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unit 2 - Image quantization and Image Transforms

1. Sampling Theorem

 Definition:

The Sampling Theorem (Shannon-Nyquist) states that a continuous signal can be perfectly reconstructed from its samples if it is sampled at twice the maximum frequency present in the signal.

 Formula:

fs≥2fmaxf_s \geq 2f_{max}

where:

  • fsf_s = sampling frequency

  • fmaxf_{max} = highest frequency component in the signal

 Application:

Used in digitizing analog images to ensure no information is lost during sampling.

2. Anti-Aliasing

Aliasing:

Occurs when sampling is done below the Nyquist rate, leading to overlapping frequency components and distortion.

Anti-Aliasing:

A process to suppress high frequencies before sampling using low-pass filters to prevent aliasing.

3. Image Quantization

 Definition:

Process of mapping a range of continuous pixel values to a finite number of levels.

 Types:

  • Scalar Quantization: Each pixel is quantized independently.

  • Vector Quantization: Blocks of pixels are quantized together.

Quantization Error:

Error=Original Pixel−Quantized Pixel\text{Error} = \text{Original Pixel} - \text{Quantized Pixel}

Too few levels → loss of detail, visible banding.

4. Orthogonal and Unitary Transforms

 Orthogonal Transform:

A linear transformation using an orthogonal matrix TT where:

TTT=IT^T T = I

  • Preserves energy.

  • Examples: DFT, DCT, Haar, Hadamard.

 Unitary Transform:

A generalization using complex numbers:

THT=IT^H T = I

where THT^H is the conjugate transpose.

5. Discrete Fourier Transform (DFT)

Formula:

F(u,v)=∑x∑yf(x,y)⋅e−j2π(uxM+vyN)F(u,v) = \sum_x \sum_y f(x,y) \cdot e^{-j2\pi \left(\frac{ux}{M} + \frac{vy}{N}\right)}

  • Converts spatial image to frequency domain.

  • Captures periodic patterns.

  • Used in filtering, compression.

6. Discrete Cosine Transform (DCT)

Formula (1D):

Xk=∑n=0N−1xn⋅cos⁡[πN(n+0.5)k]X_k = \sum_{n=0}^{N-1} x_n \cdot \cos\left[\frac{\pi}{N}(n + 0.5)k\right]

  • Like DFT, but uses only real cosine terms.

  • Energy compaction is high → used in JPEG compression.

7. Hadamard Transform

 Properties:

  • Uses only +1 and −1 (binary values).

  • Fast to compute (no multiplications).

  • Not based on sinusoidal functions.

 Matrix:

Hadamard matrix is recursively defined:

H2=[111−1],H2n=[H2n−1H2n−1H2n−1−H2n−1]H_2 = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}, \quad H_{2^n} = \begin{bmatrix} H_{2^{n-1}} & H_{2^{n-1}} \\ H_{2^{n-1}} & -H_{2^{n-1}} \end{bmatrix}

8. Haar Transform

 Properties:

  • Simplest wavelet transform.

  • Breaks signal into approximation and detail parts.

  • Useful for multi-resolution analysis.

 Steps:

  • Divide signal into pairs.

  • Calculate average and difference.

  • Recurse on averages.

9. Karhunen-Loeve Transform (KLT / PCA)

 Definition:

A statistical transform that decorrelates data. Also known as Principal Component Analysis (PCA).

 Steps:

  1. Calculate covariance matrix.

  2. Compute eigenvalues and eigenvectors.

  3. Transform data using eigenvectors.

 Advantage:

  • Optimal energy compaction.

  • Basis vectors are data-dependent.

  • Used in face recognition, compression.

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