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Entropy Coding of Video Encoded by Compressive Sensing

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## Entropy Coding of Video Encoded by Compressive Sensing

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**Entropy Coding of Video Encoded by Compressive Sensing**Yen-Ming Mark Lai, University of Maryland, College Park, MD (ylai@amsc.umd.edu) Razi Haimi-Cohen, Alcatel-Lucent Bell Labs, Murray Hill, NJ (razi.haimi-cohen@alcatel-lucent.com) August 11, 2011**Transmission is digital**101001001 101001001 channel**101001001**channel 101001001 channel 1101 1101**Input video**Output video Break video into blocks Deblock L1 minimization Take compressed sensed measurements Quantize measurements Arithmetic encode Arithmetic decode channel**=**+ - - + - + + - 5 3 4 1 3 1 13 2 2 18 CS Measurement (integers between -1275 and 1275) Input (integers between 0 and 255)**Since CS measurements contain noise from pixel quantization,**quantize at most to standard deviation of this noise 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**What to do with values outside range of quantizer?**quantizer range large values that rarely occur**CS measurements are “democratic” ****Each measurement carries the same amount of information, regardless of its magnitude ** “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)**What to do with values outside range of quantizer?**quantizer range Discard values, small PSNR loss since occurrence rare**2 second video broken into 8 blocks**60 frames (2 seconds) 288 pixels 352 pixels**PSNR**Bit Rate PSNR Bit Rate PSNR Bit 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**Fraction of CS measurements outside quantizer range**2.7 million CS measurements**How often do large values occur**theoretically? 34.13% 34.13% 2.14% 2.14% 0.135% 13.59% 13.59% 0.135%**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?**quantizer range Discard values, small PSNR loss since occurrence rare**Discarding values comes at bit rate cost**1001010110110101010011101000101 discard 0010100001010101000101001000100 0100100010110010101010010101010 1001010110110101010011101000101 discard**Best Compression (Entropy) of Quantized Gaussian Variable X**Arithmetic Coding is viable option !**Fix quantization step, vary standard deviation**Faster arithmetic encoding, less measurements**Fixed bit rate, what should we choose?**18.5 minutes, 121 bins 2.1 minutes, 78 bins 2.1 minutes, 78 bins**Fix standard deviation, vary quantization step**Increased arithmetic coding efficiency, more error**Fixed PSNR, which to choose?**Master’s Bachelor’s PhD**Future Work**• Tune decoder • to take quantization noise into account. • make use of out-of-range measurements • Improve computational efficiency of arithmetic coder**Input video**Output video Break video into blocks Deblock L1 minimization Take compressed sensed measurements For each block: 1) Output of arithmetic encoder 2) mean, variance 3) DC value 4) sensing matrix identifier Quantize measurements Arithmetic encode Arithmetic decode channel**“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**Given a discrete random variable X, the fewest number of**bits (entropy) needed to encode X is given by: For a continuous random variable X, differentialentropy is given by**Differential Entropy of Gaussian**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 where w is width of quantization interval. Then, Variance from initial quantization noise