood maximal function was developed to solve a problem in the theory of functions of complex variable. The analogue for integrals, which is required for the function theoretic applications, is derived in Hardy and Littlewood [

Definition 1. Let

where the supremum ranges over all finite cubes Q in

Now we state a Hardy-Littlewood maximal theorem.

Theorem 1. For each function

Proof. See ( [

The space BMO, i.e. bounded mean oscillation of functions is introduced by John and Nirenberg [

Definition 2. A measurable function f on

where the supremum ranges over all finite cubes Q in

is the mean value of the function f on the cube Q.

Fefferman and Stein [

Definition 3. Let f be a locally integrable function on

Of course,

Example 1. The function

After calculation it comes out to be

It is important to note that it does not matter in which

Corollary 1. For each p,

Proof. See ( [

In view of the above corollary the spaces

It is clear from the definition of the sharp function that for a pixel z, where f has almost uniform grey level region in an image,

Many isotropic nonlinear diffusivity models in physics and mechanics are governed by the nonlinear parabolic equation

depends on the gradient of the function

where

backward parabolic one, when

First of all let us prove that if only the conditions (i)-(ii) hold, then the nonlinear diffusion operator

in an appropriate Banach space

Calculating the first derivative

we conclude that

and then substituting here

if conditions (i)-(ii) hold. This means that the potential

Thus for the strong monotonicity of the nonlinear diffusion Equation (8), and hence solvability of an initial

boundary value problem related to the nonlinear diffusion equation

are sufficient. However, these conditions are not sufficient for solvability of the corresponding problem related to the 2D diffusivity model. Specifically, one needs to impose the monotonicity condition:

An analysis of the steady state diffusivity model governed by the nonlinear elliptic equation

Here

means that the diffusion flux

We will use weak solution theory for nonlinear PDE. For this, let us introduce the following well-known notations [

where

respectively.

Evidently, the norms

embedding. To define the weak solution of the nonlinear problem (10), we also need the following spaces

continuous and the embedding

space

Now we define the operators,

It is known that the operator

For a given coefficient

and the linear functional

which is well defined for

Theorem 2. Let

Proof. Let us introduce the functional

and calculate the first Gateaux derivative. We have

Hence

the nonlinear diffusion operator A. Calculating the second Gateaux derivative

Since

Substituting this in (14) and using the condition (ii) of (9), we conclude

Thus the potential

This implies the uniqueness of the weak solution of the nonlinear problem (13). Existence of the solution follows from the results given in [

Remark 1. The assertion of the above theorem holds also for the case when

Since structures in images often have the highly anisotropic features, for example, lines or corners, we propose some generalization of the presented method to the anisotropic setting. We start with an anisotropic generalization of the sharp operator.

So far we have only used isotropic nonlinear diffusion filters. In the definition (7) of the sharp operator, all integration domains Q are cubes. Therefore, the sharp function only provides information about local variations of the function, but not about the direction of these local variations. In order to allow for a quantitative descrip- tion of local variations in a certain direction, we propose to use non-symmetric sets instead of cubes. With this concept, an anisotropic extension of the sharp function can be defined as follows:

The most important in this definition is the set

In this definition, we take the supremum over all angles

For practical calculations, depending on the number of directions

Analogously to definition (7), the value

Instead of taking this mean value as function of

This changes the definition (15) to

We notice that in this definition, the difference in the integral is a difference between two functions. This offers the possibility to calculate the second function

The second step is now to write this as a convolution. Instead of an elliptical set

And lastly we replace also the outer integral with a convolution with an anisotropic Gaussian,

This measure can be evaluated in a very efficient way using the methods of Geusebroek et al. [

Now we want to use the anisotropic variant of the sharp operator to steer an anisotropic diffusion process

as it has been proposed by Weickert [

We define the diffusion tensor as follows: Let a point

where the absolute value of the anisotropic sharp operator is maximal. The eigenvectors of the diffusion tensor are then the unit vectors pointing in this direction and the orthogonal one, i.e.,

The eigenvalues are defined analogously as for edge-enhancing diffusion:

where,

is the maximal sharp value in the point x.

Having these definitions for the diffusion tensor at hand, we can use classical discretisation for anisotropic diffusion filters as described in [

To compare the sharp operator based diffusion approach with classical derivative based methods, we show filtering examples in

It is clear that the parameters of the anisotropic diffusion process have to be specified in practical situations. The time t is an inherent parameter in each diffusion process that controls the amount of simplification applied

to the data. The variance of evolving image decreases monotonically to zero in time. The contrast parameter

We have investigated the use of the sharp operator for image processing applications. We have used the sharp operator to steer diffusion filters. With the classical notion, it is suitable to be used inside the diffusivity of a Perona-Malik filter. For anisotropic filters, we have used fast anisotropic Gauss filters to extend the sharp operator to a fast directional-dependent measure of variation. With the help of this measure, we could construct an alternative diffusion tensor for an anisotropic diffusion process. The results are quite similar to classical anisotropic diffusion filters. We have seen that the sharp operator not only is of theoretical interest but also may be used in practical applications.