Color Image Compression

Color image compression is the process of reducing the size of a color image file while maintaining acceptable visual quality. Since color images contain more data than grayscale images, compression plays a critical role in reducing storage and transmission costs. Various mathematical techniques are used to eliminate redundancy in color images, enabling efficient storage and faster transmission.

Components of Color Images

A color image is typically represented using three color channels: Red (R), Green (G), and Blue (B). Each pixel in a color image contains three values, representing the intensities of these primary colors. If each channel is represented by 8 bits (1 byte), a single pixel requires 3 bytes. For a high-resolution image with millions of pixels, this leads to a large file size.

Compression techniques aim to reduce this size by removing redundant or less important information.

Types of Redundancies in Color Images

1. Coding Redundancy:
   Certain pixel values may occur more frequently than others. By encoding these frequent values with fewer bits, we can compress the image. This is achieved through variable-length coding techniques like Huffman coding.

2. Interpixel Redundancy:
   Neighboring pixels in an image often have similar color values. Compression algorithms exploit this spatial redundancy by encoding differences between neighboring pixels rather than the actual pixel values.

3. Psychovisual Redundancy:
   The human eye is less sensitive to certain color details. Compression techniques exploit this by discarding information that the eye is less likely to notice, such as small color differences or high-frequency details in the image.

Lossy vs. Lossless Compression

• Lossless Compression: This method reduces file size without any loss of image quality. It works by removing only redundant data, allowing the original image to be perfectly reconstructed. Popular algorithms include PNG and GIF.

• Lossy Compression: This method reduces file size by discarding some image data that is less perceptible to the human eye. This type of compression achieves much higher compression ratios than lossless compression, but the image cannot be perfectly reconstructed. JPEG is the most widely used lossy compression algorithm.

Mathematical Concepts in Color Image Compression

1. Transform Coding Transform coding involves transforming the image data into a different domain where it is easier to compress. A popular transformation is the Discrete Cosine Transform (DCT), which separates an image into its frequency components.

   • Discrete Cosine Transform (DCT):
     DCT converts spatial domain pixel values into frequency domain coefficients. In this representation, most of the significant image information is concentrated in a few low-frequency components, while high-frequency components (which correspond to fine details and noise) are less important.

     Mathematical Expression: For an 8×8 block of pixels, the 2D DCT is defined as:

     $$F(u, v) = \frac{1}{4} \alpha(u) \alpha(v) \sum_{x=0}^{7} \sum_{y=0}^{7} I(x, y) \cos \left( \frac{(2x+1)u\pi}{16} \right) \cos \left( \frac{(2y+1)v\pi}{16} \right)$$

     where:

     • $I(x, y)$ is the pixel value at position $(x, y)$ in the 8×8 block.
     • $F(u, v)$ is the DCT coefficient for frequency components $(u, v)$.
     • $\alpha(u)$ and $\alpha(v)$ are normalization factors: $$\alpha(u) = \begin{cases} \frac{1}{\sqrt{2}} & \text{if } u = 0 \\ 1 & \text{if } u \neq 0 \end{cases}$$

     Example: After applying DCT to an 8×8 block, most of the energy (significant information) will be concentrated in the upper-left corner of the DCT matrix (low frequencies). The high-frequency coefficients near the bottom-right corner will likely be close to zero, meaning they can be discarded during compression without significantly affecting image quality.

2. Quantization Quantization is the process of reducing the precision of the DCT coefficients by rounding them to the nearest integer. This step reduces the number of bits needed to represent each coefficient, but it introduces some loss of information, making this a lossy step.

   Mathematical Expression: The quantized DCT coefficients are obtained as:

   $$Q(u, v) = \text{round}\left( \frac{F(u, v)}{Z(u, v)} \right)$$

   where $Z(u, v)$ is a quantization matrix, which assigns larger values to high-frequency components, effectively reducing their contribution to the compressed image.

   Example: Consider a simple quantization matrix:

   $$Z = \begin{bmatrix} 16 & 11 & 10 & 16 & 24 & 40 & 51 & 61 \\ 12 & 12 & 14 & 19 & 26 & 58 & 60 & 55 \\ 14 & 13 & 16 & 24 & 40 & 57 & 69 & 56 \\ 14 & 17 & 22 & 29 & 51 & 87 & 80 & 62 \\ 18 & 22 & 37 & 56 & 68 & 109 & 103 & 77 \\ 24 & 35 & 55 & 64 & 81 & 104 & 113 & 92 \\ 49 & 64 & 78 & 87 & 103 & 121 & 120 & 101 \\ 72 & 92 & 95 & 98 & 112 & 100 & 103 & 99 \end{bmatrix}$$

   The quantized coefficients will have much smaller values, particularly for the high-frequency components, allowing for efficient encoding and compression.

3. Entropy Coding After quantization, the remaining DCT coefficients are encoded using entropy coding methods such as Huffman coding or arithmetic coding. These methods assign shorter codes to more frequent coefficients, further compressing the data.

   Mathematical Concept: In Huffman coding, a binary tree is constructed where shorter codewords are assigned to more frequent symbols (quantized DCT coefficients), and longer codewords are assigned to less frequent symbols. The compressed bitstream is generated by replacing each symbol in the original image with its corresponding codeword from the Huffman tree.

   Example: For a set of quantized DCT coefficients like {0, 0, 1, 2, 0, -1, 0, 0}, the Huffman coding algorithm would assign shorter bit codes to frequently occurring coefficients like 0 and longer codes to less frequent ones like 2 and -1.

Example: JPEG Compression Algorithm

JPEG (Joint Photographic Experts Group) is one of the most widely used color image compression methods. It uses the DCT-based compression technique to achieve high compression ratios while maintaining acceptable visual quality.

1. Step 1: Color Space Conversion JPEG first converts the image from the RGB color space to the YCbCr color space. The Y component represents the luminance (brightness), while the Cb and Cr components represent chrominance (color information). Since the human eye is more sensitive to brightness than color, the chrominance channels can be subsampled to reduce the amount of color data without affecting perceived image quality.

2. Step 2: Block Splitting The image is divided into 8×8 pixel blocks, and DCT is applied to each block.

3. Step 3: DCT Transformation DCT transforms each 8×8 block from the spatial domain into the frequency domain, where most of the significant information is concentrated in the low-frequency coefficients.

4. Step 4: Quantization The DCT coefficients are quantized using a quantization matrix, reducing the number of bits needed to represent the image. High-frequency coefficients are heavily quantized (rounded to zero), resulting in a loss of fine detail that is often imperceptible to the human eye.

5. Step 5: Entropy Coding The quantized coefficients are encoded using entropy coding techniques (like Huffman coding) to produce the final compressed image file.

Example: If we apply the JPEG compression algorithm to a 24-bit color image (RGB) with dimensions 1024×768 pixels:

• The original file size would be approximately $1024 \times 768 \times 24 \approx 18$ MB.
• After compression using JPEG with a moderate quality setting, the file size might be reduced to around 1-2 MB, representing a compression ratio of about 9:1, with minimal perceptible loss in image quality.