Entropy Coding of Video Encoded by Compressive Sensing Yen-Ming Mark Lai, University of Maryland,...

Entropy Coding of Video Encoded by Compressive Sensing

Yen-Ming Mark Lai, University of Maryland, College Park, MD([email protected])

Razi Haimi-Cohen, Alcatel-Lucent Bell Labs, Murray Hill, NJ([email protected])

August 11, 2011

mailto:[email protected]

mailto:[email protected]

Why is entropy coding important?

101001001 101001001channe

l

Transmission is digital

101001001 101001001channel

1101

channel

1101

Break video into blocks

Take compressed sensed

measurements

Quantize measurements

Arithmetic decode

Arithmetic encode

L1 minimization

Deblock

Input video

Output video

channel

Statistics of Compressed Sensed blocks

+ - - - +5 3 4 3 1

- +13

2 21

+ =Input (integers between 0 and 255)

CS Measurement(integers between -1275 and

1275)

18

Since CS measurements contain noise from pixel quantization, quantize at most to standard deviation of this noise

N12

1

standard deviation of noise from pixel quantization in CS measurements

N is total pixels in video block

We call this minimal quantization step the “normalized” quantization step

N12

1

What to do with values outside range of quantizer?

large values that rarely occur

quantizer range

** “Democracy in Action: Quantization, Saturation, and Compressive Sensing,” Jason N. Laska, Petros T. Boufounos, Mark A. Davenport, and Richard G. Baraniuk (Rice University, August 2009)

CS measurements are “democratic” **

Each measurement carries the same amount of information, regardless of its

magnitude

What to do with values outside range of quantizer?

Discard values, small PSNR loss since occurrence rare

quantizer range

Simulations

2 second video broken into 8 blocks

288 pixels

352 pixels

60 frames(2 seconds)

Ratio of CS measurements

to number of input values0.15 0.25 0.35 0.45

PSNR

Bit Rate

“Normalized” quantization step multiplier

1 10 100 200 500

PSNR

Bit Rate

Range of quantizer (std dev of measured Gaussian distribution)

1.0 1.5 2.0 3.0

PSNRBit Rate

Processing Time

• 6 cores, 100 GB RAM

• 80 simulations (5 ratios, 4 steps, 4 ranges)

• 22 hours total

– 17 minutes per simulation

– 8.25 minutes per second of video

Results

Fraction of CS measurements outside quantizer range

2.7 million CS measurements

Fraction of CS measurements outside quantizer range

How often do large values occur theoretically?

34.13%34.13%

13.59%

2.14%0.135%

2.14% 0.135

%13.59%

How often do large values occur in practice?

(theoretical)

2.7 million CS measurements

(0.135%)0.037%

What to do with large values outside range of quantizer?

Discard values, small PSNR loss since occurrence rare

quantizer range

Discarding values comes at bit rate cost

1001010110110101010011101000101

discard00101000010101010001010010

0010001001000101100101010100101010101001010110110101010011101000101 discar

d

Bits Per Measurement, Bits Per Used Measurement

Bits Per Measurement, Bits Per Used Measurement

9.4 bits

Best Compression (Entropy) of Quantized Gaussian Variable X

bitsXh 4.9log)( bitse 4.172log

2

1 2 N12

1

Arithmetic Coding is viable option !

Fix quantization step, vary standard deviationFaster arithmetic

encoding, less measurements

PSNR versus Bit Rate (10 x step size)

Fixed bit rate, what should we choose?

18.5 minutes, 121 bins



Fix standard deviation, vary quantization stepIncreased

arithmetic coding efficiency, more

error

PSNR versus Bit Rate (2 std dev)

Fixed PSNR, which to choose?

Bachelor’sMaster’s

PhD

• Tune decoder

to take quantization noise into

account.

make use of out-of-range

measurements

• Improve computational efficiency of

arithmetic coder

Future Work

Questions?

Supplemental Slides

(Overview of system)



measurements


Arithmetic decode

Arithmetic encode

L1 minimization

Deblock

Input video

Output video

For each block: 1) Output of arithmetic

encoder 2) mean, variance 3) DC value 4) sensing matrix

identifier

channel

Supplemental Slides

(Statistics of CS Measurements)

“News” Test Video Input

• Block specifications

– 64 width, 64 height, 4 frames (16,384 pixels)

– Input 288 width, 352, height, 4 frames (30 blocks)

• Sampling Matrix

– Walsh Hadamard

• Compressed Sensed Measurements

– 10% of total pixels = 1638 measurements

Histograms of Compressed Sensed Samples (blocks 1-5)

Histograms of Standard Deviation and Mean (all blocks)

Supplemental Slides

(How to Quantize)

i ii xpxpXH

)(

1log)()(

Given a discrete random variable X, the fewest number of bits (entropy) needed to encode X is given by:

dx

xpxpXh

)(

1log)()(

For a continuous random variable X, differential entropy is given by

Differential Entropy of Gaussian

22log2

1)( eXh

function of variance

maximizes entropy for fixed variance

i.e. h(X’) <= h(X) for all X’ with fixed variance

Approximate quantization noise as i.i.d. with uniform distribution

12,0~

2wUq

where w is width of quantization interval. Then,

N

ii

wXweightedVarMVar

1

22

12)()(

Variance from initial quantization noise

How much should we quantize?

N

ii

wXweightedVarMVar

1

22

12)()(

NXweightedVar12

1)(

2

input pixels are integers

measurement matrix is Walsh-Hadamard

Supplemental Slides

(Entropy of arithmetic encoder)



measurements


Arithmetic decode

Arithmetic encode

L1 minimization

Deblock

Input video

Output video

channel

What compression to expect from

arithmetic coding?

Entropy of Quantized Random Variable

log)()( XhXH

X - continuous random variable

X - uniformly quantized (discrete)X

22log2

1 e

N


log)()( XhXH

differential (continuous) entropy

X - continuous random variable

X - uniformly quantized (discrete)X

quantization step size


log)()( XhXH

- 5667^2 (average variance of video blocks)

N

bitse 5.142log2

1 2 N

2

- 16,384 pixels in video block

bits5.7 7 bit savings from quantization

Supplemental Slides

(Penalty for wrong distribution)



measurements


Arithmetic decode

Arithmetic encode

L1 minimization

Deblock

Input video

Output video

channel

What is penalty for using wrong probability

distribution?

),(~ 2ppNp

),(~ 2qqNq

p

q

q

qpqpqpD

ln2

)()||( 2

222

Let p and q have normal distributions

Then,

)||()(1)( qpDXHXLE

Entropy of random variable

Expected length of wrong codeword

Kullback-Leiber divergence (penalty)

Assume random variable X has probability distribution p but we encode with distribution q

Worst case scenario for video blocks of “News”

)2800,340(~ 2Np

)11200,360(~ 2Nq

bits

qpDp

q

q

qpqp

33.1

ln2

)()||( 2

222

bitsxH

qpDXHXLE

33.2)(

)||()(1)(

Summary



measurements


Arithmetic decode

Arithmetic encode

L1 minimization

Deblock

Input video

Output video

channel

What are statistics of

measurements?

Gaussian with different means and variances



measurements


Arithmetic decode

Arithmetic encode

L1 minimization

Deblock

Input video

Output video

channel

How much should we quantize?

By square root of total number of pixels in video

block



measurements


Arithmetic decode

Arithmetic encode

L1 minimization

Deblock

Input video

Output video

channel

What compression to expect from

arithmetic coding?

•14.5 bits/measurement (integer quantization)•7 bits/measurement (quantization)



measurements


Arithmetic decode

Arithmetic encode

L1 minimization

Deblock

Input video

Output video

channel

What is penalty for using wrong probability

distribution?

For “News”, must send extra 2

bits/measurement

Entropy Coding of Video Encoded by Compressive Sensing Yen-Ming Mark Lai, University of Maryland,...

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