**3. Temperature distribution analysis**

The plantar foot region is divided into four regions covering the toe region, metatarsal heads, medial arch and heel region to quantitatively analyze the temperature distribution. Histogram plot of the temperature distributions is accomplished by computing the mean temperature for each row of the plantar raw temperature profile for both left and right foot. The temperature values which are greater than 0°C are alone used to compute the mean temperature while those temperature values that are equal to 0°C are related to background and hence eliminated in each row. The histogram plot of the temperature distributions was offset corrected and normalized. The balance of every single distribution is eliminated by deducting the smallest temperature value either at the start or termination of the distribution. Later it is normalized by selecting a precision of three decimal figures to differentiate deviations in temperature from one pixel to another pixel.

#### **3.1 Kernel density estimation**

Once the final histogram is obtained, an approximation of density function is estimated using Kernel Density Estimation (KDE). A kernel distribution is a nonparametric representation of the probability density function (pdf) of a random variable. Let *x*1, *x*2, … , *xn* be observations drawn independently from a distribution P with density p. The kernel density estimate ^*gh*ð Þ *x* is defined as

$$\hat{\mathbf{g}}\_h(\mathbf{x}) = \frac{1}{nh} \sum\_{i=1}^n K\left(\frac{\mathbf{x} - \mathbf{x}\_i}{h}\right) \tag{1}$$

Where *K*ð Þ� denotes the smoothing kernel function, and h > 0 is the smoothing bandwidth criterion which regulates the amount of smoothing. The kernel function is balanced and unimodal about the origin. The Gaussian kernel with normal distribution is utilized in this work. Bandwidth controls the smoothness of the density estimation. A smaller value of *h* will result in a rough estimation, while a higher value of *h* will result in a remarkably smooth estimate. The KDE smoothens every observation into a smaller density and the summation of these smaller densities

together is used to attain the ultimate density estimate. The KDE is used to smooth the histogram plot of the temperature distribution and investigate the statistical significance [23, 24]. The temperature distributions of the four ROI (toes, metatarsal heads, medial arch and heel) in each foot are estimated using kernel density estimation. The histogram of the mean temperature distribution was smoothened using kernel density estimation which follows the non-parametric distribution. **Figures 7** and **8** (a)-(d)

**Figure 7.** *Smoothened temperature distribution - control group.*

**Figure 8.** *Smoothened temperature distribution - DM group.*

illustrates the histogram and smoothened histogram of the left and right foot for the control and DM group.

The smoothening is repeated for all the four regions of each left and right foot of the control and DM group. The region-wise smoothened temperature distribution is superimposed on the whole foot smoothened output for the control and DM groups depicted in **Figures 9** and **10**, respectively. From the mean temperature distribution histogram plot it was observed that the control group have a larger mean temperature around 21–22°C with a smaller number of occurrences of temperatures distribution while the DM group have a larger mean temperature around 23–24°C with a larger

**Figure 9.**

*Region wise smoothened temperature distribution superimposed on the whole foot - control group.*


#### **Table 2.**

*Median and interquartile range for whole foot.*


#### **Table 3.**

*Region wise median and interquartile range.*

number of occurrences of particular temperatures values. Similar kinds of deviations are observed in all the four ROIs of the left foot and right foot in the control and DM groups.

Since the KDE, which is a non-parametric distribution utilized to smoothen the temperature distribution, the central tendency among the control and DM groups was calculated by arranging the temperature distribution in ascending order to determine median and interquartile range for statistical analysis. Similarly, the median and interquartile range are also obtained for all the four ROIs with the corresponding smoothened temperature distribution for both control and DM groups. The computed median and interquartile range extracted from whole foot and region-wise for both groups are shown in **Tables 2** and **3**.

Thus, between **Tables 2** and **3** it was observed that the control group has a lower median temperature than the DM group. Since the mean temperature distribution is having a higher value in the lower quartile and upper quartile for the whole foot and region-wise, the interquartile range for the DM group is lesser than the control group.

The Chi-square goodness-of-fit [25] test was employed to assess the null hypothesis statistically for all the smoothened temperature distribution of left foot and right foot among control and DM group as shown in **Table 4**. This test is utilized to determine whether the variables are attained from the particular group temperature distribution or not and also to evaluate whether the sample data is representative of the full population of the temperature distribution. For both control and DM groups, the test returned an h value equivalent to one which specifies that chi-square goodness-of-fit rejects the null hypothesis at 5% significance level. Hence, these median and interquartile ranges are utilized to detect the diabetic foot in the classification process.


#### **Table 4.**

*Chi-square goodness-of-fit test results.*
