Local edge detection

The main objective of local edge detection by the proposed method, is to collect all candidate edge details obtained by forward wavelet decomposition at a given resolution leveljusing an adaptive thresholdtL. In the proposed study this adaptive thresholdtL

is determined by using the maximum likelihood estimate (MLE) of wavelet histograms and its intensity distribution P for selected confidence level α1. An optimal adaptive threshold is defined by its capability to segregate a multimodal distribution without causing inadequate clustering of image data and is simple to compute. Histogram of wavelet coefficients usually follow a mixture-model distribution and are characteristic of grey-level distribution within the image. Depending on the image this distribution may be multimodal; in this study as the coefficients Cψ are obtained from single coil

magnitude MR image the distribution is considered to be Rician in nature.

In single coil magnitude MR image, due to the nature of image reconstruction from raw k-space data, the inherent Gaussian noise in the image data causes the distribution of final image to be non-linear and signal dependent with Rice distribution (Gudbjartsson and Patz,2005). For the Rician noise affected MR images; at low intensities such as the background, the histogram has a Rayleigh distribution whereas for high intensity region the distribution is more Gaussian, given by Eq. (3.29).

3D Multiscale UWT decomposition

3D Image volume

3D Multiscale UWT reconstruction

Start

3D processed image volume

Global multiscale threshold

Automatic Adaptive local

threshold

Figure 3.5: Proposed method for cartilage edge detection

In this study, for simplicity of analysis we consider the wavelet histogram to follow a bi- modal distribution characterized mainly by low intensity background information and high intensity edge information which may also contain the desired cartilage details. For local edge detection we need to isolate coefficients which are under high intensity Gaussian distribution PG and can be done using a threshold tL. By computing the

lower iGL, iRL and higher intensity iGH, iRH confidence limits of individual histogram

we can compute a single threshold value tL for the parent histogram, by using linear

Figure 3.6: Proposed threshold estimation using histogram

3.3.1.1 Estimation of intensity interval using MLE

To compute the local threshold value tL, we first need to determine the MLE and

intensity interval of the bimodal histogram. The foreground image details are assumed to have Gaussian distributionPGand a local threshold valuetLis implemented to isolate

wavelet coefficientsCˆψG belonging to this distribution. The background distribution in

the parent bimodal histogram is assumed to be Rayleigh and is given as (Siddiqui,1964),

PR(Cψi|µR, σR) = Cψi σ2 R e− (C2 ψi) 2σ2 R (3.29)

where PR is the probability distribution function, µR is the mean of the Rayleigh dis-

tribution, σR is the variance of the distribution and Cψi is an independent wavelet

coefficients. The MLE for variance of Rayleigh distribution is given as (Siddiqui,1964),

ˆ σR2 ≈ 1 2Np Np X i=1 C_ψ2_i (3.30)

whereNp represents number of samples and are the histograms of edge details obtained

due to wavelet decomposition for a given resolution. The complete derivation of MLE for Rayleigh is given in appendix A. The confidence intervalCI for Rayleigh in terms of

unknown mean intensity, is sampled with respect to normal distribution for large number of samples and follows t-distribution. The intensity interval for background coefficients can then be given as follows (Canavos,1984),

ˆ CψR −t1−α/2 SR p Np ≤µR≤CˆψR+t1−α/2 SR p Np (3.31)

where µR is true unknown mean for Rayleigh distribution, SR is sample variance for

Rayleigh distribution,CˆψRis estimated Rayleigh mean computed with MLE for Rayleigh

distribution given as,

ˆ

CψR = ˆσR r

π

2 (3.32)

The values of respectivet1−α/2 can be obtained from t-distribution table withNpdegrees

of freedom for a selected confidence levelα1(Canavos,1984). Higher confidence levelα1 ensures more area is covered under the given distribution. On solving the above equation we can obtain lower and higher intensity values for histogram of Rayleigh distribution. These confidence limits for intensity are represented as iRL and iRH respectively as

shown in Fig. (3.6).

Similarly, the probability distribution of wavelet coefficients for foreground are modelled as Gaussian distribution given asPG(Cψi|µG, σG) =

1 √ 2πσ2 G e(−Cψi−µG 2σ2 G ) wherePG is the

probability function,µGis the mean of Gaussian distribution, variance of the distribution

is given as σ_G2 and where Cψi is the variable of histogram distribution (Forbes et al., 2011). The maximum likelihood estimate in terms of mean for Gaussian distribution is given as (Forbes et al.,2011).

ˆ CψG= 1 Np Np X i=1 Cψi (3.33)

WhereNp is the number of samples andCˆψG is the mean of the samples (Forbes et al., 2011). In order to determine confidence interval for intensity with respect to mean we use the sampling distribution of the given distribution for unknown mean with respect to the normal distribution. This sampling distribution follows a t-distribution with (Np−1)

degree of freedom and is thus given as follows (Canavos,1984),

t= CˆψG−µG SG/

p Np

whereµG is the true mean of the normal distribution andSGis the sample variance. tis

symmetric around meanCˆψG and image details relevant to the foreground are likely to

lie within this confidence interval of the histogram. The intensity values are once again characterized as higher foreground and lower foreground and are represented asiGH and

iGL respectively as shown in Fig. (3.6). Appendix 2 gives the mathematical derivation

for MLE of Gaussian distribution and computation of intensity interval.

Algorithm 1:Adaptive Local Edge Detection

Input: Input Image Volume Ivol =f(x, y, z)

1 Select wavelet familyW ={ϕ_j,k, ψ_j,k} 2 forwavelet scale j=1:J do,

3 3D un-decimated forward wavelet decomposition, Wj=ψj,k ∗Ivol

4 3D histogram for wavelet coefficients H_Wj HWj = histogram(Wj)

5 Maximum Likelihood Estimation (MLE) for mean for multimodal distribution using HWj,

6 Intensity interval, using 99% confidence level a. Foreground distribution igaussian = [iGL:iGH]

b. background distribution irayleigh = [iRL :iRH] .

7 Linear interpolation to compute local thresholdt_l.

8 Selecting wavelet coefficient under foreground distribution with proposed local thresholdtl.

9 end for

Output: Local thresholded wavelet coefficientsWs

3.3.1.2 Local threshold

As the parent histogram is a continuous distribution, a linear interpolation of the esti- mated intensity values is used to compute threshold tL. A linear interpolation can be

obtained by using the higher intensity limit of the background distribution iRH and its

corresponding pixel informationbRH with that of the lower intensity limit of foreground

distribution iGL and its corresponding pixel information bGL as shown in Fig. (3.6).

The estimated local threshold value is given as (Wikipedia),

tL=iRH+ (iGL−iRH)

b−bRH

bGL−bRH

where tL is the required local threshold for a given resolution scale s. All the relative

pixel information are obtained from the 3D wavelet histogram itself and b can be any user entered value anywhere from within the histogram. The pixel information within each bin act like weights when computing the final threshold value.

The local threshold value tL is computed from wavelet histograms generated for each

scale and thus adapts to the data provided by wavelet coefficients at each scales. Cor- responding histograms are used to separate the foreground edge details usingtL value,

which are further used to determine optimum coefficients by implementing multiscale edge detection. Local edge detection by the proposed method can thus be summarized in algorithm 1.