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Am learning the concept of Coding Redundancy in image compression and am finding it hard to understand the table on page 3 of this document. All other explanations seem to follow the same symbols and methodology but don't explain the concept clearly.

Question1:
In the first column, r1=1/7, r2=2/7 etc and r7 is 1. Why is that? According to the column "Code1", the Code1 is created based on the r value which should range from 0 to 7. Why then are there fractions like 1/7, 2/7 etc?

Question2:
What is "Code2"? At least for Code1, when the value is 000, I know it is the binary representation of 0. But the same thing in Code2, is 11. I don't understand what and how they are trying to represent with Code2.

Question3:
Even if 0 is represented by 11 and 6 is represented by 000001 in variable length encoding, how is the algorithm supposed to know which 0 starts where and which 1 ends where? For example, to represent 60, I'd have to encode it as 00000111. When decoding, how is the algorithm supposed to know that it's 60?

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  • $\begingroup$ There should be one question per post. $\endgroup$ – Evil Nov 28 '17 at 12:08
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In the first column, $r_k$ is the gray level. It ranges from 0 to 1, where 0 is 0% white (i.e., black) and 1 is 100% white. There are 8 gray levels arranged in regular intervals of 1/7.

Code 1 and Code 2 are two different ways of encoding the gray level. Code 1 is a fixed-length code (all codewords have the same length) whereas Code 2 is a variable-length code, specifically a prefix code, presumably a Huffman code for the distribution in the second column. Code 2 is better because the expected codeword length is 2.7 bits, which is smaller than the 3 bits that you get with Code 1.

Since Code 2 is a prefix code, decoding it is actually very simple, and requires no lookahead. (That's why prefix codes are sometimes called instantaneous codes.)

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