Thresholding
Global Thresholding
Adaptive Thresholding
Otsu's Method
Edge detection comparison (Wikimedia Commons CC)
Thresholding: Grayscale pixels → Binary output based on threshold T
Definition
Thresholding converts a grayscale image into a binary image (0 or 1) by comparing each pixel intensity to a threshold value \(T\).
Definition
$$
g(x,y) = \begin{cases} 1 & \text{if } f(x,y) > T \\ 0 & \text{if } f(x,y) \leq T \end{cases}
$$
Where \(f(x,y)\) is the input pixel intensity and \(g(x,y)\) is the output binary pixel.
How it works:
- Convert image to grayscale (if color)
- Compute histogram of pixel intensities
- Select a threshold value \(T\)
- Apply the formula above to every pixel
Key: Global thresholding uses a single value of \(T\) for the entire image. Works only when lighting is uniform.
When to use:
- Image has clearly separable foreground and background
- Histogram shows two distinct peaks (bimodal)
- Lighting is uniform across the image
Exam: If asked "Draw the binary output" — just apply the threshold to each pixel and draw 0/1 grid.
Adaptive (Local) Thresholding
Computes a different threshold \(T\) for each pixel based on its local neighbourhood.
$$
T(x,y) = \frac{1}{|N|}\sum_{(s,t) \in N_{xy}} f(s,t) - C
$$
Where:
- \(N_{xy}\) = neighbourhood of pixel \((x,y)\) (e.g., 15×15 window)
- \(|N|\) = number of pixels in neighbourhood
- \(C\) = constant subtracted (e.g., 5–15)
Variants:
| Method | T Calculation |
|---|---|
| Mean | Average of neighbourhood |
| Weighted (Gaussian) | Gaussian-weighted average (center pixels weigh more) |
| Median | Median of neighbourhood values |
Window Size Effect:
- Small window (e.g., 3×3): Captures fine details, more noise sensitive
- Large window (e.g., 31×31): Smoother result, may miss small objects
Advantage: Handles non-uniform illumination, shadows, and varying contrast across the image.
Disadvantage: Computationally expensive. Window size \(k\) is critical — too small = noisy, too large = lose detail.
Otsu's Method (Automatic Global)
Fully automatic global thresholding. Finds optimal \(T\) that maximizes between-class variance and minimizes within-class variance.
Exam ImportantPrerequisites:
- Histogram must be bimodal (two peaks with a valley between)
- Two classes: foreground (dark) and background (light)
Algorithm:
- Compute histogram of image, normalize to get probabilities \(p_i\) for intensity \(i\)
- For each candidate threshold \(k = 0, 1, \ldots, L-1\):
$$
\omega_0(k) = \sum_{i=0}^{k} p_i \quad \text{(class 0 probability)}
$$
$$
\omega_1(k) = \sum_{i=k+1}^{L-1} p_i = 1 - \omega_0(k) \quad \text{(class 1 probability)}
$$
$$
\mu_0(k) = \frac{\sum_{i=0}^{k} i \cdot p_i}{\omega_0(k)} \quad \text{(class 0 mean)}
$$
$$
\mu_1(k) = \frac{\sum_{i=k+1}^{L-1} i \cdot p_i}{\omega_1(k)} \quad \text{(class 1 mean)}
$$
$$
\mu_T = \sum_{i=0}^{L-1} i \cdot p_i \quad \text{(global mean)}
$$
Between-Class Variance:
$$
\sigma_B^2(k) = \omega_0(k)\left[\mu_0(k) - \mu_T\right]^2 + \omega_1(k)\left[\mu_1(k) - \mu_T\right]^2
$$
Simplified form:
$$ \sigma_B^2(k) = \omega_0(k) \cdot \omega_1(k) \cdot \left[\mu_0(k) - \mu_1(k)\right]^2 $$Within-Class Variance:
$$
\sigma_W^2(k) = \omega_0(k)\sigma_0^2(k) + \omega_1(k)\sigma_1^2(k)
$$
Total variance: \(\sigma_T^2 = \sigma_B^2 + \sigma_W^2\) (constant)
Optimal threshold \(T^*\): value of \(k\) that maximizes \(\sigma_B^2(k)\) (equivalently minimizes \(\sigma_W^2\)).
Exam: "What is Otsu's method?" = Automatic threshold selection for bimodal histograms by maximizing between-class variance. "When does it fail?" = Non-bimodal histograms, uneven illumination.
