Why Image compression is Necessary?

Goal
 Store image data as efficiently as possible
 Ideally, want to
– Maximize image quality
– Minimize storage space and processing resources
 Can’t have best of both worlds
 What are some good compromises?

Why is it possible to compress images?
 Data != information/knowledge
 Data >> information
 Key idea in compression: only keep the info.
 But why is data != info? Answer: Redundancy
 Statistical redundancy
– Spatial redundancy and coding redundancy
 Phychovisual redundancy
– Greyscale redundancy and frequency redundancy

Spatial redundancy
 Pixel values are not spatially independent
 High correlation among neighbor pixels

Coding redundancy
 Redundancy when mapping from the pixels (symbols)
to the final compressed binary code (Information theory)
 Example:
 Lavg,1 = 3 bits/symbol
 Lavg,2 = 4x0.1+2x0.2+0.5+4x0.05+3x0.15 = 1.95 bits/sym
 Code 2 is also unique and shorter

Phychovisual redundancy
 The ”end-user” is a human => only represent the info.
which can be perceived by the Human Visual System
(HSV)
 From a data’s point of view => lossy
 From the HSV’s point of view => lossless
 Intensity redundancy
 Frequency redundancy
Low-level
Processing
unit
Incident
light
Preceived visual
information
High-level
Processing
unit

Intensity redundancy
 Weber’s law: ∆I / I1 ~ constant
∆I = I1 – I2, where just recognizable
 The high (bright) values need
a less accurate representation
compared to the low (dark) values
 Weber’s law holds for all
human senses!
I2
I1
Sound
∆I
∆I
Noise level
I2
I2
I1
I1

Frequency redundancy
 Human eye functions as a lowpass filter =>
– High frequencies in an image can be ”ignored” without the
HVS noticing
– Key issue in lossy image compression
 Now we know why image compression is possible:
Redundancies
 Investigate how to implement these redundancies
in algorithms

Two main schools of image
compression
 Lossless
– Stored image data can
reproduce original image
exactly
– Takes more storage
space
– Uses entropy coding only
(or none at all)
– Examples: BMP, TIFF,
GIF
 Lossy
– Stored image data can
reproduce something that
looks “close” to the
original image
– Uses both quantization
and entropy coding
– Usually involves transform
into frequency or other
domain
– Examples: JPEG, JPEG-
2000

Concepts
 Lossless compression
– Astronomy, medicine => details
 Lossy compression
– ”normal” images => overall
 An image: 2D array of pixels (rows, columns)
– A pixel value:
 Greyscale/intensity: [8 bits]
 Color: [3 x 8 bits] [Red, Green, Blue] [R,G,B]
– Data:
 Greyscale: 300x300 = 90.000 pixels = 90Kbyte = 720Kbit
 Color: x3 = 270Kbyte = 2.16Mbit

BMP (Bitmap)
 Use 3 bytes per pixel, one
each for R, G, and B
 Can represent up to 224
= 16.7
million colors
 No entropy coding
 File size in bytes =
3*length*height, which can be
very large
 Can use fewer than 8 bits per
color, but you need to store
the color palette
 Performs well with ZIP, RAR,
etc.

GIF (Graphics Interchange Format)
 Can use up to 256 colors
from 24-bit RGB color space
– If source image contains
more than 256 colors, need
to reprocess image to fewer
colors
 Suitable for simpler images
such as logos and textual
graphics, not so much for
photographs
 Uses LZW lossless data
compression

JPEG (Joint Photographic Experts
Group)
 Most dominant image
format today
 Typical file size is about
10% of that of BMP (can
vary depending on
quality settings)
 Unlike GIF, JPEG is
suitable for photographs,
not so much for logos
and textual graphics

JPEG Encoding Steps
 Preprocess image
 Apply 2D forward DCT
 Quantize DCT coefficients
 Apply RLE, then entropy encoding

JPEG Block Diagram
FDCT
Source
Image
Quantizer
Entropy
Encoder
TableTable
Compressed
image data
DCT-based encoding
8x8 blocks
R
B
G

Preprocess
 Shift values [0, 2P
- 1] to [-2P-1
, 2P-1
- 1]
– e.g. if (P=8), shift [0, 255] to [-127, 127]
– DCT requires range be centered around 0
 Segment each component into 8x8 blocks
 Interleave components (or not)
– may be sampled at different rates

Interleaving
 Non-interleaved: scan from left to right, top to
bottom for each color component
 Interleaved: compute one “unit” from each
color component, then repeat
– full color pixels after each step of decoding
– but components may have different resolution

Color Transformation (optional)
 Down-sample chrominance components
– compress without loss of quality (color space)
– e.g., YUV 4:2:2 or 4:1:1
 Example: 640 x 480 RGB to YUV 4:1:1
– Y is 640x480
– U is 160x120
– V is 160x120

Interleaving
 ith color component has dimension (xi, yi)
– maximum dimension value is 216
– [X, Y] where X=max(xi) and Y=max(yi)
 Sampling among components must be integral
– Hi and Vi; must be within range [1, 4]
– [Hmax, Vmax] where Hmax=max(Hi) and Vmax=max(Vi)
 xi = X * Hi / Hmax
 yi = Y * Vi / Vmax

Forward DCT
 Convert from spatial to frequency domain
– convert intensity function into weighted sum of
periodic basis (cosine) functions
– identify bands of spectral information that can be
thrown away without loss of quality
 Intensity values in each color plane often
change slowly

Understanding DCT
 For example, in R3
, we can write (5, 2, 9) as the
sum of a set of basis vectors
– we know that [(1,0,0), (0,1,0), (0,0,1)] provides one
set of basis functions in R3
(5,2,9) = 5*(1,0,0) + 2*(0,1,0) + 9*(0,0,1)
 DCT is same process in function domain

DCT Basic Functions
 Decompose the intensity function into a
weighted sum of cosine basis functions

1D Forward DCT
 Given a list of n intensity values I(x),
where x = 0, …, N-1
 Compute the N DCT coefficients:
1...0,
2
)12(
cos)()(
2
)(
1
0
−=
+
= ∑
−
=
nu
n
x
xIuC
n
uF
n
x
µπ




=
=
otherwise
ufor
uCwhere
1
,0
2
1
)(

1D Inverse DCT
 Given a list of n DCT coefficients F(u),
where u = 0, …, n-1
 Compute the n intensity values:




=
=
otherwise
ufor
uCwhere
1
,0
2
1
)(
1...0,
2
)12(
cos)()(
2
)(
1
0
−=
+
= ∑
−
=
nx
n
x
uCuF
n
xI
n
u
µπ

Extend DCT from 1D to 2D
 Perform 1D DCT on each
row of the block
 Again for each column of
1D coefficients
– alternatively, transpose
the matrix and perform
DCT on the rows
X
Y

Equations for 2D DCT
 Forward DCT:
 Inverse DCT:





 +





 +
= ∑∑
−
=
−
= m
vy
n
ux
yxIvCuC
nm
vuF
m
y
n
x 2
)12(
cos*
2
)12(
cos*),()()(
2
),(
1
0
1
0
ππ





 +





 +
= ∑∑
−
=
−
= m
vy
n
ux
vCuCuvF
nm
xyI
m
v
n
u 2
)12(
cos*
2
)12(
cos)()(),(
2
),(
1
0
1
0
ππ

Visualization of Basis FunctionsIncreasingfrequency
Increasing frequency

Quantization
 Divide each coefficient by integer [1, 255]
– comes in the form of a table, same size as a block
– multiply the block of coefficients by the table, round result to
nearest integer
 In the decoding process, multiply the quantized
coefficients by the inverse of the table
– get back a number close to original
– error is less than 1/2 of the quantization number
 Larger quantization numbers cause more loss

De facto Quantization Table
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
Eye becomes less sensitive
Eyebecomeslesssensitive

Entropy Encoding
 Compress sequence of quantized DC and AC
coefficients from quantization step
– further increase compression, without loss
 Separate DC from AC components
– DC components change slowly, thus will be
encoded using difference encoding

DC Encoding
 DC represents average intensity of a block
– encode using difference encoding scheme
– use 3x3 pattern of blocks
 Because difference tends to be near zero, can
use less bits in the encoding
– categorize difference into difference classes
– send the index of the difference class, followed by
bits representing the difference

AC Encoding
 Use zig-zag ordering of coefficients
– orders frequency components from low->high
– produce maximal series of 0s at the end
 Apply RLE to ordering

Huffman Encoding
 Sequence of DC difference indices and values
along with RLE of AC coefficients
 Apply Huffman encoding to sequence – Exploits
sequence’s statistics by assigning frequently used
symbols fewer bits than rare symbols
 Attach appropriate headers
 Finally have the JPEG image!

JPEG Decoding Steps
 Basically reverse steps performed in encoding
process
– Parse compressed image data and perform
Huffman decoding to get RLE symbols
– Undo RLE to get DCT coefficient matrix
– Multiply by the quantization matrix
– Take 2-D inverse DCT of this matrix to get
reconstructed image data

Reconstruction Error
 Resulting image is “close” to original image
 Usually measure “closeness” with MSE (Mean
Squared Error) and PSNR (Peak Signal to
Noise Ratio) – Want low MSE and high PSNR

Example - One everyday photo with
file size of 2.76 MB

file size of 600 KB

file size of 350 KB

file size of 240 KB

file size of 144 KB

file size of 88 KB

Analysis
 Near perfect image at 2.76M, so-so image at 88K
 Sharpness decreases as file size decreases
 False contours visible at 144K and 88K
– Can be fixed by dithering image before quantization
 Which file size is the best?
– No correct answer to this question
– Answer depends upon how strict we are about image quality, what
purpose image is to be used for, and the resources available

Why Image compression is Necessary?

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