3. Describe Fourier transforms, Two-dimensional Fourier transforms (wiener filtering and Discrete cosine transform). Mention Applications.
Answer:
1. Fourier Transforms
Fourier Transform is a mathematical tool that transforms a spatial domain signal/image into its frequency domain representation. It is used to analyze the frequency content of images and filters.
Where:
- M,N: Image width and height
All 1D Fourier properties extend to 2D using vector inner products.
Used in:
- JPEG, MPEG, H.264
- Block-based compression (8×8 blocks)
DCT Basis Functions:
- First (DC) component: constant
- Higher: encode horizontal/vertical frequencies
- Used in image/video compression standards
5. Applications of Fourier Transforms
5.1. Sharpening and Noise Removal
- Performed using neighborhood processing.
- Traditionally done via linear filters (e.g., convolution).
- Modern techniques favor non-linear filters:
- Weighted median, bilateral filter, anisotropic diffusion, non-local means.
- Variational methods (e.g., total variation using L1 norm) are also used.
- Recently, deep neural networks dominate denoising tasks (Section 10.3).
5.2. Effectiveness Measurement of Denoising
- Common metric: Peak Signal-to-Noise Ratio (PSNR) — used when clean image is known.
- Better metrics:
- SSIM (Structural Similarity Index).
- FLIP (perceptual image difference).
- Neural perceptual similarity metrics — prefer texture-preserving results over overly smooth outputs.
- In absence of clean image: use no-reference image quality assessment.
5.3. Blur Removal and Super-resolution
- Covered in detail in Section 10.3.
- Use advanced models (often deep learning) for:
- Removing motion or defocus blur.
- Enhancing image resolution.
These show how filters attenuate or enhance different frequency ranges.