## 3 Proposed Tumor Visualization and Identification Framework
\right\}\text{, }B:\text{ }\left\{\begin{array}{l}\frac{P_{s+1}}{1-P_{s}},\text  
Note that the term in the parenthesis in equation 2 encodes the entropy of one part \(\{P_{1},P_{2},P_{3},...,P_{s}\}\) of the original distribution \(\{P_{1},P_{2},P_{3},...,P_{s}\}\) and should be represented by \(H_{s}\), for simplicity, and, thus, the quantity \(H(A)\) takes the following form:  
\[H(A)=\ln P_{s}+\frac{H_{s}}{P_{s}} \tag{3}\]  
In a similar way,  
\[H(B)=-\sum_{s+1}^{n}\frac{p_{i}}{1-P_{s}}\ln\frac{p_{i}}{1-P_{s}}=\ln(1-P_{s}) +\frac{H_{n}-H_{s}}{1-P_{s}} \tag{4}\]  
where \(H_{n}\) refers to the entropy of the original distribution. The total entropy of the two individual distributions is defined by \(\psi\)(s) and is given as:  
\[\psi(s)=H(A)+H(B){=}\ln P_{s}(1-P_{s})+\frac{H_{s}}{P_{s}}+\frac{H_{n}-H_{s}}{ 1-P_{s}} \tag{5}\]  
The value of \"\(s\)\" that maximizes entropy value (maximum information retrieved) then accounts for the optimal threshold. The third segmentation technique employed in our proposed algorithmic scheme, the mean shift clustering, attempts to extract complementary information at a different level of data organization by closely analyzing the probability distribution of the sample fluorescence image. This method has been developed on the grounds of statistical clustering as applied in a feature space of the image itself. These features may reflect the intensity distribution (1D histogram space) alone or in combination with the spatial pixel distribution (defining a 3D space), or even the color distribution (in a 3D space). The principal motivation beyond mean shift clustering is the modeling and approximation of input data points as an empirical probability density function, according to which a large number of neighboring samples in feature space (dense regions) are translated into local maxima or modes (most frequent value met in a set of data points) of the corresponding statistical representation (distribution). For every individual (data point) in the processed population, a gradient ascent process is iteratively applied to the locally estimated density until the algorithm converges to an optimal solution. The data points associated with the same stationary point (mode) formulate dominant regions within the image (clusters that contain pixels close to each other and with similar intensity values). In our application, the mean shift approach operates in the 1D space of intensities. It initially defines a window around each point in the space of intensities and calculates its mean as the \"center of gravity\" based on the included intensities. In the second step, it translates (\"shifts\") this midpoint (center of the current window) to the mean value and repeats the procedure until the stopping criterion is met (convergence is achieved). Conceptually, in each step of the iterative calculations, the window shifts to a denser region of grayscale intensities, until it approximates a local maximum of the total distribution function. This iterative scheme repeatedly assigns every pixel (intensity point) to a class center (mean point) and intuitively calculates the number of dominant regions (segments) within the input signal/image via the final number of the cluster-centroids determined. The principal concepts of the mathematical background of the mean shift algorithm are briefly presented below. Kernel density estimation can estimate the density function of a random variable in a non-parametric mode. A kernel \(\varphi(\mathbf{x})\) is a non-negative function of a multidimensional feature vector, with which the vector is integrated over the domain of its definition. This is usually implemented as a Parzen window with its kernel \(K\)() representing a rectangular window operating in the space of the n data points \(\{x_{1},x_{2},\ldots,x_{n}\}\) and a bandwidth parameter \(h\) notation that sustains the physical meaning of the window size indicator.