Q14Digital Image Processing

Image Segmentation is the strict architectural process of mathematically separating objects of interest (Foreground) from the rest of the image (Background). Global Thresholding is the absolute fastest and most fundamental segmentation algorithm. It utilizes exactly one single, rigid mathematical constant ($T$) applied to every single pixel in the entire matrix, completely ignoring the pixel's physical spatial location.

The Mathematical Execution

For an input image $f(x,y)$, the algorithm violently generates a segmented binary image $g(x,y)$ using the strict logic:

$$g(x,y)=\{\begin{array}{ll}1\text{ (White)},& \text{if }f(x,y)>T\\ 0\text{ (Black)},& \text{if }f(x,y)\le T\end{array}$$

Determining the Global Threshold ($T$)

The absolute critical challenge is mathematically calculating the perfect value for $T$. The engineer must analyze the Image Histogram.

• Bimodal Histograms: If the image consists of a bright object on a dark background, the histogram will physically display two massive, distinct peaks separated by a deep valley. The engineer simply places $T$ exactly at the absolute lowest mathematical point of the valley, perfectly separating the two classes.
• Otsu's Method: A highly advanced algorithmic approach. Instead of guessing, Otsu's algorithm iteratively tests every single possible value from $0$ to $255$. It mathematically calculates the Within-Class Variance and violently selects the exact threshold $T$ that absolutely minimizes the variance inside the foreground and background classes, ensuring mathematical perfection without human intervention.

Catastrophic Limitations

Global Thresholding mathematically fails completely if the image suffers from uneven illumination (e.g., a shadow covers half the object). Because $T$ is globally rigid, the shadowed foreground will be misclassified as background. In such cases, Local Adaptive Thresholding is required.