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I know that JPEG uses 2D dct and splits the images in 8x8 blocks. Why doesn't it simply split the image in one-dimensional vectors in $\mathbb{R}^{64}$? Wouldn't it simplify the math? My guess is that this way doesn't take advantage of pixel correlation. Is this right or is there any other reason?

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If JPEG used a 1d DCT it would only be eliminating the redundancy in a single direction. You could compress the image with a 1d DCT then turn it sideways and compress it again. Since the DCT is separable doing this double 1d compression procedure would be (roughly) the same as just using a 2d DCT transform to begin with.

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