Arithmetic Operations in Image Processing - Part 1

🧠 Core idea
Arithmetic operations treat an image as a matrix of values.
By combining pixel values in different ways, we can control brightness, contrast, noise, and details.

This post summarizes the most commonly used arithmetic operations in image processing,
with formulas written in kramdown / MathJax-safe Markdown.

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🖼️ Image as Numbers

An image can be viewed as:

• Grayscale: \(I(x,y) = [0,255] \; or \; [0,1]\)
• Color: operations are applied per channel (R, G, B)

All arithmetic operations are applied pixel-wise.

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1️⃣ Addition (Add)

Formula

\[I_{out}(x) = I_1(x) + I_2(x)\]

Effect

• Increases brightness
• Combines information from two images
• Noise is also amplified

Typical Uses

• Brightness adjustment
• Frame accumulation / averaging
• Original image + edge map (sharpening)

Notes

• Values may overflow → clipping required

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2️⃣ Subtraction (Subtract)

Formula

\[I_{out}(x) = I_1(x) - I_2(x)\]

Effect

• Highlights differences
• Removes common background
• Emphasizes changes

Typical Uses

• Background subtraction
• Frame differencing
• Defect detection

Notes

• Negative values may appear
• Often followed by absolute value

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3️⃣ Absolute Difference

Formula

\[I_{out}(x) = | I_1(x) - I_2(x) |\]

Effect

• Direction-independent difference
• Clear visualization of change

Typical Uses

• Motion detection
• Change detection maps

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4️⃣ Multiplication (Multiply)

Formula

\[I_{out}(x) = I(x) \cdot c\]

Effect

• Adjusts contrast
• Dark regions get darker faster

Typical Uses

• Contrast scaling
• Masking (ROI weighting)

Notes

• Requires normalization

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5️⃣ Division (Divide)

Formula

\[I_{out}(x) = \frac{I_1(x)}{I_2(x) + \epsilon}\]

Effect

• Illumination normalization
• Enhances relative contrast

Typical Uses

• Flat-field correction
• Illumination compensation

Notes

• ( \epsilon ) avoids division by zero
• Noise can be amplified in dark areas

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6️⃣ Weighted Addition (Blending)

Formula

\[I_{out}(x) = \alpha I_1(x) + (1-\alpha) I_2(x)\]

Effect

• Smooth blending
• Controlled combination

Typical Uses

• Image blending
• Overlay visualization

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7️⃣ Power / Square Operation

Formula

\[I_{out}(x) = I(x)^2\]

Effect

• Enhances strong responses
• Suppresses weak noise

Typical Uses

• Energy maps
• Feature emphasis

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🧠 Practical Summary

Operation	Core Effect	Typical Use
Add	Brightness ↑	Accumulation
Subtract	Difference	Background removal
Abs Diff	Change only	Motion detection
Multiply	Contrast	Masking
Divide	Normalize	Illumination fix
Blend	Mix images	Visualization
Square	Emphasize peaks	Feature energy

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✨ Final Takeaway

Arithmetic operations are simple but powerful.
When combined with filtering and morphology,
they form the foundation of most classical image-processing pipelines.

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