Image Compression in MATLAB: Implementing the JPEG Compression Algorithm

# 2.1 Discrete Cosine Transform (DCT)

### 2.1.1 Mathematical Principles of DCT

The Discrete Cosine Transform (DCT) is an orthogonal transformation that converts time-domain signals (image pixels) into frequency-domain signals (DCT coefficients). The mathematical principles of DCT are as follows:

F(u, v) = α(u)α(v)
    ∑∑ f(x, y) cos[(2x + 1)uπ/2N] cos[(2y + 1)vπ/2N]

where:

* `F(u, v)` are the DCT coefficients
* `f(x, y)` are the image pixels
* `α(u)` and `α(v)` are normalization constants
* `N` is the image size

### 2.1.2 Practical Applications of DCT

DCT plays a crucial role in image compression because it can concentrate the energy of the image in the low-frequency DCT coefficients. By quantizing and encoding these low-frequency coefficients, the size of the image file can be effectively reduced.

# 2. Principles of JPEG Compression Algorithm

The JPEG compression algorithm is a lossy compression algorithm that achieves compression by discarding unimportant information from the image. The JPEG compression algorithm is mainly divided into three steps: Discrete Cosine Transform (DCT), Quantization, and Encoding.

### 2.1 Discrete Cosine Transform (DCT)

#### 2.1.1 Mathematical Principles of DCT

The Discrete Cosine Transform (DCT) is a mathematical transformation that converts an image from the spatial domain (pixel values) to the frequency domain (cosine coefficients). The mathematical formula for DCT is as follows:

F(u, v) = \frac{1}{4}C(u)C(v)\sum_{x=0}^{N-1}\sum_{y=0}^{N-1}f(x, y)\cos\left[\frac{(2x+1)u\pi}{2N}\right]\cos\left[\frac{(2y+1)v\pi}{2N}\right]

where:

* `f(x, y)` is the spatial domain image
* `F(u, v)` is the frequency domain image
* `N` is the size of the image
* `C(u)` and `C(v)` are normalization constants

DCT decomposes the image into a weighted sum of cosine functions. The low-frequency components (such as the average of the image) are located in the upper left corner of the frequency domain, while the high-frequency components (such as the edges and textures of the image) are located in the lower right corner.

#### 2.1.2 Practical Applications of DCT

DCT is used in the JPEG compression algorithm to transform the image from the spatial domain to the frequency domain. By decomposing the image into a weighted sum of cosine functions, the JPEG compression algorithm can identify and discard unimportant information in the image.

### 2.2 Quantization

Quantization is the second step of the JPEG compression algorithm. The quantization process divides the DCT coefficients by values in a quantization table, thereby reducing the precision of the coefficients. The larger the values in the quantization table, the more information is discarded.

#### 2.2.1 Design of Quantization Table

A quantization table is an 8x8 matrix that specifies the quantization values for each DCT coefficient. The design of the quantization table greatly affects the performance of the JPEG compression algorithm. A good quantization table can minimize image distortion while maintaining a high compression rate.

#### 2.2.2 Quantization Process

The quantization process divides the DCT coefficients by the values in the quantization table. The coefficients after quantization are called quantized coefficients. Quantized coefficients are usually integers, which helps reduce the number of bits required for encoding.

### 2.3 Encoding

Encoding is the third step of the JPEG compression algorithm. The encoding process converts the quantized coefficients into a bitstream. The JPEG compression algorithm supports two encoding methods: lossless Huffman coding and arithmetic coding.

#### 2.3.1 Lossless Huffman Coding

Lossless Huffman coding is a lossless data compression algorithm. It assigns bit codes to quantized coefficients based on their frequency. The more frequently occurring quantized coefficients are assigned shorter bit codes.

#### 2.3.2 Arithmetic Coding

Arithmetic coding is a more efficient lossless data compression algorithm than Huffman coding. It represents the quantized coefficients as a fraction and encodes the fraction using arithmetic operations.

# 3.1 Usage of DCT Function

**3.1.1 dct2 Function**

The `dct2` function provided by MATLAB is used to calculate the Discrete Cosine Tran