Mammogram Diagnostics Using Robust Wavelet-Based Estimator of Hurst Exponent

Abstract

Breast cancer is one of the leading causes of death in women. Mammography is an effective method for early detection of breast cancer. Like other medical images, mammograms demonstrate a certain degree of self-similarity over a range of scales, which can be used in classifying individuals as cancerous or non-cancerous. In this paper, we study the robust estimation of Hurst exponent (self-similarity measure) in two-dimensional images based on non-decimated wavelet transforms (NDWT). The robustness is achieved by applying a general trimean estimator on non-decimated wavelet detail coefficients of the transformed data, and the general trimean estimator is derived as a weighted average of the distribution’s median and quantiles, combining the median’s emphasis on central values with the quantiles’ attention to the extremes. The properties of the proposed estimators are studied both theoretically and numerically. Compared with other standard wavelet-based methods (Veitch and Abry (VA) method, Soltani, Simard, and Boichu (SSB) method, median based estimators MEDL and MEDLA, and Theil-type (TT) weighted regression method), our methods reduce the variance of the estimators and increase the prediction precision in most cases. We apply proposed methods to digitized mammogram images, estimate Hurst exponent, and then use it as a discriminatory descriptor to classify mammograms to benign and malignant. Our methods yield the highest classification accuracy around 65%.

Appendix

1.1 Proof of Theorem 5.1

Proof

A single wavelet coefficient in a non-decimated wavelet transform of a 2-D fBm of size N × N with Hurst exponent H is normally distributed, with variance depending on its level j. The four coefficients in each set

$$\displaystyle \begin{aligned}\{d_{j,j;(k_{i1},k_{i2})}, d_{j,j;(k_{i1},k_{i2}+\frac{N}{2})}, d_{j,j;(k_{i1}+\frac{N}{2},k_{i2})}, d_{j,j;(k_{i1}+\frac{N}{2},k_{i2}+\frac{N}{2})}\}\end{aligned}$$

are assumed to be independent and follow the same normal distribution.

$$\displaystyle \begin{aligned} &d_{j,j;(k_{i1},k_{i2})}, d_{j,j;(k_{i1},k_{i2}+\frac{N}{2})}, d_{j,j;(k_{i1}+\frac{N}{2},k_{i2})}, d_{j,j;(k_{i1}+\frac{N}{2},k_{i2}+\frac{N}{2})}\\ &\quad \sim \mathcal{N}\left(0, 2^{-\left(2H+2\right)j}\sigma^2\right).\end{aligned} $$

Then the mid-energies in D _j defined in (5.9) and (5.8) can be readily shown to have exponential distribution with scale parameter $\lambda _j=\sigma ^2\cdot 2^{-\left (2H+2\right )j}$. Therefore at each detail level j, the mid-energies in D _j are i.i.d. $\mathcal {E}xp\left (\lambda _j^{-1}\right )$, and when applying general trimean estimator $\hat {\mu }_{j}$ on D _j, following the derivation in Sect. 5.3, we have

$$\displaystyle \begin{aligned}\boldsymbol{\xi}=\left[\log\left(\frac{1}{1-p}\right)\lambda_j\ \ \ \log\left(2\right)\lambda_j\ \ \ \log\left(\frac{1}{p}\right)\lambda_j\right]^T,\end{aligned}$$

and

therefore, the asymptotic distribution of $\hat {\mu }_{j,i}$ is normal with mean

and variance

Since the Hurst exponent can be estimated as

$$\displaystyle \begin{aligned} \hat{H}=-\frac{\hat{\beta}}{2}-1, \end{aligned} $$

(5.27)

where $\hat {\beta }$ is the regression slope in the least square linear regression on pairs $\left (j, \log _2\left (\hat {\mu }_{j}\right )\right )$ from level J ₁ to J ₂, J ₁ ≤ j ≤ J ₂. It can be easily derived that $\hat {\beta }$ is a linear combination of $\log _2\left (\hat {\mu }_{j}\right )$,

$$\displaystyle \begin{aligned} \hat{\beta}=\sum_{j=J_1}^{J_2}a_j\log_2\left(\hat{\mu}_{j}\right), \ \ a_j=\frac{j-(J_1+J_2)/2}{\sum_{j=J_1}^{J_2}\left(j-(J_1+J_2)/2\right)^2}. \end{aligned}$$

We can check that $\sum _{j=J_1}^{J_2}a_j=0$ and $\sum _{j=J_1}^{J_2}a_j j=1$. Also, if $X\sim \mathcal {N}(\mu , \sigma ^2)$, the approximate expectation and variance of g(X) are

$$\displaystyle \begin{aligned} \mathbb{E}\left(g(X)\right)=g(\mu)+\frac{g''(\mu)\sigma^2}{2}, \ \ \mbox{and}\ \ \operatorname{\mathrm{Var}}\left(g(X)\right)=\left(g'(\mu)\right)^2\sigma^2, \end{aligned}$$

based on which we calculate

$$\displaystyle \begin{aligned} \mathbb{E}\left(\log_2\left(\hat{\mu}_{j}\right)\right)=-(2H+2)j+\mbox{Constant},\ \mbox{and}\ \operatorname{\mathrm{Var}}\left(\log_2\left(\hat{\mu}_{j}\right)\right)=\frac{\frac{2}{M^2}f\left(\alpha,p\right)}{(\log2)^2c^2\left(\alpha, p\right)}. \end{aligned}$$

Therefore

$$\displaystyle \begin{aligned} \mathbb{E}\left(\hat{\beta}\right) &=\sum_{j=J_1}^{J_2}a_j\mathbb{E}\left(\log_2\left(\hat{\mu}_{j}\right)\right)=-(2H+2),\ \mbox{and}\ \operatorname{\mathrm{Var}}\left(\hat{\beta}\right)\\ &=\sum_{j=J_1}^{J_2}a^2_j\operatorname{\mathrm{Var}}\left(\log_2\left(\hat{\mu}_{j}\right)\right):=4V1, \end{aligned} $$

and

$$\displaystyle \begin{aligned} \mathbb{E}\left(\hat{H}\right)=H,\ \mbox{and}\ \operatorname{\mathrm{Var}}\left(\hat{H}\right)=V1, \end{aligned} $$

(5.28)

where the asymptotic variance V ₁ is a constant number independent of simple size N and level j,

$$\displaystyle \begin{aligned} V_1=\frac{6f(\alpha,p)}{(\log2)^2M^2c^2\left(\alpha,p\right)q(J_1,J_2)}, \end{aligned}$$

and

$$\displaystyle \begin{aligned} q(J_1,J_2)=(J_2-J_1)(J_2-J_1+1)(J_2-J_1+2). \end{aligned}$$

1.2 Proof of Theorem 5.2

Proof

We have stated that each mid-energy in D _j follows $\mathcal {E}xp\left (\lambda _j^{-1}\right )$ with scale parameter $\lambda _j=\sigma ^2\cdot 2^{-\left (2H+2\right )j}$. If we denote the kth element in $\log \left (D_{j}\right )$ as y _j,k for $k=1,\ldots ,\frac {M^2}{2}$ and j = 1, …, J, the pdf and cdf of y _j,k are

$$\displaystyle \begin{aligned}f\left(y_{j,k}\right)=\lambda_j^{-1}e^{-\lambda_j^{-1}e^{y_{j,k}}}e^{y_{j,k}},\end{aligned}$$

and

$$\displaystyle \begin{aligned}F\left(y_{j,k}\right)=1-e^{-\lambda_j^{-1}e^{y_{j,k}}}.\end{aligned}$$

The p-quantile can be obtained by solving $F\left (y_p\right )=1-e^{-\lambda _j^{-1}e^{y_p}}=p$, and $y_p=\log \left (-\lambda _j\log \left (1-p\right )\right )$. Then it can be shown that $f\left (y_p\right )=-\left (1-p\right )\log \left (1-p\right )$. When applying the general trimean estimator $\hat {\mu }_{j}$ on $\log \left (D_{j}\right )$, following the derivation in Sect. 5.3, we get

and

thus, the asymptotic distribution of $\hat {\mu }_{j,i}$ is normal with mean

and variance

where

$$\displaystyle \begin{aligned} \begin{aligned} g_1\left(p\right)=&\frac{p}{\left(1-p\right)\left(\log\left(1-p\right)\right)^2}+\\ &\frac{1-p}{p\left(\log p\right)^2}+\frac{2p}{\left(1-p\right)\log\left(1-p\right)\log p},\\ \end{aligned} \end{aligned}$$

and

$$\displaystyle \begin{aligned}g_2\left(p\right)=\frac{2p}{\left(1-p\right)\log\left(1-p\right)\log\frac{1}{2}}+\frac{2}{\log\frac{1}{2}\log p}.\end{aligned}$$

Since the Hurst exponent can be estimated as

$$\displaystyle \begin{aligned} \hat{H}=-\frac{1}{2\log2}\hat{\beta}-1, \end{aligned} $$

(5.29)

where $\hat {\beta }$ is the regression slope in the least square linear regressions on pairs $\left (j, \hat {\mu }_{j}\right )$ from level J ₁ to J ₂, J ₁ ≤ j ≤ J ₂. It can be easily derived that $\hat {\beta }$ is a linear combination of $\hat {\mu }_{j}$,

$$\displaystyle \begin{aligned} \hat{\beta}=\sum_{j=J_1}^{J_2}a_j\hat{\mu}_{j}, \ \ a_j=\frac{j-(J_1+J_2)/2}{\sum_{j=J_1}^{J_2}\left(j-(J_1+J_2)/2\right)^2}. \end{aligned}$$

Again, we can check that $\sum _{j=J_1}^{J_2}a_j=0$ and $\sum _{j=J_1}^{J_2}a_j j=1$. Therefore

$$\displaystyle \begin{aligned} \mathbb{E}\left(\hat{\beta}\right)&=\sum_{j=J_1}^{J_2}a_j\mathbb{E}\left(\hat{\mu}_{j,i}\right)=-(2H+2)\log2,\ \mbox{and}\ \operatorname{\mathrm{Var}}\left(\hat{\beta}\right)\\ &=\sum_{j=J_1}^{J_2}a^2_j\operatorname{\mathrm{Var}}\left(\hat{\mu}_{j,i}\right):=4(\log2)^2V_2, \end{aligned} $$

and

$$\displaystyle \begin{aligned} \mathbb{E}\left(\hat{H}\right)=H,\ \mbox{and}\ \operatorname{\mathrm{Var}}\left(\hat{H}\right)=V_2, \end{aligned} $$

(5.30)

where the asymptotic variance V ₂ is a constant number independent of simple size N and level j,

$$\displaystyle \begin{aligned} V_2=\frac{6f(\alpha,p)}{(\log2)^2M^2q(J_1,J_2)}, \end{aligned}$$

and q(J ₁, J ₂) is given in Eq. (5.13).

1.3 Proof of Lemma 5.2

Proof

When applying Tukey’s trimean estimator $\hat {\mu }_{j}^T$ on D _j, following the derivation in Sect. 5.3.1, we have

and

therefore, the asymptotic distribution of $\hat {\mu }_{j}^T$ is normal with mean

and variance

$$\displaystyle \begin{aligned}\operatorname{\mathrm{Var}}\left(\hat{\mu}_{j,i}^T\right)=\frac{2}{M^2}A_T\varSigma_T A_T^T=\frac{5}{3M^2}\lambda_j^2.\end{aligned}$$

When applying Gastwirth estimator $\hat {\mu }_{j}^G$ on D _j, following the derivation in Sect. 5.3.2, we have

and

therefore, the asymptotic distribution of $\hat {\mu }_{j}^G$ is normal with mean

and variance

$$\displaystyle \begin{aligned}\operatorname{\mathrm{Var}}\left(\hat{\mu}_{j,i}^G\right)=\frac{2}{M^2}A_G\varSigma_G A_G^T=\frac{1.67}{M^2}\lambda_j^2.\end{aligned}$$

Based on Eq. (5.28), we have

(5.31)

where the asymptotic variances $V^T_1$ and $V^G_1$ are constant numbers,

$$\displaystyle \begin{aligned} V^T_1=\frac{5}{(\log2)^2M^2c^2_1q(J_1,J_2)}, \end{aligned}$$

$$\displaystyle \begin{aligned} V^G_1=\frac{5.01}{(\log2)^2M^2c^2_2q(J_1,J_2)}. \end{aligned}$$

The function q(J ₁, J ₂) is the same as Eq. (5.13) in Theorem 5.1.

1.4 Proof of Lemma 5.3

Proof

When applying Tukey’s trimean estimator $\hat {\mu }_{j}^T$ on $\log \left (D_{j}\right )$, following the derivation in Sect. 5.3.1, we have

and

therefore, the asymptotic distribution of $\hat {\mu }_{j}^T$ is normal with mean

and variance

When applying Gastwirth estimator $\hat {\mu }_{j}^G$ on $\log \left (D_{j,i}\right )$, following the derivation in Sect. 5.3.2, we have

and

therefore, the asymptotic distribution of $\hat {\mu }_{j}^G$ is normal with mean

and variance

Based on Eq. (5.30), we can easily derive

(5.32)

where the asymptotic variances $V^T_2$ and $V^G_2$ are constant numbers,

$$\displaystyle \begin{aligned} V^T_2=\frac{3V_T}{(\log2)^2q(J_1,J_2)}, \end{aligned}$$

$$\displaystyle \begin{aligned} V^G_2=\frac{3V_G}{(\log2)^2q(J_1,J_2)}. \end{aligned}$$

The function q(J ₁, J ₂) is provided in Eq. (5.13).