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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