Mathematical analysis of histogram equalization techniques for medical image enhancement: a tutorial from the perspective of data loss

Abstract

This tutorial demonstrates a novel mathematical analysis of histogram equalization techniques and its application in medical image enhancement. In this paper, conventional Global Histogram Equalization (GHE), Contrast Limited Adaptive Histogram Equalization (CLAHE), Histogram Specification (HS) and Brightness Preserving Dynamic Histogram Equalization (BPDHE) are re-investigated by a novel mathematical analysis. All these HE methods are widely employed by researchers in image processing and medical image diagnosis domain, however, this has been observed that these HE methods have significant limitation of data loss. In this paper, a mathematical proof is given that any kind of Histogram Equalization method is inevitable of data loss, because any HE method is a non-linear method. All these Histogram Equalization methods are implemented on two different datasets, they are, brain tumor MRI image dataset and colorectal cancer H and E-stained histopathology image dataset. Pearson Correlation Coefficient (PCC) and Structural Similarity Index Matrix (SSIM) both are found in the range of 0.6-0.95 for overall all HE methods. Moreover, those results are compared with Reinhard method which is a linear contrast enhancement method. The experimental results suggest that Reinhard method outperformed any HE methods for medical image enhancement. Furthermore, a popular CNN model VGG-16 is implemented, on the MRI dataset in order to prove that there is a direct correlation between less accuracy and data loss.

Appendix

Lemma1: For any contrast enhancement method, (with having transformation function which has number of roots 1),

$$\begin{aligned} If,\hspace{0.1 cm} \frac{p_r(r)}{p_s(s)}\approx 1,\hspace{0.1 cm} then \hspace{0.1 cm} Corr_{sr}\approx 1 \end{aligned}$$

(53)

where, $p_{r}$ (r) is the PDF of source image, $p_{s}$ (s) is the PDF of the processed image, $Corr_{sr}$ is the correlation co-efficient between processed image and source image.

From (6), we got (as the number of roots of the transformation function is 1),

$$\begin{aligned} p_s(s)= p_r(r)|\frac{dr}{ds}| \end{aligned}$$

(54)

Now if $p_{r}$ (r)$\approx p_{s}$ (s), according to the Lemma1, then from (54), we got, (for simplicity of calculation let’s assume $p_{r}$ (r)=$p_{s}$ (s))

$$\begin{aligned} \int {dr} =\int {ds} \end{aligned}$$

(55)

$$\begin{aligned} or, \hspace{0.1 cm} s = r+k \end{aligned}$$

(56)

where k is integration constant.

From (56), this is concluded that the transformation function of such contrast enhancement method will be linear.

Taking global standard deviation both of the sides in (56), we got,

$$\begin{aligned} \sigma _{gl}(s)= {\sigma _{gl}(r)} \end{aligned}$$

(57)

Similarly, by taking global mean both side of the (56), we got,

$$\begin{aligned} \mu _{gl}(s)= {\mu _{gl}(r)}+k \end{aligned}$$

(58)

Now, Covariance between processed image (s) and source image (r) is given by the following equation. (from 13)

$$\begin{aligned} \sigma _{sr}=\frac{1}{MN}{\sum _{i=1}^{M}{\sum _{j=1}^{N}{(s_i-\mu _{gl}(s_i))*(r_j-\mu _{gl}(r_j))}}} \end{aligned}$$

(59)

Now, substituting the value from (56) and (58) into (59), we got,

$$\begin{aligned} \sigma _{sr}=\frac{1}{MN}{\sum _{i=1}^{M}{\sum _{j=1}^{N}{(r_j-\mu _{gl}(r_j))}^{2}}} \end{aligned}$$

(60)

$$\begin{aligned} or, \hspace{0.1 cm} \sigma _{sr}={\sigma _{gl}^{2}(r)} \end{aligned}$$

(61)

Now the correlation coefficient between processed image and original image is given by following equation. (from 17)

$$\begin{aligned} Corr_{sr}= \frac{\sigma _{sr}}{\sigma _{gl}(s)*\sigma _{gl}(r)} \end{aligned}$$

(62)

Substituting values from (57) and (61) into (62) we got,

$$\begin{aligned} Corr_{sr}= 1 \hspace{0.2 cm} (Hence \hspace{0.1 cm} Proved) \end{aligned}$$

(63)

Lemma2: For any contrast enhancement method,

$$\begin{aligned} If,\hspace{0.1 cm} p_r(r)=\frac{1}{c}{p_s(s)},\hspace{0.1 cm} then \hspace{0.1 cm} Corr_{sr}\approx 1. \end{aligned}$$

(64)

whereas, c is a real constant, $p_r(r)$ is the PDF of source image, $p_s(s)$ is the PDF of processed image, $Corr_{sr}$ is the correlation co-efficient between processed image and source image. In other words, if the transformation function of contrast enhancement method is linear, then there will be no data loss.

For simplicity of calculation let’s assume $p_r (r)=1/c* p_s (s))$ then from (54) we got,

$$\begin{aligned} \int {dr} =\frac{1}{c}*{\int {ds}} \end{aligned}$$

(65)

$$\begin{aligned} s =\frac{1}{c}*(r+k) \end{aligned}$$

(66)

where k is an integration constant.

From (66), this is concluded that the transformation function of such contrast enhancement method is linear.

Taking global standard deviation both of the sides in (66), we got,

$$\begin{aligned} \sigma _{gl}(s)=\frac{1}{c} *{\sigma _{gl}(r)} \end{aligned}$$

(67)

Similarly, by taking global mean both side of the (66), we got,

$$\begin{aligned} \mu _{gl}(s)= \frac{1}{c}*{\mu _{gl}(r)}+k_1 \end{aligned}$$

(68)

Now, substituting the value from (66) and (68) into (59), we got,

$$\begin{aligned} \sigma _{sr}=\frac{1}{c}*{\frac{1}{MN}{\sum _{i=1}^{M}{\sum _{j=1}^{N}{(r_j-\mu _{gl}(r_j))}^{2}}}} \end{aligned}$$

(69)

$$\begin{aligned} or, \hspace{0.1 cm} \sigma _{sr}=\frac{1}{c}*{\sigma _{gl}^{2}(r)} \end{aligned}$$

(70)

Now, substituting values from (67) and (70), into (62) we got,

$$\begin{aligned} Corr_{sr}= 1 \hspace{0.2 cm} (Hence \hspace{0.1 cm} Proved) \end{aligned}$$

(71)

Hence, it is proved that a linear transformation doesn’t prone to data loss.