# Quantization Uniform

Let the output of image sensor take values between 0 to 10. If the samples are quantized uniformly to 256 levels, show that transition and reconstruction levels are

$$t_k=\dfrac{10(k-1)}{256},\, k=1,\cdots,257$$ $$r_k=t_k+\dfrac{5}{256},\, k=1,\cdots,256,$$

respectively.

I asked the first formula, but the second I don't know how i will be able to do ...

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I don't know the answer with absolute certainty, but I'm fairly certain that the answer follows immediately, if you know what is meant by A) uniform quantization, B) transition level, C) reconstruction level. Have you looked those up? – Jyrki Lahtonen Dec 16 '12 at 18:00

It seems "transition levels" refers to the output levels at which there is a transition in the quantized value. Quantizing the interval from $0$ to $10$ uniformly with $256$ levels leads to $256$ intervals of length $10/256$ each, and the $257$ values $t_k$ are the boundary values of those intervals: $0,10/256,20/256,\dotsc,10$. Then when you want to reconstruct an output level from a quantized level, your best guess (e.g. in the sense of mean error or root mean square error) is the midpoint of the interval, and indeed the $256$ values $r_k$ are the midpoints of the intervals: $5/256,15/256,\dotsc$.