Anisotropic diffusion using oriented Laplacians

Anisotropic diffusion can be performed using a number of mathematical formulations. Tschumperl´e showed that there are links between the three formulations of regularising functional minimisation, divergence-based diffusion, and oriented Laplacian diffusion, that is, diffusion using using directional second derivatives [Tsch05] [Tsch02].

6.4 Denoising using anisotropic diffusion 155

the image I can be decomposed into two orthogonal components7 _I

uuand Ivv where the

the second derivatives of I are in the directions u and v (u, v∈ R2and u⊥ v) and Iuu= uTHu and Ivv = vTHv

where H is the Hessian of I given by Hi,j = ∂

2_I

∂xi∂yj

This decomposition leads to the oriented Laplacians equation ∂I

∂t = g1Iuu+ g2Ivv (6.7)

where g1, g2 > 0.

This decomposition allows the anisotropic diffusion to be thought of as the combination of two oriented heat flows or two oriented diffusion Laplacians with different conductivity coefficients (smoothing weights) in the directions (g2) of the gradient, and orthogonal to

the gradient (g1). The g1 and g2 are usually functions of the image gradient and define

a large range of anisotropic diffusion methods. For example, using g1 = 1, g2 = 0, v = ∇I

|∇I| and u = v⊥the oriented Laplacian equation Eqn. (6.7) becomes

∂I ∂t = Iuu

This diffusion along the lines orthogonal to the gradient is called the mean curvature flow [Weic08, p39].

One of the advantages of adopting the oriented Laplacians formulation is that they have a geometric meaning that make it easier to understand the nature of the smoothing [Tsch05]. The other major advantage is that it can be shown that

∂I

∂t = g1Iuu+ g2Ivv= trace(TH) (6.8)

where T = g1uuT + g2vvT. This is referred to as the trace-based form for the anisotropic

diffusion.

The 2x2 symmetric matrix T is called a diffusion tensor and has eigenvalues g1 and g2 for

respective eigenvectors u and v. Diffusion tensors are a structure tensor in which the eigen- values of the structure tensor have been used to regularise the diffusion, that is, change the local behaviour of the anisotropic diffusion so that it is not ill-posed.

The role of the diffusion tensor can be seen by first starting with the original Perona–Malik

formulation. The Perona–Malik form of the diffusion equation uses an edge-stopping func- tion based on a scalar quantity. In this regards it uses a nonlinear function, but is isotropic (the same behaviour in every direction for a given gradient|∇I|). The diffusion tensor, on the other hand, allows the local orientation of the gradients to influence diffusion because of the link with the structure tensor.

The (isotropic) nonlinear diffusion uses a gradient magnitude based on∇IT_{∇I. Although}

it is possible to work with gradients directly, this is usually not practical because if one wants to regularise or smooth gradient values to make them useful, cancellation of gradi- ents can occur. Brox et al. present an example of a thin line with a positive gradient on one side and a negative gradient on the other [Brox06]. If a smoothing operation is performed, then the two gradients cancel each other out. The way to get around this is to form a tensor S_{(in this case a matrix) from the outer product∇I∇I}T _{[Brox06] [Weic08, p56]. This tensor} can then be smoothed without the cancellation effects.

Performing an eigen decomposition of the tensor S =∇I∇IT _{gives eigenvalues λ}

1= |∇I|2

and λ2 = 0. In this situation little has been gained from using the outer product. The

advantage arises when smoothing is applied to the tensor. Gaussian smoothing Gσ of a

structure tensor results in a tensor Sσ = S ∗ Gσwhere the value of the tensor at point (x, y)

is influenced by the neighbourhood of (x, y). This improves the orientation information that can be derived from the structure tensor [Brox06].

The smoothed structure tensor S can be used to provide direction-specific smoothing for the diffusion by using the eigenvalues of the tensor in the edge-stopping function g(). This gives the so-called diffusion tensor [Weic08]. Directions in which there are large variations in local structure have a correspondingly large eigenvalue λ+in the structure tensor. When these large values are used in the edge-stopping function, g(λ+_{) is small, so diffusion in}

the direction of the corresponding eigenvector θ+slows or stops. This stops the diffusion at discontinuities and other locations with large gradients. Orthogonal to this direction the eigenvector θ− _{with eigenvalue λ}− _{will be small, so g(λ}−_{) is large and diffusion will}

occur in the direction of θ−. This results in little blurring across edges, and a dominance of blurring along edges. This results in anisotropic diffusion that is based on orientation of the edges.

Critical to the success of the anisotropic diffusion is the selection of an edge-stopping func- tion. This is now a function of the diffusion tensor eigenvalues λi rather than the magni-

tude of the image gradients|∇Ii|. Eqn. (6.8) can be written [Tsch06a]

∂I

∂t = trace(TH)

= trace ((g_(λ−+_,λ−₎θ−θ−T + g+_(λ+_,λ−₎θ+θ+T) H)

6.4 Denoising using anisotropic diffusion 157

where the eigenvectors and eigenvalues are pointwise within the image.

Although theoretical developments are occurring for the selection of an optimal edge- stopping function for a given application, it still remains mostly a matter of trying dif- ferent functions for the application. The function used in this research was based on the published successful use of the function [Tsch06a], including being used for the detection of wire structures in x-rays [Tsch06b]. The function is of the form

g_(λ++_,λ−₎=

1

(1 + λ+_{+ λ}−₎p2 and g−(λ+_,λ−₎=

1

(1 + λ+_{+ λ}−₎p1 with p1 < p2. (6.10)

This function allows for control of the diffusion in both the direction of maximum variation and the direction of minimum variation, with the additional couple of the two terms via the eigenvalues.

6.4.2.1 Incorporating a signal-dependent noise model into the anisotropic diffusion

The signal-dependent noise model developed in Section 4.3.1 showed that higher pixel intensities in the image have a larger associated noise. The parameters for this model were found for each radiograph using the method described in Section 6.3. The resulting standard deviation of the noise is of the form 10b·I(x,y) where I(x, y) is the pixel intensity at point (x, y) in the image. Given the estimate of b from the noise estimation method, it is possible to estimate the standard deviation of the noise. The point in question is what to do with this estimate.

Ideally, the parameters p1 and p2 should be set according to the noise content of the im- age. However, the author found that these parameters had a significant effect upon the quality of the anisotropic diffusion - the results were too sensitive to the values chosen. It is likely than another type of diffusivity function will be required to reduce this sensi- tivity, and this has been reserved for future research. In the meantime, to incorporate the signal-dependent noise model into the anisotropic diffusion method, a new noise slowing function s(I, t): s∈ [0, 1] was incorporated into Eqn. (6.9)

∂I

∂t = s(σ, t) trace ((g

−

(λ+_,λ−₎θ−θ−T + g_(λ++_,λ−₎θ+θ+T) H). (6.11)

As with the stopping function g() for the Perona–Malik anisotropic diffusion, the inten- tion was to slow the diffusion in areas of low noise so that the diffusion time t could be extended and the diffusion would favour noise reduction in high noise areas. The form of such a function is not obvious. Furthermore, the noise model of Eqn. (6.4) only applies to the radiographic image before denoising. This means that once denoising commences the model of the relationship between noise and pixel intensity will no longer apply. Although

it should theoretically be possible to track the influence of the denoising on local statistics, there is currently no theoretical development to support this. As an alternative, the ap- proach was to have the function s() depend on the diffusion time in such a way that by time T , s(I, T ) = 1, corresponding to the normal diffusion equation. The chosen function is illustrated in Figure 6.27 and is of the form

s(I, t) = t T + 10b·I 10b_·Imax(1 − t T) (6.12)

where T is the total diffusion time (total number of iterations for the denoising), I ∈ [0, 4095], and Imax = 4095, the maximum pixel intensity for the DICOM images.

0 500 1000 1500 2000 2500 3000 3500 4000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Pixel intensity

s() − noise slowing function

Figure 6.27 The noise slowing function for the anisotropic diffusion. This is an example of 50

iterations with the lines shown every 10 iterations. The function is for the theoretical noise of b = 1/2000