Fourth order diffusion model based edge map extraction of infrared breast images

Abstract

Breast thermography plays a significant role in early detection of breast cancer. In this work, an attempt has been made to extract the edge map in Type-I and Type-II images using You and Kaveh (YK) and Nonlinear Nonlocal (NLNL) diffusion filter. NLNL is an enhanced fourth order diffusion model. The extracted edge map is compared with the YK fourth order diffusion filtered edge map. Two gradient based metrics Edge preservation Index (EPI) and Gradient Magnitude Scale Deviation (GMSD) are used to validate the extracted edge maps. Result shows, the varying characteristics edges are found to be distinct, clear and enhanced by YK fourth order model. Compared to an enhanced fourth order method the EPI is found to be high for YK model edge map images. The index value is found to be 40% and 28% respectively. Similarly minimum GMSD is observed for YK fourth order model compared to the nonlinear nonlocal diffusion model. Hence the edge map extracted from YK fourth order filter shall aid precise segmentation and shall be used for early screening and clinical interpretation.

Keywords

Nonlinear nonlocal diffusion filtering YK model infrared breast images AQI EFI and GMSD

1. Introduction

Breast thermography is a non-invasive and radiation free procedure which records and displays thermal patterns emitted by human skin [1]. Statistical incidence of breast cancer among women reveals that there is a need for early detection of the disease. Mammography which is the current gold standard technique has high false positive rate and less sensitivity in dense breast tissue. Thermography, being a functional imaging technique is reported as an adjunct tool due to its improved sensitivity and specificity [2].

Generally thermal images are low contrast in nature. Further the quality of image is degraded due to atmospheric and ambient conditions which results in low signal to noise ratio and absence of clear edges [3]. For early detection of cancer, segmentation of breast thermal images is inevitable [4]. Poor contrast and low SNR are the additional constraint to edge detection. The inferior edge map affects the accuracy of the segmentation in the breast thermal images [5]. Denoising is necessary to reduce noise and improve the contrast of thermal images. Gaussian smoothing averages pixel along all directions instead of orientations. Due to this property, the edge information are blurred which makes detection and localization of edges a difficult task [6]. Smoothing operation performed by second order anisotropic non linear diffusion is to preserves the edges [7]. But they fail to consider the integration scale. In literature it is reported, Nonlinear Nonlocal (NLNL) and YK fourth order diffusion filters are used to remove noise. NLNL filter is an enhanced fourth order diffusion model, effective in regularizing the diffusion and eliminating the noise [8]. The non linearity intensity reduction obtained using fractional derivative in the NLNL filter is used to enhance selectivity and robustness in the presence of noise. The YK fourth order model uses laplacian operator. The boundaries are approximated by piecewise inclined planes.

In this work, NLNL and YK fourth order diffusion filter are used to extract the edge map from the infrared breast images [9]. The edge map is extracted for various gradient threshold values. To choose appropriate edge map the gradient threshold parameters are optimized using Anisotropy Quality index (AQI) [10]. To aid precise segmentation the parameter optimized edge map are compared and evaluated using gradient based metric EPI [11, 12] and GMSD [13].

2. Methodology

2.1 Database

Breast thermal images for this study are obtained from online database of the project [14]. The total of 55 images is considered for this study and analysis. The characteristic of images in the database varies in terms of edge. The database images are observed with two main characteristics of edges, such as clear, distinguishable lower breast boundaries and inframammary fold and images with vague lower breast boundaries and indistinguishable inframammary fold.

2.2 YK and nonlinear nonlocal fourth order diffusion filtering

In order to avoid the undesired blurring of edges by the ordinary linear filters and non-linear anisotropic diffusing filtering, nonlinear nonlocal diffusion filtering is adopted to preserve the edges. In order to do region specific smoothing on the image the conduction diffusion coefficient is considered. It is a function of brightness function ‘ $c$ ’. The appropriate choice of the function ‘ $c$ ’ preserve, sharpen and brighten the edges.

The YK fourth order diffusion equation proposed by You and Kaveh is given by

$\displaystyle\frac{\partial u}{\partial t}=-\nabla^{2}\left[{c\left({|\nabla^{% 2}u|}\right)\nabla^{2}u}\right]$ (1)

The Laplacian operator ‘ $\nabla^{2}u$ ’ approximates the images with piecewise planar image. The diffusion coefficient $c$ (.) controls the diffusion and distinguishes the noise and edge pixel of an image.

The Euler equation mentioned in Eq. (1) is minimized by finding the laplacian of the nosiy pixel using

$\displaystyle\nabla^{2}u^{n}_{i,j}=\frac{u_{i+1,j}^{n}+u_{i-1,j}^{n}+u_{i,j+1}% ^{n}+u_{i,j-1}^{n}-4u_{i,j}^{n}}{h^{2}}$ (2)

with symmetric boundary conditions

$\displaystyle u_{-1,j}^{n}=u_{0,j}^{n},u_{I+1,j}^{n}=u_{I,j}^{n}\quad j=0,1,2\ldots J$

and

$\displaystyle u_{i,-1}^{n}=u_{i,0}^{n},u_{i,J+1}^{n}=u_{i,J}^{n}\quad i=0,1,2\ldots I$

where $h$ is size of window, $I$ and $J$ are the size of image.

The edge distortion introduced by the fourth order diffusion filter is more than second order filter. You and Kaveh proposed the diffusivity function based on the absolute value of the laplacian of the image intensity function to overcome edge distortion and it is obtained using

$\displaystyle g(\nabla^{2}u)=c(|\nabla^{2}u|)\nabla^{2}u\quad\textit{where}% \quad g_{i,j}^{n}=g(\nabla^{2}u_{i,j}^{n})$ (3)

Further the Laplacian of $g$ (.) is calculated using

$\displaystyle\nabla^{2}g_{i,j}^{n}=\frac{g_{i+1,j}^{n}+g_{i-1,j}^{n}+g_{i,j+1}% ^{n}+g_{i,j-1}^{n}-4∼{}g_{i,j}^{n}}{h^{2}}$ (4)

with the symmetric boundary condition

$\displaystyle g_{-1,j}^{n}=g_{0,j}^{n},g_{I+1,j}^{n}=g_{I,j}^{n}\quad j=0,1,2\ldots J$ $\displaystyle g_{i,-1}^{n}=g_{i,0}^{n},g_{i,J+1}^{n}=g_{i,J}^{n}\quad i=0,1,2\ldots I$

Finally the numerical approximation of the Eq. (1) is given by

$\displaystyle u_{i,j}^{n+1}=u_{i,j}^{n}-\Delta t\nabla^{2}g_{i,j}^{n}$ (5)

NLNL filter is an enhanced fourth order diffusion model. The laplacian operator in Eq. (1) is replaced by gradient operator to obtain NLNL filter. Similarly, to regularize the diffusion coefficient ‘ $c$ ’ the conventional regularizer such as spatial regularization, temporal regularization and space time convolution are replaced by Fractional derivative regularization. The enhanced fourth order filter based on the fractional derivative regularization is shown in Eq. (2.2)

$\displaystyle\frac{\partial u}{\partial t}=-\Delta(c|\nabla^{1-\varepsilon}u|^% {2})\Delta{u})=0,\quad\Omega∼{}{\rm for}∼{}{t}>0$ $\displaystyle{u},{\rm periodic}\qquad\qquad\qquad\qquad\qquad∼{}\;\;{\rm for}∼% {}{t}>0$ (6) $\displaystyle{u}(0)={u}_{0}∼{}{\rm in}∼{}\Omega\qquad\qquad\qquad\qquad∼{}∼{}% \;\;{\rm for}∼{}{t}=0$

where $\Omega$ is the unit square and $\varepsilon\in$ (0, 1). When $\varepsilon$ tends to be zero, the Eq. (5) is similar to anisotropic diffusion filter. The fractional derivative with initial and boundary equation is effective in regularizing the diffusion for edge detection and to eliminate the noise.

The quality of images is associated with the edges in the images [15]. The edge map extracted from the infrared breast images using YK model and new enhanced fourth order diffusion filter for various gradient threshold values are validated using Anisotropy Quality Index (AQI). Degradation of edges results in the anisotropy of the image. The optimum gradient threshold values are chosen based on AQI value. Anisotropy index of the image is measured by evaluating the entropy at possible directions. The normalized pseudo Wigner distribution and Renyi entropy are used to estimate the entropy at predefined direction using Eqs (7) and (8)

$\displaystyle W[n,k]=2\sum\limits_{m=\frac{N}{2}}^{\frac{N}{2}-1}{z[n,m]}∼{}{% \ast}∼{}z[n-m]e^{-2i\left({\frac{2\pi n}{N}}\right)k}$ (7) $\displaystyle R_{\alpha}=\frac{1}{1-\alpha}\log_{2}\sum\limits_{n}{\sum\limits% _{k}{P^{\alpha}[n,k]}}$ (8)

The expectation value of the entropy for different orientations is measured as Anisotropy quality index using Eq. (9)

$\displaystyle\textit{AQI}=\sigma(t)=\sqrt{\sum\limits_{s=1}^{S}(\mu_{t}-\bar{R% }[t,\theta_{s}])^{2}}/S\quad\textit{where}\quad\mu_{t}=\sum\limits_{s=1}^{S}{% \bar{R}}(t,\theta_{s})/S$ (9)

The pixel level variation in the entropy reflects the disturbances in the edges. Maximum AQI is obtained for good quality edge map images.

Figure 1.

Representative set of input images. (a and b) Clear, visible lower breast boundaries and inframammary fold and (c and d) Indistinguishable lower breast boundaries and inframammary fold images.

2.3 Validation of edge map

To choose proper edge map for precise segmentation, the parameter optimized edge map are further validated using EPI and GMSD.

Edge Preservation Index (EPI) is used to measure the resemblances of edge images derived from original and filtered images using

$\displaystyle\textit{EPI}=\frac{\sum{(\Delta x-\overline{\Delta x}})(\Delta y-% \overline{\Delta y})}{\sum{(\Delta x-\overline{\Delta x})^{2}(\Delta y-% \overline{\Delta y})^{2}}}$ (10)

where $\Delta x$ and $\Delta y$ are the high pass filtered version of images $x$ and $y$ obtained using Laplacian operator with 3 $\times$ 3 mask. The mean values of $\Delta x$ and $\Delta y$ are represented by $\overline{\Delta x}$ and $\overline{\Delta y}$ .

GMSD is the metric to compare the gradient magnitude maps of the input and filtered images using

$\displaystyle\textit{GMSD}=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}{(\textit{GMS% }(i)-\textit{GMSM})^{2}}}$ (11)

where $N$ is the total number of pixel, GMS represents the gradient magnitude similarity map which depends on the gradient of original and diffused image and GMSM is the mean gradient magnitude similarity map.

3. Results and discussion

The representative set of images are shown in Fig. 1. Images with clear lower breast boundaries and inframammary fold edges are observed in Fig. 1a. The images with indistinguishable lower breast boundaries and inframammary fold are shown in Fig. 1b. The edge maps are extracted from representative set of images using nonlinear nonlocal diffusion filter and YK fourth order diffusion model.

Using NLNL and YK fourth order filter the edge map are extracted by varying the gradient threshold value. The edge map extracted using YK fourth order filter exhibits the tradeoff between the noise removal and edge preservation. The time step of 0.25 and gradient threshold are the parameter used to obtain the edge map. The representative set of input image with and extracted edge maps for threshold values of 2, 5 and 20 are shown in Fig. 2.

Table 1
Anisotropic quality index of YK and NLNL edge map for various gradient thresholds

Edge map	Gradient threshold	AQI values
YK	2	0.0157
	5	0.0132
	20	0.0120
NLNL	10 ${}^{-5}$	0.00137
	10 ${}^{-3}$	0.00112
	10 ${}^{-1}$	0.00097

Figure 2.

Representative results of YK fourth order filter. (a and b) Input images and (c–h) edge map images for gradient threshold value of 2, 5 and 20 respectively.

Figure 3.

Representative results of NLNL filter. (a and b) Input images (c–h) edge map images for gradient threshold of 10 ${}^{-5}$ , 10 ${}^{-3}$ and 10 ${}^{-1}$ respectively.

Figure 4.

Representative results of YK and NLNL fourth order. (a and b) Filtered image (c and d) edge map for gradient threshold value of 2 and (e and f) edge map for gradient threshold value of 10 ${}^{-5}$ .

Figure 5.

Validation index of YK and NLNL diffusion model edge map (a) EPI and (b) GMSD.

The edge map extracted from breast images using YK fourth order diffusion filter is shown in Fig. 2. The edge map obtained from both categories of images for the gradient threshold value of 2 is shown in Fig. 2c and d. It is found to have sharp lower breast boundaries and inframammary fold. The edges are observed with distinct boundary due to piecewise planar image approximation. The low frequency and high frequency noise is removed while the edges are preserved for threshold value of 2. When the gradient threshold is greater than 2 there is significant increase in the filtering of noise. But the edges near the inframammary fold and lower breast boundaries are blurred to greater extent. For threshold value of 5 and 20 the edges with undistinguishable speckle, with black and white spots are smoothened to greater extent and found to be blurred and shown in Fig. 2e–h.

The edge map extracted using non local non linear fourth order diffusion filter is shown in Fig. 3. The choice of parameter of NLNL fourth order filter are, regularizer $\varepsilon=$ 0.1, time step $h=5\times 10^{-12}$ and gradient threshold of $c=10^{-5},10^{-3},10^{-1}$ with 12 iterations. When gradient threshold value is fixed to 10 ${}^{-5}$ , the filter is efficient in preserving the edges and also removes noise. When the gradient threshold value is greater than 10 ${}^{-5}$ the edges are observed to be blurred in both categories of images and shown in Fig. 3e–h. The extracted edge map shown in Fig. 3c and d fails to show significant edge information. The non linearity intensity reduction obtained using fractional derivative in the fourth order filter reduced the contrast of the images.

Hence for accurate segmentation the edge map with sharp and speckle free edges are necessary. To choose appropriate edge map the AQI value extracted from the edge map of NLNL and YK fourth order diffusion filter are tabulated in Table 1.

From Figs 2 and 3, as the gradient threshold increases, the sharpness of edges is found to be decreased. To eliminate the visual inspection error and to aid accurate segmentation an AQI value is obtained and analyzed to select the parameter optimized edge map. The AQI value extracted from the edge map reflected the characteristics of edges. From Table 1 the AQI value is found to be very less when gradient threshold value is greater than 2 and 10 ${}^{-5}$ . The noise left out by the diffusion filter reduces the AQI values when gradient threshold is fixed to 5 and 20. The AQI value is observed to be low when gradient threshold value increases to 10 ${}^{-3}$ , 10 ${}^{-1}$ . This may be due to blurred, spurious and over smoothed edges. For the gradient threshold value 2 and 10 ${}^{-5}$ the values are observed to be high. Among the YK fourth order diffusion filter and NLNL filter the AQI is observed to be high for YK fourth order filter. This value states that the edge map extracted using YK fourth order diffusion filter eliminates the noise and tends to preserve the content of edges for threshold value of 2. The filtered and edge map images of both categories extracted using YK and NLNL fourth order diffusion model for gradient threshold value of 2 and 10 ${}^{-5}$ are shown in Fig. 4. The edge map is further validated using EPI and GMSD metrics.

It is observed from Fig. 5a the YK fourth order diffusion filter shows high EPI values compared to NLNL filter in breast images. This may be due to the presence of laplacian operator in the diffusion equation. The laplacian operator removes the noise and preserves the edges. Due to the presence of gradients in the YK model it is found to preserve 40% of the edges and only 28% of edges are preserved by NLNL method.

From Fig. 5b the values of GMSD of YK fourth order filtered images are minimum compared to NLNL filter. The GMSD of YK fourth order diffusion filter is found to be 0.005. This may be due to less deviation in gradient magnitude of filtered and input image. The GMSD of NLNL fourth order diffusion filter is observed to be 0.006. This may be due to more gradient change compared to YK diffusion filter.

4. Conclusion

The detection of edges at lower breast boundaries and inframammary fold is an inevitable processing technique for accurate segmentation of breast tissues. In this work nonlinear nonlocal enhanced fourth order diffusion model is used to extract the edge maps from varying edge characteristics infrared thermal breast images. The edge maps are compared with the edge map of YK fourth order diffusion model. The edge maps are validated using EPI and GMSD. The result shows, the edge map images obtained using YK fourth order diffusion model are found to be enhanced. Compared to NLNL filter the edge map of YK fourth order filter are distinct and clear for weak edge images also. The elimination of the traditional second order forward and backward diffusion is found to be efficient in preserving the edges by reducing the noise. The EPI is high for YK fourth order diffusion filtered image. It is observed to be 40% and only 28% for NLNL fourth order diffusion filtered images. The minimum GMSD is observed for YK model compared to NLNL method. The values are found to be 0.005 and 0.006 respectively. Hence, the edge map extracted using the YK fourth order diffusion filter shall aid accurate segmentation and clinical interpretation.

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