CHAPTER 5 AND 6

Digital Image Processing tutorial · PySpark.in

CHAPTER 5 – IMAGE RESTORATION & RECONSTRUCTION

(From pages 317–398)

Image Restoration = Recovering an image that has been degraded
Objective, model-based, mathematical.
Different from enhancement (which is subjective).

5.1 IMAGE DEGRADATION / RESTORATION MODEL

The degradation process is modeled as:

Spatial domain:

$$g(x,y)=h(x,y)\ast f(x,y)+\eta (x,y)$$

Where:

• f(x,y) → Original image

• h(x,y) → Degradation (blur) function

• * (convolution)

• η(x,y) → Additive noise

• g(x,y) → Degraded image

Frequency domain:

$$G(u,v)=H(u,v)F(u,v)+N(u,v)$$

This is the foundation of all restoration techniques.

⭐ 5.2 NOISE MODELS

(From pp. 318–327)

Common noise types:

1. Gaussian Noise

• Distributed normally.

• Smooth appearance.

• Most common in sensors.

2. Rayleigh Noise

Occurs in radar images.

3. Gamma / Erlang Noise

Occurs when many small processes add.

4. Exponential Noise

5. Salt–and–Pepper Noise

• Random black & white dots.

• Caused by faulty memory, bit errors.

6. Uniform Noise

⭐ 5.3 RESTORATION USING SPATIAL FILTERS

(Noise only)

1. Mean Filters

• Arithmetic mean

• Harmonic mean

• Geometric mean
  Smooth image but blur edges.

2. Order-Statistics Filters

• Median filter → Very good for salt-pepper

• Max/Min filters

• Midpoint filter

• Alpha-trimmed mean

⭐ 5.4 PERIODIC NOISE REMOVAL (Frequency Domain)

(From p. 340)

Periodic noise appears as sharp spikes in Fourier domain.

Use:

• Notch reject filters

• Bandstop filters

Steps:

1. Compute DFT

2. Identify spikes

3. Place notches

4. Apply filter

5. Compute IDFT

⭐ 5.5 LINEAR, POSITION-INVARIANT DEGRADATION

(From pp. 348–351)

If blur is uniform and linear,

$$g(x,y)=h(x,y)\ast f(x,y)$$

Important: Spatial invariance → convolution → simple frequency multiplication.

⭐ 5.6 ESTIMATING THE DEGRADATION FUNCTION

(From p. 352)

Methods:

1. Observation – look at PSF

2. Experimentally – use test patterns

3. Mathematical modeling – motion blur, out-of-focus blur

4. Using inverse filtering concepts

⭐ 5.7 INVERSE FILTERING

(From p. 356)

Idea:

F ^ ( u , v ) = G ( u , v ) H ( u , v ) \hat F(u,v)= \frac{G(u,v)}{H(u,v)} F^(u,v)=H(u,v)G(u,v)

Very sensitive to noise → not used alone.

⭐ 5.8 WIENER FILTER (MMSE FILTER)

(From p. 358)

A robust alternative.

F ^ ( u , v ) = H ∗ ( u , v ) ∣ H ( u , v ) ∣ 2 + S η ( u , v ) S f ( u , v ) G ( u , v ) \hat F(u,v)= \frac{H^*(u,v)}{|H(u,v)|^2 + \frac{S_\eta(u,v)}{S_f(u,v)}}G(u,v) F^(u,v)=∣H(u,v)∣2+Sf(u,v)Sη(u,v)H∗(u,v)G(u,v)

Where:

• Uses power spectra of noise and image

• Minimizes mean square error

• Most widely used practical restoration method.

⭐ 5.9 CONSTRAINED LEAST SQUARES FILTER

(From p. 363)

Uses Laplacian for smoothness.

F ^ ( u , v ) = H ∗ ( u , v ) ∣ H ( u , v ) ∣ 2 + γ P ( u , v ) G ( u , v ) \hat F(u,v)= \frac{H^*(u,v)}{|H(u,v)|^2+\gamma P(u,v)} G(u,v) F^(u,v)=∣H(u,v)∣2+γP(u,v)H∗(u,v)G(u,v)

Where P(u,v) is Laplacian in frequency domain.

Better than inverse; slightly different than Wiener.

⭐ 5.10 GEOMETRIC MEAN FILTER

(From p. 367)

Generalizes Wiener + Inverse:

F ^ = [ H ∗ ∣ H ∣ 2 ] a [ H ∗ ∣ H ∣ 2 + S η S f ] 1 − a G \hat F= \left[\frac{H^*}{|H|^2}\right]^a \left[\frac{H^*}{|H|^2 + \frac{S_\eta}{S_f}}\right]^{1-a} G F^=[∣H∣2H∗]a∣H∣2+SfSηH∗1−aG

When:

• a=1 → inverse filter

• a=0 → parametric Wiener

• a=0.5 → geometric mean

⭐ 5.11 IMAGE RECONSTRUCTION FROM PROJECTIONS

(From pp. 368–398)

Important for Computed Tomography (CT).

Key steps:

1. Take multiple projections (raysums)

2. Backprojection – smear each projection

3. Filtered Backprojection (FBP)

   • Convolution with a filter

   • Removes blurring

Used in medical imaging widely.

⭐ CHAPTER 6 – COLOR IMAGE PROCESSING

(Pages 399–458)

Two motivations:

1. Humans perceive thousands of colors

2. Color is a powerful descriptor → segmentation, recognition

6.1 COLOR FUNDAMENTALS

(From pp. 400–404)

Visible Spectrum: 400–700 nm

• Violet → Blue → Green → Yellow → Orange → Red

• No sharp boundaries (smooth transitions)

Achromatic Light

• No color (only intensity): black → gray → white

Chromatic Light

Three quantities:

1. Radiance – energy leaving source

2. Luminance – perceived brightness

3. Brightness – subjective sensation

⭐ 6.2 COLOR MODELS

(From pp. 405–420)

Popular color models:

1. RGB – hardware, monitors

2. CMY/CMYK – printing

3. HSI (Hue, Saturation, Intensity) – human perception

4. CIE XYZ, Chromaticity diagram

   • Based on color matching experiments

   • Fundamental for representing color

   • Equation:

     $$x+y+z=1$$

Color Gamut

Triangle inside chromaticity diagram shows all colors a device can display (Fig. 6.6).

⭐ 6.3 PSEUDOCOLOR IMAGE PROCESSING

(From p. 420)

Assigning colors to grayscale images.

Methods:

1. Intensity Slicing

   • Assign color to specific ranges.

2. Color Coding

   • Map grayscale value → color based on lookup table.

Used in: Medical imaging, astronomy, satellite imagery.

⭐ 6.4 BASICS OF FULL-COLOR IMAGE PROCESSING

(From pp. 429–430)

Two approaches:

1. Per-component processing

   • Process R, G, B separately

   • Works only if operations are independent

2. Vector processing

   • Treat a pixel as a 3D vector

   • Needed for:

     • Median filtering

     • Smoothing

     • Sharpening

     • Segmentation

⭐ 6.5 COLOR TRANSFORMATIONS

(From p. 430)

General model:

s i = T ( r i ) s_i = T(r_i) si=T(ri)

Includes:

• Contrast stretching

• Tone correction

• Intensity/color correction

• Gamma correction

Used for fixing over/under-saturation, key type, etc.

⭐ 6.6 COLOR IMAGE SMOOTHING & SHARPENING

(From p. 442)

Two ways:

1. Apply filters on each channel (R, G, B)

2. Use vector filters on color voxels

⭐ 6.7 USING COLOR IN IMAGE SEGMENTATION

(color-based clustering)

Popular methods:

• K-means in color space

• Thresholding in HSI

• Region growing based on chromaticity

⭐ 6.8 NOISE IN COLOR IMAGES

(From p. 452)

• Noise may be independent across channels or correlated
• Use:

• Vector median

• Alpha-trimmed vector filter

• K-nearest color neighborhood

⭐ 6.9 COLOR IMAGE COMPRESSION

(From p. 455)

• RGB converted to YCbCr (JPEG standard)
• Chrominance subsampling (4:2:0) to reduce data
• Quantization + entropy coding

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