JPEG Image Compression

• JPEG stands for Joint Photographic Experts Group
• JPEG was the first international static image compression standard, published in 1992
• The quality of its output for a relatively good compression ratio is the main reason for JPEG's success
• JPEG is a lossy image compression method that uses a transform coding method with the DCT (Discrete Cosine Transform)
• An image is a function of i and j (or x and y) in the spatial domain
• In JPEG, the 2D DCT is used to produce a frequency response, represented as a function F(u, v) in the spatial frequency domain, indexed by integers u and v

Observations for JPEG Image Compression

• The effectiveness of the DCT transform coding method in JPEG relies on three main observations:
• Observation 1: Useful image contents change slowly across the image; intensity values do not vary widely several times in a small area, such as within an 8×8 image block, giving rise to "spatial redundancy" due to the high amount of repeated info
• Observation 2: Humans are less likely to notice the loss of very high spatial frequency components than the loss of lower Ones, so spatial redundancy can be reduced by reducing the high spatial frequency contents
• Observation 3: Visual acuity (the ability to distinguish closely spaced lines) is greater for grayscale ("black and white") than for color

Main Steps in JPEG Image Compression

• Transform RGB to YIQ or YUV and subsample color
• Perform DCT on image blocks
• Apply Quantization
• Zigzag Ordering
• DPCM on DC coefficients
• RLE on AC coefficients
• Perform entropy coding
• Reversing the arrows in the block diagram for a JPEG encoder yields a JPEG decoder

Optional Converting RGB to YUV

• YUV color mode stores color in terms of luminance (brightness) and chrominance (hue)
• Human eye is less sensitive to chrominance than luminance
• YUV is not required for JPEG compression, but it gives a better compression rate
• It's simple arithmetic to convert RGB to YUV, based on the relative contributions of red, green, and blue to luminance and chrominance factors
• Multiple different formulas exist for this conversion, depending on the target monitor

Example RGB to YUV formulas

• Y = 0.299 * R + 0.587 * G + 0.114 * B
• U = -0.1687 * R – 0.3313* G + 0.5 * B + 128
• V = 0.5 * R – 0.4187 * G – 0.813 * B + 128
• JPEG process operates on YCbCr color space

DCT on Image Blocks

• Each image is divided into 8 × 8 blocks called data units
• The 2D DCT is applied to each block image f(i, j), generating DCT coefficients F(u, v) for each block
• These coefficients represent the average pixel value and successive higher-frequency changes
• Since DCT involves the transcendental function cosine, some information loss should be expected due to the limited precision of computer arithmetic which can cause some image quality loss
• Before computing the DCT of the 8×8 block, its values are shifted from a positive range to a range centered on zero
• For an 8-bit image, each entry in the original block falls in the range [0,255]
• The midpoint of the range (128) is subtracted from each entry to produce a data range centered on zero, so the modified range is [-128,127]
• This step reduces the dynamic range requirements in the DCT processing stage

Quantization

• Each of the 64 frequency components in a data unit is divided by a separate number called its quantization coefficient (QC), then rounded to an integer
• This is where information is irretrievably lost
• Large QCs cause more loss, and high-frequency components typically have larger QCs
• Each of the 64 QCs is a JPEG parameter that can be specified by the user
• Most JPEG implementations use the QC tables recommended by the JPEG standard for the luminance and chrominance image components
• Formula expressing JPEG compression: F(u, v) = round(F(u, v) / Q(u, v))
• Human eye is good at seeing small differences in brightness over a relatively large area, but not so good at distinguishing the exact strength of a high frequency brightness variation
• This allows one to greatly reduce the amount of information in the high frequency components by dividing each component in the frequency domain by a constant for that component, and then rounding to the nearest integer
• This rounding operation is the only lossy operation in the whole process (other than chroma subsampling) if the DCT computation is performed with sufficiently high precision
• High frequency components are typically rounded to zero, and the rest become small positive or negative numbers, which take many fewer bits to represent
• The elements in the quantization matrix control the compression ratio, with larger values producing greater compression
• Changing the compression ratio is achieved by scaling the numbers in the Q(u,v) matrix
• The quality factor, a user choice in every JPEG implementation, is linearly tied to the scaling factor
• JPEG introduces more loss if the image has quickly changing details