Image Processing and Computer Vision

Image Compression using JPEG

• JPEG (Joint Photographic Experts Group) is a standard for compressing image files.
• The main goal of compression is to reduce the file size, which is proportional to the storage requirement.

Color Space Conversion

• In JPEG images, RGB (Red, Green, Blue) is converted to YCbCr (Luminance and Chrominance) color space.
• This is because humans are more sensitive to brightness than color differences.
• The chroma (color) channels are subsampled to reduce the amount of data allocated to the image.

DCT (Discrete Cosine Transform)

• DCT is used to express the image as a linear combination of basis functions.
• The first step in DCT is to split the image into 8x8 blocks.
• DCT coefficients are generated using a sample code.
• The zigzag scan is used to map the 8x8 matrix to a 1x64 vector.

Quantization

• Quantization reduces the precision of the DCT coefficients.
• The JPEG algorithm uses a quantization matrix to eliminate high-frequency components.

DCT Coefficients Encoding

• AC (Alternating Current) coefficients are encoded using run-length encoding and Huffman coding.
• DC (Direct Current) coefficients are encoded using Huffman coding.
• Huffman coding represents frequent patterns with low-bit codes and less frequent patterns with high-bit codes.

Decoding

• Decoding involves inverse quantization, inverse DCT, and color space conversion back to RGB.

Compression Ratio and Artifacts

• Compression ratio is the ratio of the original image size to the compressed image size.
• Artifacts (distortions) appear after lossy compression and are proportional to the compression ratio and inversely proportional to image quality.

Fourier Transform

• Fourier transform expresses a signal as a weighted sum of sinusoids.
• It transforms a signal from the time domain to the frequency domain.

1D Fourier Transform

• The discrete Fourier transform is given by X(k) = Σx(i) * e^(-j2πki/N)
• X(k) is a complex number that encodes information about amplitude and phase.

2D Discrete Fourier Transform (2D DFT)

• The 2D DFT is used for image processing and is given by X(k, L) = Σx(i, j) * e^(-j2π(ki/M + Lj/N))### Fourier Transform
• 2D discrete Fourier transform (2D DFT) is used for image reconstruction.
• Top-left coefficients in 2D DFT represent low-frequency components containing information on the general shape of the image.
• Low-frequency components retain a high percentage of the signal energy, resulting in larger amplitude compared to high-frequency components.

Image Reconstruction using 2D DFT

• Considering top-left coefficients (low-frequency components) yields better reconstruction results.
• Considering bottom-right coefficients (high-frequency components) yields minor reconstruction results.

Calculating Amplitude and Phase

• 𝑋𝑑 is a complex number encoding amplitude and phase information of a complex sinusoidal component.
• Amplitude (𝐴𝑚𝑝) and Phase (𝑃ℎ) are calculated using 𝑅𝑒(𝑋𝑑) and 𝐼𝑚(𝑋𝑑).

Normalization and Visualization

• Original image phase information does not provide significant information.
• Amplitude image is mostly black due to out-of-range values.
• Normalization is done using 𝒇 𝒙, 𝒚 = 𝒍𝒐𝒈(𝟏 + 𝑨𝒎𝒑 𝒙, 𝒚) to improve visualization.
• Low-frequency components are shifted to the center for better visualization.

Comparison with DCT

• JPEG uses DCT (Discrete Cosine Transform) instead of DFT because DCT concentrates energy in a small number of coefficients.
• Fast Fourier Transform (FFT) is an efficient version of DFT.

Low and High Pass Filters

• Low pass filter in the frequency domain is used for smoothing the image by attenuating high-frequency components.
• Ideal low pass filter is designed using a cut-off frequency C and applied element-wise to the transformed image.

Designing Ideal Low Pass Filter

• Filter H(u, v) is designed to exclude values greater than C, where C is the cut-off frequency.
• Distance 𝑫(𝒖, 𝒗) = 𝒖² + 𝒗² is used to determine the filter values.

High Pass Filter

• High pass filter in the frequency domain is used for detecting edges of the image by attenuating low-frequency components.
• Ideal high pass filter is the inverse of the ideal low pass filter.