13-09-2012, 10:26 AM
Lossy Compression
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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)
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
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
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
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.