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    8 x 8 JPEG Image

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    Consider the following 8x8 8-bits image:
    26 25 30 30 37 37 37 37
    7 28 29 30 37 37 37 37
    6 9 31 28 37 37 37 37
    4 5 10 30 37 37 37 37
    9 9 9 9 26 25 30 30
    9 9 9 9 7 28 29 30
    9 9 9 9 6 9 31 28
    9 9 9 9 4 5 10 30

    a) Calculate the compression ratio if we encode the image using the baseling JPEG standard.
    b) Calculate the RMSE for part (a).
    c) Suppose we want to compress the image using Block Truncation Coding (BTC) with 4x4 sub-blocks, answer the following:
    i) What are the Bit maps and reconstruction levels used by the BTC technique?
    ii) Reconstruct the image from the information obtained in (i)
    iii) Describe the effects of BTC on the image.
    iv) Calculate the PSNR.
    v) Suppose you have a 64 gray-level 512x512 image.What is the compression ratio achieved by the BTC with 8x8 sub-blocks?

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    Solution Preview

    a) To encode the above 8 x 8 image using baseline JPEG standard, first we need to shift each pixel from unsigned integer value with range [0, 255] to signed integer value with range [-128, 127]. To do so, we subtract 128 from each pixel in the above picture (Figure 1). Figure 2 shows the adjusted pixels.

    After shifting all pixel values in the 8 x 8 image, next we will input to the forward DCT (FDCT). The one-dimension (1-D) FDCT is mathematically defined as (see attached).

    Note that we will refer to each pixel using zero-base index ...

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