Digital Image Processing HW#1 Solution
Digital Image Processing
Example of Compandor
02/18/2004
TA: Jessie Hsu yh2117@columbia.edu
The idea of a compandor:
[pic]
We want to quantize u, but through transforming it into a uniformly distributed random variable plus uniform quantization, we are doing this in a much less complex way.
From the lecture notes, the transformation function g from u to w is given by
[pic]
Suppose we have pdf of u
|[pic] |Given by [pic], [pic] |
Plugging in the definition of g(u), we have
|[pic] |[pic] |
Suppose we do 2-level quantization,
|Then we have [pic] in w |[pic] |
The goal is now to take these thresholds (or decision levels) and reconstruction levels back to u, so we derive g-1
[pic]
Therefore we have
[pic] in u
The quantization is thus determined for u without extensive computation on u and pu(u).
Remark: How do we make sure w is uniformly distributed?
To obtain the pdf of a transformed random variable, one starts with its cdf after transformation,
[pic]
and the pdf is obtained by taking derivative with respect to w0
[pic]
In other words, [pic], [pic]. w is indeed uniformly distributed
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