Image Compression part 2

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Image Compression (part 2)
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Image Compression

• Everyday an enormous amount of information is
  stored, processed, and transmitted
• Financial data
• Reports
• Inventory
• Cable TV
• Online Ordering and tracking

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Image Compression

• Because much of this information is graphical or
  pictorial in nature, the storage and
  communications requirements are immense ( great
  ).
• Image compression addresses the problem of
  reducing the amount of data requirements to
  represent a digital image.
• Image Compression is becoming an enabling
  technology HDTV (high Digital TV).
• Also it plays an important role in Video
  Conferencing, remote sensing, satellite TV, FAX,
  document and medical imaging.

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Image Compression

• Outline
• Fundamentals
• Coding Redundancy
• Interpixel Redundancy
• Psychovisual Redundancy
• Fidelity Criteria
• Error-Free Compression
• Variable-length Coding
• LZW Coding
• Predictive Coding
• Lossy Compression
• Transform Coding
• Wavelet Coding
• Image Compression Standards

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Fundamentals

• The term data compression refers to the process
  of reducing the amount of data required to
  represent a given quantity of information
• Data Information
• Various amount of data can be used to represent
  the same information
• Data might contain elements that provide no
  relevant information data redundancy
• Data redundancy is a central issue in image
  compression. It is not an abstract concept but
  mathematically quantifiable entity

Some Images are adopted from R. C. Gonzalez R.
E. Woods
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Model

• We want to remove redundancy from the data
• Mathematically

Transformation
Statistically Uncorrelated data
2D array Of pixels
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Data Redundancy

• Let n1 and n2 denote the number of information
  carrying units in two data sets that represent
  the same information
• The relative redundancy RD is define as
• where CR, commonly called the compression ratio,
  is

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Data Redundancy

• If n1 n2 , CR1 and RD0 no redundancy
• If n1 gtgt n2 , CR and RD high
  redundancy
• If n1 ltlt n2 , CR and RD undesirable
• A compression ratio of 10 (101) means that the
  first data set has 10 information carrying units
  (say, bits) for every 1 unit in the second
  (compressed) data set.
• In Image compression , 3 basic redundancy can be
  identified
• Coding Redundancy
• Interpixel Redundancy
• Psychovisual Redundancy

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Coding Redundancy

• How to assign codes to alphabet
• In digital image processing
• Code gray level value or color value
• Alphabet is used conceptually
• General approach
• Find the more frequently used alphabet
• Use fewer bits to represent the more frequently
  used alphabet, and use more bits for the less
  frequently used alphabet

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Coding Redundancy

• Focus on gray value images
• Histogram shows the frequency of occurrence of a
  particular gray level
• Normalize the histogram and convert to a pdf
  representation let rk be the random variable
• pr(rk) nk/n k 0, 1,2 ., L-1, where L is
  the number of gray level values
• l(rk) number of bits to represent rk
• Lavg ?k0 to L-1 l(rk) pr(rk) average number
  of bits to encode one pixel. For M x N image,
  bits required is MN Lavg
• For an image using an 8 bit code, l(rk) 8, Lavg
  8.
• Fixed length codes.

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Coding Redundancy

• Recall from the histogram calculations
• where p(rk) is the probability of a pixel to
  have a certain value rk
• If the number of bits used to represent rk is
  l(rk), then

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Coding Redundancy

• Example

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Coding Redundancy
Variable-Length Coding
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Inter-pixel Redundancy
Here the two pictures have Approximately the
same Histogram. We must exploit Pixel
Dependencies. Each pixel can be estimated From
its neighbors.
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Run-Length Encoding
Example of Inter-pixel Redundancy removal
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Psycho-visual Redundancy
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Psycho-visual Redundancy
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Psycho-visual Redundancy
The human visual system is more sensitive to
edges Middle Picture Uniform quantization from
256 to 16 gray levels CR 2 Right picture
Improved gray level quantization (IGS) CR 2
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Performance Measures

• Compression Ratio (CR) gt 1
• Bit Rate (BR) bits per pixel
• Lossy compression distortion measures M x N
  image with P bits per pixel
• Mean Squared Error (MSE)
• (Smaller the better)
• Peak Signal to Noise Ratio (PSNR)
• (Greater the better)
• MSE and PSNR do not always correlate with quality
  as perceived by the human eye!

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Fidelity Criteria (mean square error)
The error between two functions is given
by So, the total error between the two images
is The root-mean-square error averaged over
the whole image is
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Again

• mean square error MSE
• RMSE

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Fidelity Criteria (PSNR)

• A closely related objective fidelity criterion is
  the mean square signal to noise ratio of the
  compressed-decompressed image

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Fidelity Criteria
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Compression Types
Compression
Error-Free Compression (Loss-less)
Lossy Compression
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Compression Model
The source encoder is responsible for removing
redundancy (coding, inter-pixel,
psycho-visual) The channel encoder ensures
robustness against channel noise.
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What is Predictive Coding (This slide is not
included)
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Predictive Coding (This slide is not included)

• Make use of the past history of the data being
  encoded to provide more efficient compression.
• For example

Prediction Add 2 to the previous number and find
the residual.
Original sequence
Transmitted sequence (residual sequence)
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What is DCT transformation?
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DCT Transform (used in JEPG)
8x8 DCT Transform
8x8 Image sub-block
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Quantization
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Coefficient Ordering and Run Length Coding
Low frequencies
Zig-zag Scan
High frequencies
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Facts about JPEG

• JPEG - Joint Photographic Experts Group
• International standard 1992
• Most popular format
• Other formats (.bmp) use similar techniques
• Lossy image compression
• transform coding using the DCT
• JPEG 2000
• New generation of JPEG
• DWT (Discrete Wavelet Transform)

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Observations

• The effectiveness of the DCT transform coding
  method in JPEG relies on 3 major observations
• Observation 1
• Useful image contents change relatively slowly
  across the image, i.e., it is unusual for
  intensity values to vary widely several times in
  a small area, for example, within an 88 image
  block.
• - much of the information in an image is
  repeated, hence spatial redundancy".

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Observations

• Observation 2
• Psychophysical experiments suggest that humans
  are much less likely to notice the loss of very
  high spatial frequency components than the loss
  of lower frequency components.
• - the spatial redundancy can be reduced by
  largely reducing the high spatial frequency
  contents.
• Observation 3
• Visual acuity (accuracy in distinguishing closely
  spaced lines) is much greater for gray (\black
  and white") than for color.
• - chroma subsampling (420) is used in JPEG.

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8x8 DCT Example
or v
or u
DC Component
Corresponding DCT coefficients (in
frequency domain)
Original values of an 8x8 block (in spatial
domain)
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JPEG Steps

• Block Preparation From RGB to YUV (YIQ) planes
• Transform Two-dimensional Discrete Cosine
  Transform (DCT) on 8x8 blocks.
• Quantization Compute Quantized DCT Coefficients
  (lossy).
• Encoding of Quantized Coefficients
• Zigzag Scan
• Differential Pulse Code Modulation (DPCM) on DC
  component
• Run Length Encoding (RLE) on AC Components
• Entropy Coding Huffman or Arithmetic

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JPEG Diagram
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JPEG Block Preparation
RGB Input Data
After Block Preparation
Input image 640 x 480 RGB (24 bits/pixel)
transformed to three planes Y (640 x 480,
8-bit/pixel) Luminance (brightness) plane. U, V
(320 X 240 8-bits/pixel) Chrominance (color)
planes.
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Block Effect

• Using blocks, however, has the effect of
  isolating each block from its neighboring
  context.
• choppy (blocky") with high compression ratio

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JPEG Quantized DCT Coefficients
q(u,v)
Uniform quantization Divide by constant N and
round result. In JPEG, each DCT Fu,v is
divided by a constant q(u,v). The table of
q(u,v) is called quantization table.
Fu,v
Rounded Fu,v/ Q(u,v)
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More about Quantization

• quantization is the main source for loss
• Q(u, v) tend to have larger values towards the
  lower right corner. This aims to introduce more
  loss at the higher spatial frequencies
• - a practice supported by Observations 1
  and 2.
• Q(u,v) are obtained from psychophysical studies
  with the goal of maximizing the compression ratio
  while minimizing perceptual losses in JPEG
  images.

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JPEG Encoding of Quantized DCT Coefficients

• DC Components
• DC component of a block is large and varied, but
  often close to the DC value of the previous
  block.
• Encode the difference of DC component from
  previous 8x8 blocks using Differential Pulse Code
  Modulation (DPCM).
• AC components
• The 1x64 vector has lots of zeros in it.
• Using RLE, encode as (skip, value) pairs, where
  skip is the number of zeros and value is the next
  non-zero component.
• Send (0,0) as end-of-block value.

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JPEG Zigzag Scan
Maps an 8x8 block into a 1 x 64 vector Zigzag
pattern group low frequency coefficients in top
of vector.
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Why ZigZag Scan

• RLC aims to turn the block values into sets
• lt-zeros-to-skip , next non-zero
  valuegt.
• ZigZag scan is more effective

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Entropy Coding

• Huffman/arithmetic coding
• Lossless
• Read textbook p.260-262

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JPEG Modes

• Sequential Mode
• default JPEG mode, implicitly assumed in the
  discussions so far. Each graylevel image or color
  image component is encoded in a single
  left-to-right, top-to-bottom scan.
• Progressive Mode.
• Hierarchical Mode.
• Lossless Mode

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Progressive Mode

• Progressive
• Delivers low quality versions of the image
  quickly, followed by higher quality passes.
• Method 1. Spectral selection
• - Takes advantage of the spectral"
  (spatial frequency spectrum) characteristics of
  the DCT coeffcients
• - higher AC components provide detail
  information.
• Scan 1 Encode DC and rst few AC components,
  e.g., AC1, AC2.
• Scan 2 Encode a few more AC components, e.g.,
  AC3, AC4, AC5.
• ...
• Scan k Encode the last few ACs, e.g., AC61,
  AC62, AC63.

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Progressive Mode contd

• Method 2 Successive approximation
• - Instead of gradually encoding spectral bands,
  all DCT coeffcients are encoded simultaneously
  but with their most significant bits (MSBs)
  first.
• Scan 1 Encode the rst few MSBs, e.g., Bits 7, 6,
  5, 4.
• Scan 2 Encode a few more less signicant bits,
  e.g., Bit 3.
• ...
• Scan m Encode the least signicant bit (LSB), Bit
  0.

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Hierarchical Mode

• Encoding
• First, lowest resolution picture (using low-pass
  filter)
• Then, successively higher resolutions
• additional details (encoding differences)
• Transmission
• transmitted in multiple passes
• progressively improving quality
• Similar to Progressive JPEG

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Hierarchical Encoding
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Example 3-Level Encoding
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Decoding
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Lossless Mode

• Using prediction and entropy coding
• Forming a differential prediction
• A predictor combines the values of up to three
  neighboring pixels as the predicted value for the
  current pixel
• Seven schemes for combination
• Encoding
• The encoder compares the prediction with the
  actual pixel value at the position X' and
  encodes the difference using entropy coding

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7 Predictors
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Comparison with Other Lossless
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JPEG Bitstream
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JPEG 2000 vs JPEG

• Original image

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JPEG2000 vs JPEG
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Image Compression Revisiting JEPG
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Image Compression Basics

• Model driven
• Reduce data redundancy
• Neighboring values on a line scan in an image
• DPCM, predictive coding
• Human perception properties
• Human visual system eye/brain is more sensitive
  to some information as compared to others low
  frequencies vs high frequencies be
  careful..edges are often critical
• Enhancement approaches

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Entropy

• Entropy measurement of the uncertainty of the
  input. Higher the uncertainty the higher the
  entropy.

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Compression Issues

• Progressive display
• Display partially decompressed images
• User begins to see parts of the image, does not
  have to wait for complete decompression
• Hierarchical encoding
• Encode images at multiple resolution levels.
• Display images at lower resolution level and then
  incrementally improve the quality
• Asymmetry
• Time for encoding
• Time for decoding

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JPEG is based on

• Huffman coding
• Optimal entropy encoding
• Run length encoding
• Used in G3, fax
• Discrete Cosine Transform
• Frequency based
• Apply perception rules in the frequency domain
• The fidelity and level of compression can be
  controlled 151 or even better

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Discrete Cosine Transform

• Real cousin of Fourier transform
• Complexity
• NN
• Fast DCT similar to FFT
• To reduce cost
• Divide image into 8 x 8 blocks
• Compute DCT of blocks
• Reduce the size of the object to be compressed

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Quantization

• The eye is more sensitive to the lower
  frequencies.
• Divide each frequency component by a constant
• Divide higher frequency components with a larger
  value
• Truncate, and this will reduce the non-zero
  values
• Four quantization matrices are available in JPEG

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Color

• RGB planes
• Transform RGB into YUV
• Y luminance
• U,V chrominance
• UV have lower spatial resolutions
• Down sampled to take advantage of lower resolution

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Overview of JPEG

• RGB YUV
• Down sample UV
• Original data is 8 bits per pixel, all positive
  0,255. Shift to -128, 127.
• Divide image into 8x8 blocks
• DCT on each block
• Use quantization table to quantize values in each
  block Reducing high freq content
• Use zig-zag scanning to order values in each
  block
• Organize data into bands DC, low f, mid f, high
  f
• Run length encoding
• Huffman encoding

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Examples
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Compression Ratio
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Compression Ratio

• The reduction in file size is necessary to meet
  the bandwidth requirements for many transmission
  systems, and for the storage requirements in
  computer databases
• Also, the amount of data required for digital
  images is enormous

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Compression Speed

• This number is based on the actual transmission
  rate being the maximum, which is typically not
  the case due to Internet traffic, overhead bits
  and transmission errors

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Compression Speed

• Additionally, considering that a web page might
  contain more than one of these images, the time
  it takes is simply too long
• For high quality images the required resolution
  can be much higher than the previous example

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• Example 10.1.5 applies maximum data rate to
  Example 10.1.4

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• Now, consider the transmission of video images,
  where we need multiple frames per second
• If we consider just one second of video data that
  has been digitized at 640x480 pixels per frame,
  and requiring 15 frames per second for interlaced
  video, then

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Entropy

• The entropy for an N x N image can be calculated
  by this equation

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Entropy

• Entropy is the measurement of the average
  information in an image.
• It provides a theoretical minimum for the average
  number of bits per pixel to code the image
• It can also be used as a metric for judging the
  success of a coding scheme, as it is
  theoretically optimal

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Entropy

• Examples10.2.1 and 10.2.2 illustrate the range of
  the entropy
• The examples also illustrate the information
  theory perspective regarding information and
  randomness
• The more randomness that exists in an image, the
  more evenly distributed the gray levels, and more
  bits per pixel are required to represent the data

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Entropy Examples
c) Image after binary threshold, entropy
0.976 bpp
a) Original image, entropy 7.032 bpp
b) Image after local histogram equalization,
block size 4, entropy 4.348 bpp
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Entropy Examples
f) Circle with a radius of 32, and a linear
blur radius of 64, entropy 2.030 bpp
d) Circle with a radius of 32, entropy
0.283 bpp
e) Circle with a radius of 64, entropy
0.716 bpp
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• Figure 10.2.1 depicts that a minimum overall file
  size will be achieved if a smaller number of bits
  is used to code the most frequent gray levels
• Average number of bits per pixel (Length) in a
  coder can be measured by the following equation

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Huffman Coding

• The Huffman code (D. Huffman,1952) is a minimum
  length code
• Given the statistical distribution of the gray
  levels (the histogram), the Huffman algorithm
  will generate a code that is as close as possible
  to the minimum bound, the entropy
• It results in an unequal (or variable) length
  code, where the size of the code words can vary

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Huffman Coding Algorithm

• Find the gray level probabilities for the image
  by finding the histogram
• Order the input probabilities (histogram
  magnitudes) from smallest to largest
• Combine the smallest two by addition
• Repeat 2), until only two probabilities are left
• By working backward along the tree, generate
  code by alternating assignment of 0 and 1

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Run-Length Coding (RLC)

• RLC counts adjacent pixels with the same gray
  level value called the run-length, which is then
  encoded and stored
• It is the best for binary, two-valued, images
• It also works with complex images that have been
  preprocessed by thresholding to reduce the number
  of gray levels to two
• It can be implemented in various ways, but the
  first step is to define the required parameters

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Run-Length Coding (RLC)

• Horizontal RLC (counting along the rows) or
  vertical RLC (counting along the columns) can be
  used
• In basic horizontal RLC, the number of bits used
  for the encoding depends on the number of pixels
  in a row
• If the row has 2n pixels, then the required
  number of bits is n, so that a run that is the
  length of the entire row can be encoded

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• The next step is to define a convention for the
  first RLC number in a row does it represent a
  run of 0's or 1's?

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Arithmetic Coding

• In practice, this technique may be used as part
  of an image compression scheme, but is
  impractical to use alone
• It is one of the options available in the JPEG
  standard

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