Transcript Lossy Compression

Lossy Compression
CIS 465
Spring 2013
Lossy Compression
• In order to achieve higher rates of
compression, we give up complete
reconstruction and consider lossy
compression techniques
• So we need a way to measure how good
the compression technique is
 How close to the original data the
reconstructed data is
Distortion Measures
• A distortion measure is a mathematical
quality that specifies how close an
approximation is to its original
 Difficult to find a measure which corresponds
to our perceptual distortion
 The average pixel difference is given by the
Mean Square Error (MSE)
Distortion Measures
• The size of the error relative to the signal
is given by the signal-to-noise ratio (SNR)
• Another common measure is the peaksignal-to-noise ratio (PSNR)
Distortion Measures
• Each of these last two measures is
defined in decibel (dB) units
 1 dB is a tenth of a bel
 If a signal has 10 times the power of the error,
the SNR is 20 dB
 The term “decibels” as applied to sounds in
our environment usually is in comparison to a
just-audible sound with frequency 1kHz
Rate-Distortion Theory
 We trade off rate (number of bits per symbol)
versus distortion this is represented by a ratedistortion function R(D)
Quantization
• Quantization is the heart of any scheme
 The sources we are compressing contains a
large number of distinct output values (infinite
for analog)
 We compress the source output by reducing
the distinct values to a smaller set via
quantization
 Each quantizer can be uniquely described by
its partition of the input range (encoder side)
and set of output values (decoder side)
Uniform Scalar Quantization
• The inputs and output can be either scalar
or vector
• The quantizer can partition the domain of
input values into either equally spaced or
unequally spaced partitions
• We now examine uniform scalar
quantization
Uniform Scalar Quantization
• The endpoints of partitions of equally
spaced intervals in the input values of a
uniform scalar quantizer are called
decision boundaries
 The output value for each interval is the
midpoint of the interval
 The length of each interval is called the step
size 
 A UQT can be midrise or midtread
Uniform Scalar Quantization
Uniform Scalar Quantization
• Midtread quantizer
 Has zero as one of its output values
 Has an odd number of output values
• Midrise quantizer
 Has a partition interval that brackets zero
 Has an even number of output values
• For  = 1
Uniform Scalar Quantization
• We want to minimize the distortion for a
given input source with a desired number
of output values
 Do this by adjusting the step size  to match
the input statistics
 Let B = {b0, b1, …, bM } be the set of decision
boundaries
 Let Y = {y1, y2, …, yM } be the set of
reconstruction or output values
Uniform Scalar Quantization
• Assume the input is uniformly distributed
in the interval [-Xmax, Xmax]
 Then the rate of the quantizer is
 R = log2 M
 R is the number of bits needed to code the M
output values
 The step size  is given by
 = 2Xmax/M
Quantization Error
• For bounded input, the quantization error
is referred to as granular distortion
 That distortion caused by replacing a whole
range of values from a maximum values to ∞
(and also on the negative side) is called the
overload distortion
Quantization Error
Quantization Error
• The decision boundaries bi for a midrise
quantizer are
 [(i - 1), i], i = 1 .. M/2 (for positive data X)
• Output values yi are the midpoints
 i - /2, i = 1.. M/2 (for positive data)
• The total distortion (after normalizing) is
twice the sum over the positive data
Quantization Error
• Since the reconstruction values yi are the
midpoints of each interval, the quantization
error must lie within the range [-/2, /2]
 As shown on a previous slide, the quantization
error is uniformly distributed
 Therefore the average squared error is the same
as the variance (d)2 of from just the interval [0,
] with errors in the range shown above
Quantization Error
• The error value at x is e(x) = x - /2, so the
variance is given by
(d)2 = (1/) 0 (e(x) - e)2 dx
(d)2 = (1/) 0 [x - (/2) - 0]2 dx
(d)2 = 2/12
Quantization Error
• In the same way, we can derive the signal
variance (x)2 as (2Xmax)2/12, so if the
quantizer is n bits, M = 2n then
SQNR = 10log10[(x)2/(d)2]
SQNR = 10log10 {[(2Xmax)2/12][12/2]}
SQNR = 10log10 {[(2Xmax)2/12][12/ (2Xmax)2]}
SQNR = 10log10M2 = 20n log102
SQNR = 6.02n dB
Nonuniform Scalar Quantization
• A uniform quantizer may be inefficient for
an input source which is not uniformly
distributed
 Use more decision levels where input is
densely distributed
 This lowers granular distortion
 Use fewer where sparsely distributed
 Total number of decision levels remains the same
 This is nonuniform quantization
Nonuniform Scalar Quantization
• Lloyd-Max quantization
 iteratively estimates optimal boundaries based on
current estimates of reconstruction levels then
updates the level and continues until levels converge
• In companded quantization
 Input mapped using a compressor function G then
quantized using a uniform quantizer
 After transmission, quantized values mapped back
using an expander function G-1
Companded Quantization
• The most commonly used companders are
u-law and A-law from telephony
 More bits assigned where most sound occurs
Transform Coding
• Reason for transform coding
 Coding vectors is more efficient than coding
scalars so we need to group blocks of
consecutive samples from the source into
vectors
 If Y is the result of a linear transformation T of
an input vector X such that the elements of Y
are much less correlated than X, then Y can
be coded more efficiently than X.
Transform Coding
• With vectors of higher dimensions, if most
of the information in the vectors is carried
in the first few components we can roughly
quantize the remaining elements
• The more decorrelated the elements are,
the more we can compress the less
important elements without affecting the
important ones.
Discrete Cosine Transform
• The Discrete Cosine Transform (DCT) is a
widely used transform coding technique
 Spatial frequency indicates how many times
pixel values change across an image block
 The DCT formalizes this notion in terms of
how much the image contents change in
correspondence to the number of cycles of a
cosine wave per block
Discrete Cosine Transform
• The DCT decomposes the original signal
into its DC and AC components
 Following the techniques of Fourier Analysis,
any signal can be described as a sum of
multiple signals that are sine or cosine
waveforms at various amplitudes and
frequencies
• The inverse DCT (IDCT) reconstructs the
original signal
Definition of DCT
• Given an input function f(i, j) over two input variables, the
2D DCT transforms it into a new function F(u, v), with u
and v having the same range as i and j. The general
definition is
• Where i,u = 0, 1, … M-1, j, v = 0, 1, … N-1; and the
constants C(u) and C(v) are defined by:
Definition of DCT
• In JPEG, M = N = 8, so we have
• The 2D IDCT is quite similar
• with i, j, u, v = 0, 1, …,7
Definition of DCT
• These DCTs work on 2D signals like
images.
• For one dimensional signals we have
Basis Functions
• The DCT and IDCT use the same set of
cosine functions - the basis functions
Basis Functions
DCT Examples
DCT Examples
• The first example on the previous slide
has a constant value of 100
 Remember - C(0) = sqrt(2)/2
 Remember - cos(0) = 1
 F1(0) = [sqrt(2)/(22)]  (1100 +1100 +
1100 + 1100 + 1100 +1100 + 1100)
 283
DCT Examples
• For u = 1, notice that cos(/16) = -cos(15/16),
cos(3/16)= -cos(13/16), etc. Also, C(1). So we have:
• F1(1) = (1/2)  [cos(/16)  100 + cos(3/16)  100 +
cos(5/16)  100 + cos(7/16)  100 + cos(9/16)  100
+ cos(11/16)  100 + cos(13/16)  100 + cos(15/16)
 100] = 0
• The same holds for F1(2), F1(3), …, F1(7) each = 0
DCT Examples
• The second example shows a discrete
cosine signal f2(i) with the same frequency
and phase as the second cosine basis
function and amplitude 100
 When u = 0, all the cosine terms in the 1D
DCT equal 1.
 Each of the first four terms inside of the
parentheses has an opposite (so they cancel).
 E.g. cos /8 = -cos 7/8
DCT Examples
• F2(0) = [sqrt(2)/(22)]  1  [100cos(/8) + 100cos(3/8)
+ 100cos(5/8) + 100cos(7/8) + 100cos(9/8) +
100cos(11/8) + 100cos(13/8) + 100cos(15/8)]
• =0
• Similarly, F2(1), F2(3), F2(4), …, F2(7) = 0
• For u = 2, because cos(3/8) = sin(/8)
• we have cos2(/8) + cos2(3/8) = cos2(/8) + sin2(/8) =
1
• Similarly, cos2(5/8) + cos2(7/8) = cos2(9/8) +
cos2(11/8) = cos2(13/8) + cos2(15/8) = 1
DCT Examples
• F2(2) =1/2  [cos(/8)  cos(/8) + cos(3/8)  cos(3/8)
+ cos(5/8)  cos(5/8) + cos(7/8)  cos(7/8) +
cos(9/8)  cos(9/8) + cos(11/8)  cos(11/8) +
cos(13/8)  cos(13/8) + cos(15/8) + cos(15/8)] 
100
• = 1/2  (1 + 1 + 1 + 1)  100
DCT Examples
DCT Characteristics
• The DCT produces the frequency
spectrum F(u) corresponding to the spatial
signal f(i)
 The 0th DCT coefficient F(0) is the DC
coefficient of f(i) and the other 7 DCT
coefficients represent the various changing
(AC) components of f(i)
 0th component represents the average value of f(i)
DCT Characteristics
• The DCT is a linear transform
 A transform is linear iff
 Where  and  are constants and p and q are
any functions, variables or constants.
Cosine Basis Functions
• For better decomposition, the basis
functions should be orthognal, so as to
have the least amount of redundancy
• Functions Bp(i) and Bq(i) are orthognal if
 Where “.” is the dot product
Cosine Basis Functions
• Further, the functions Bp(i) and Bq(i) are
orthonormal if they are orthognal and
 Orthonormal property guarantees
reconstructability
 It can be shown that
Graphical Illustration of 2D DCT
Basis Functions
2D Separable Basis
• With block size 8, the 2D DCT can be
separated into a sequence of 2 1D DCT
steps (Fast DCT)
 This algorithm is much more efficient (linear
vs. quadratic)
Discrete Fourier Transform
• The DCT is comparable to the more widely
known (in mathematical circles) Discrete
Fourier Transform
Other Transforms
• The Karhunen-Loeve Transform (KLT) is a reversible
linear transform that optimally decorrelates the input
• The wavelet transform uses a set of basis functions
called wavelets which can be implemented in a
computationally efficient manner by means of multiresolution analysis