🖼️ JPEG Image Compression

🖼️ JPEG Image Compression

B.Sc. Computer Science Honours – Image Processing & Data Compression

📚 1. Introduction to JPEG

JPEG stands for Joint Photographic Experts Group. It is one of the most widely used image compression standards for photographic and natural images.

JPEG compression primarily exploits two important properties of images:

• Spatial redundancy between neighbouring pixels.
• Psychovisual redundancy caused by limitations of human vision.

JPEG is generally associated with lossy compression. The quantization stage intentionally discards some image information that is considered less visually significant.

📷 2. Interactive JPEG Demonstration

Upload an image to inspect the major stages of JPEG-style processing.

🖼️ 3. Original and Grayscale Image

Original Image

Grayscale Image

🔲 4. 8 × 8 Image Block

JPEG processes an image in small blocks. The classical JPEG transform uses an 8 × 8 pixel block.

The selected block below is converted to grayscale values and shifted by 128 before applying the DCT.

📐 5. Discrete Cosine Transform (DCT)

The DCT converts spatial pixel information into frequency-domain coefficients.

F(u,v) = 1/4 C(u)C(v) Σₓ₌₀⁷ Σᵧ₌₀⁷ f(x,y) cos[(2x+1)uπ/16] cos[(2y+1)vπ/16]

where:

• f(x,y) = spatial-domain pixel value.
• F(u,v) = DCT coefficient.
• u,v = frequency indices.
• C(0) = 1/√2.
• C(k) = 1 for k > 0.

The upper-left coefficient F(0,0) is the DC coefficient. The remaining coefficients are called AC coefficients.

🎯 6. Quantization

Quantization is the main lossy stage of JPEG compression. Each DCT coefficient is divided by a corresponding value in a quantization matrix and rounded to an integer.

Q(u,v) = round( F(u,v) / T(u,v) )

where T(u,v) is the quantization matrix.

Larger quantization values produce stronger compression and generally more information loss. Smaller values preserve more detail but produce less compression.

↘️ 7. Zig-Zag Scanning

After quantization, the 8 × 8 coefficient matrix is rearranged using a zig-zag scan.

The zig-zag ordering places low-frequency coefficients near the beginning and high-frequency coefficients toward the end.

High-frequency coefficients are often quantized to zero, creating long runs of zeros that can subsequently be compressed efficiently.

🔄 8. Complete JPEG Compression Pipeline

RGB Image → Color Space Conversion → Chroma Subsampling → 8×8 Blocks → Level Shift → DCT → Quantization → Zig-Zag Scan → Run-Length Coding → Entropy Coding → JPEG Bitstream

The exact JPEG process contains additional details, but these stages represent the main conceptual pipeline taught in image processing and data compression courses.

📖 9. Detailed Theory for B.Sc. Computer Science Honours

9.1 Why Does JPEG Compress Images?

Digital images contain a large amount of redundant information. Neighbouring pixels in natural images are often strongly correlated. JPEG transforms this spatial redundancy into frequency coefficients and then represents visually less important information with reduced precision.

9.2 RGB to YCbCr

JPEG commonly uses the YCbCr colour representation. The image is separated into:

• Y – luminance component.
• Cb – blue-difference chrominance component.
• Cr – red-difference chrominance component.

Human vision is generally more sensitive to changes in brightness than to fine chrominance detail. Therefore, chroma components can often be sampled at a lower resolution.

9.3 Chroma Subsampling

JPEG can reduce the spatial resolution of chrominance components using schemes such as:

• 4:4:4
• 4:2:2
• 4:2:0

For example, 4:2:0 uses fewer chroma samples relative to luminance. This reduces the amount of information that needs to be encoded.

9.4 Blocking

The image is divided into blocks, traditionally 8 × 8 pixels. Each block is processed independently by the DCT.

8 × 8 = 64 samples per block

Block processing makes the transform computationally manageable and allows frequency information to be represented locally.

9.5 Level Shifting

For an 8-bit image, pixel values normally range from 0 to 255. Before applying the DCT, JPEG conceptually shifts the values by 128:

f'(x,y) = f(x,y) - 128

Therefore the transformed values approximately range from −128 to +127.

9.6 Frequency Components

The DCT represents the block using different spatial frequencies.

• Low frequencies: gradual changes and broad image structures.
• High frequencies: rapid changes, edges and fine details.

The DC coefficient represents the average level of the block, while AC coefficients describe variations around that average.

9.7 DC and AC Coefficients

DC → F(0,0)

AC → F(u,v), where (u,v) ≠ (0,0)

The DC coefficient is normally encoded relative to the DC coefficient of the previous block. AC coefficients are processed using the zig-zag order and run-length representation.

9.8 Quantization and Loss

Quantization is responsible for the principal information loss in baseline JPEG.

Consider:

DCT coefficient = 83
Quantization value = 10

83 / 10 = 8.3
round(8.3) = 8

The original value 83 cannot generally be reconstructed exactly from the quantized value 8 and the quantization value 10.

Important: The DCT itself is a reversible transform in the ideal mathematical sense. The major intentional loss in JPEG compression occurs during quantization.

9.9 Zig-Zag Ordering

A typical zig-zag sequence starts near the DC coefficient and gradually moves toward higher-frequency coefficients.

This arrangement tends to group many zero-valued high-frequency coefficients near the end of the sequence after quantization.

9.10 Run-Length Coding in JPEG

After zig-zag scanning, quantized AC coefficients are represented using run-length concepts.

A typical representation describes:

(RUNLENGTH, SIZE, AMPLITUDE)

where RUNLENGTH specifies the number of preceding zero coefficients.

The special EOB (End of Block) symbol can indicate that the remaining coefficients in the block are zero.

9.11 Entropy Coding

JPEG commonly uses entropy coding after quantization and zig-zag processing. In baseline JPEG, Huffman coding is commonly used.

Thus, JPEG combines transform coding, quantization, run-length representation and entropy coding.

9.12 JPEG Compression Ratio

The compression ratio can be expressed as:

Compression Ratio = Original Data Size / Compressed Data Size

Higher compression usually requires stronger quantization, which can increase visible distortion.

9.13 JPEG Quality Factor

Many JPEG software tools expose a quality setting. A higher quality setting generally corresponds to less aggressive quantization and larger files, while lower quality settings generally use stronger quantization and produce smaller files.

The exact mapping between a software quality value and the resulting JPEG quantization tables is implementation-dependent.

9.14 JPEG Artifacts

Strong JPEG compression can introduce visible artifacts, including:

• Blocking artifacts
• Ringing near sharp edges
• Loss of fine texture
• Blurring
• Colour distortions

9.15 Advantages of JPEG

• Excellent for photographic and natural images.
• Provides substantial file-size reduction.
• Supported by almost all image-processing platforms.
• Allows adjustable compression quality.
• Uses well-studied transform and entropy coding techniques.
• Suitable for web images, photography and digital media.

9.16 Limitations of JPEG

• Standard JPEG compression is lossy.
• Repeated recompression can accumulate visible degradation.
• Not ideal for line drawings or images containing sharp text.
• Strong compression can produce blocking artifacts.
• Fine details may be lost during quantization.

9.17 JPEG vs Lossless Compression

Feature	JPEG	Lossless Compression
Information loss	Normally lossy in photographic JPEG compression	No information loss
Compression	Can achieve high compression	Usually lower compression for natural images
Quantization	Yes	No lossy quantization
Typical use	Photographs and natural images	Medical, technical or archival data where exact recovery matters

9.18 Important Examination Points

• JPEG stands for Joint Photographic Experts Group.
• JPEG is primarily designed for photographic and natural images.
• JPEG commonly uses the 8 × 8 DCT.
• DCT converts spatial information into frequency coefficients.
• The DC coefficient is F(0,0).
• Other coefficients are AC coefficients.
• Quantization is the principal lossy stage.
• Zig-zag scanning orders quantized coefficients.
• Run-length and entropy coding further reduce the representation.
• JPEG commonly uses YCbCr and may use chroma subsampling.
• High compression can produce blocking and other visual artifacts.

📊 10. Image Information

Upload an image to display its dimensions and processing information.

🧠 11. JPEG Concept Summary

Image → YCbCr → Chroma Subsampling → 8×8 Blocks → Level Shift → DCT → Quantization → Zig-Zag → Run-Length Coding → Entropy Coding → JPEG Bitstream

The central idea is to transform image data into frequency components, reduce the precision of less visually significant information, and then encode the resulting coefficients efficiently.

CREATED BY BIJAN KRISHNA PAUL B.Sc. Computer Science Honours • Image Processing • Data Compression • JPEG

```