🔄 Iterative Thresholding

B.Sc. Computer Science Honours | Digital Image Processing & Image Segmentation

📘 1. Introduction

Iterative Thresholding, also called the Iterative Selection Method, is an image segmentation technique used to automatically determine a suitable threshold value from the intensity distribution of an image.

Instead of choosing the threshold manually, the algorithm starts with an initial estimate and repeatedly improves the threshold until the value becomes stable.

Core idea: Divide the image into two groups using an initial threshold, calculate the mean intensity of both groups, and use those means to calculate a new threshold.

Grayscale
Image

→

Divide into
Two Groups

→

Calculate
Means

→

New
Threshold

↻

Stable
Threshold

⚙️ 2. Basic Principle

Let the grayscale image contain pixel intensities represented by f(x,y).

Choose an initial threshold T. The pixels are divided into two groups:

G₁ = {f(x,y) > T}

G₂ = {f(x,y) ≤ T}

Calculate the mean intensity of each group:

μ₁ = Mean(G₁)      μ₂ = Mean(G₂)

Then calculate a new threshold:

T_new = (μ₁ + μ₂) / 2

The process continues until the threshold becomes stable.

🧠 3. Iterative Thresholding Algorithm

Step 1: Convert the image into a grayscale image.

Step 2: Select an initial threshold T.

Step 3: Divide the pixels into two groups G₁ and G₂.

Step 4: Calculate the mean intensity μ₁ of G₁.

Step 5: Calculate the mean intensity μ₂ of G₂.

Step 6: Calculate the new threshold: Tnew = (μ₁ + μ₂) / 2.

Step 7: Compare the new threshold with the previous threshold.

Step 8: If the difference is sufficiently small, stop. Otherwise repeat the process.

🔁 4. Complete Flow of Iterative Thresholding

Initial
T

→

G₁ & G₂

→

μ₁ & μ₂

→

Tnew

→

Compare

If the threshold has not converged: repeat the process. If it has converged: produce the segmented image.

🔢 5. Numerical Example

Consider the following simplified set of grayscale pixel values:

20, 30, 40, 50, 60, 150, 160, 170, 180, 190

Assume the initial threshold is:

T₀ = 100

Iteration 1

Using T = 100:

Group	Pixel Values	Mean
G₁ > 100	150, 160, 170, 180, 190	170
G₂ ≤ 100	20, 30, 40, 50, 60	40

New threshold:

T₁ = (170 + 40) / 2 = 105

Iteration 2

Using T = 105, the groups remain unchanged.

μ₁ = 170      μ₂ = 40

T₂ = (170 + 40) / 2 = 105

Since: T₂ = T₁ = 105, the threshold has converged.

Final Threshold = 105

🎯 6. Convergence Condition

The algorithm stops when the difference between consecutive threshold values becomes sufficiently small.

|T_new − T_old| < ε

Here, ε is a small tolerance value.

In a simple implementation, the algorithm can also stop when Tnew = Told.

📐 7. Mathematical Formulation

Let the image contain N pixels and let the threshold at iteration k be T_k.

The two groups are:

G₁(Tₖ) = {f(x,y) | f(x,y) > Tₖ}

G₂(Tₖ) = {f(x,y) | f(x,y) ≤ Tₖ}

The group means are:

μ₁(Tₖ) = Mean[G₁(Tₖ)]

μ₂(Tₖ) = Mean[G₂(Tₖ)]

Then:

Tₖ₊₁ = [μ₁(Tₖ) + μ₂(Tₖ)] / 2

🧪 8. Interactive Iterative Thresholding Calculator

Enter grayscale pixel values: Initial Threshold:

Enter pixel values and click Calculate Iterations.

🧩 9. 8×8 Image Matrix Demonstration

The following interactive matrix represents a simplified grayscale image. Iterative thresholding will automatically calculate a threshold from the matrix.

Generate a matrix and run the algorithm.

🐍 10. Beginner Python Implementation

The following Python program implements iterative thresholding without using a built-in automatic thresholding function.

import cv2 import numpy as np # Read image image = cv2.imread("input.jpg") # Convert to grayscale gray = cv2.cvtColor(image, cv2.COLOR_BGR2GRAY) # Initial threshold T = 128 while True: # Group 1: pixels greater than threshold G1 = gray[gray > T] # Group 2: pixels less than or equal to threshold G2 = gray[gray <= T] # Calculate means if len(G1) == 0 or len(G2) == 0: break mean1 = np.mean(G1) mean2 = np.mean(G2) # Calculate new threshold new_T = (mean1 + mean2) / 2 print("Old T:", T) print("Mean G1:", mean1) print("Mean G2:", mean2) print("New T:", new_T) print("----------------") # Check convergence if abs(new_T - T) < 0.5: T = new_T break T = new_T # Convert threshold to integer T = int(round(T)) print("Final Threshold:", T) # Create binary image _, binary = cv2.threshold( gray, T, 255, cv2.THRESH_BINARY ) cv2.imshow("Original", image) cv2.imshow("Iterative Threshold", binary) cv2.waitKey(0) cv2.destroyAllWindows()

🐍 11. Python Code Explanation

cv2.imread(): Reads the input image.

cv2.cvtColor(): Converts the image into grayscale.

G1: Contains pixels greater than the current threshold.

G2: Contains pixels less than or equal to the current threshold.

np.mean(): Calculates the average intensity of each group.

new_T: New threshold obtained from the two group means.

cv2.threshold(): Produces the final binary image.

📊 12. Example Iteration Table

Iteration	Old T	μ₁	μ₂	New T
1	100	170	40	105
2	105	170	40	105

The algorithm stops because the threshold has become stable.

📊 13. Global vs Iterative Thresholding

Feature	Global Thresholding	Iterative Thresholding
Threshold	Usually manually selected	Calculated iteratively
Process	Single threshold operation	Repeated refinement
Automatic Selection	Not necessarily	Yes, from image statistics
Computation	Low	Higher than one-pass thresholding
Result	Binary segmentation	Binary segmentation using converged T

📊 14. Iterative Thresholding vs Otsu's Method

Feature	Iterative Thresholding	Otsu's Method
Basic Idea	Repeatedly updates T using group means	Selects T using a histogram-based class-separation criterion
Starting Value	Usually requires an initial T	Evaluates candidate thresholds
Iterations	Yes	Not iterative in the same sense
Statistics	Two class means	Class probabilities and variances
Automatic	Yes	Yes

🌍 15. Applications

📄 Document Processing

Separating text and background in scanned documents.

🔢 OCR

Preparing characters for optical character recognition.

🧬 Medical Image Analysis

Separating regions of interest based on intensity.

🏭 Industrial Inspection

Separating objects or defects from backgrounds.

🛰️ Remote Sensing

Intensity-based separation of image regions.

🔬 Scientific Images

Segmenting objects with distinguishable intensity ranges.

✅ 16. Advantages

1. Automatically estimates a threshold from image statistics.

2. Simple mathematical concept.

3. Does not require testing every possible threshold as its basic operation.

4. Easy to implement using NumPy and OpenCV.

5. Useful when foreground and background have reasonably distinct intensity distributions.

⚠️ 17. Limitations

1. The method depends on the initial threshold and image intensity distribution.

2. It may perform poorly when foreground and background intensities overlap strongly.

3. Uneven illumination can reduce segmentation quality.

4. Noise can influence the calculated group means.

5. It is fundamentally a two-class thresholding approach in its basic form.

⏱️ 18. Computational Consideration

If the image contains N pixels and the algorithm performs K iterations, a straightforward implementation requires approximately:

O(N × K)

In practice, the number of iterations is usually relatively small for many simple images, but it depends on the image distribution, initial threshold and stopping condition.

📝 19. Algorithm in Short

Choose
T₀

→

Create
G₁,G₂

→

Find
μ₁,μ₂

→

Tnew = (μ₁+μ₂)/2

→

Converged?

No → Repeat      Yes → Segment Image

🎓 20. Important Examination Points

1. Iterative thresholding automatically estimates a threshold from image intensity statistics.

2. The image is divided into two groups using the current threshold.

3. The means of the two groups are calculated.

4. The new threshold is: Tnew = (μ₁ + μ₂) / 2.

5. The process continues until the threshold converges.

6. It is useful for image segmentation when foreground and background have reasonably different intensity distributions.

📌 21. Quick Revision Table

Concept	Key Point
Initial Threshold	Starting estimate of T.
G₁	Pixels greater than T.
G₂	Pixels less than or equal to T.
μ₁	Mean intensity of G₁.
μ₂	Mean intensity of G₂.
New Threshold	Tnew = (μ₁ + μ₂) / 2.
Stopping Condition	|Tnew − Told| < ε.
Final Result	Thresholded / segmented image.

CREATED BY BIJAN KRISHNA PAUL