$ The given image^C

$

$ Splitting of the image into the red, green and blue channels.

$

$ * JPEG compression does not support alpha channels.

$ Transformation from the RGB color space to YCbCr.

$

$ Y - luma

$ Cb - chroma-blue

$ Cr - chroma-red

$

$ Y  =       0.299 * R + 0.587 * G + 0.144 * B

$ Cb = 128 - 0.168 * R - 0.331 * G +   0.5 * B

$ Cr = 128 +   0.5 * R - 0.418 * G - 0.081 * B

$ Encode images by implementing less resolution for chroma information than from luma information.

$

$ * Takes advantage of the human visual system lower acuity for color differences than for luminance

$ Splitting of the channels into streams of 8x8 chunks to be passed to the DCT.

$ Process of dividing the frequency matrix element-wise by a quantization matrix in order to provide more resolution to more perceivable frequency components (low frequency) and to get rid of less perceivable components (high frequency).

$

$ Reduction factor can be varied by changing the quantization matrix.

$

$ * Lossy process

$ Order coefficients in a zig-zag manner to maximize the chance of a large sequence of 0s, which can be encoded more efficiently.

$

$ * Lossless process

$ JPEG specific run-length encoding turns the stream of coefficients into a stream of tuples of the form:

$

$ [(r, s), c]

$ r - preceding 0s

$ s - bit size of coefficient

$ c - coefficient value

$

$ * [(0, 0)] - End of block

$ Construct a Huffman tree from the tuples and encode the stream using the tree.

$ soi - start of image marker

$ app0 - app specific metadata

$ dqt - specifies one or more quantization tables

$ dht - specifies one or more huffman table

$ sof0 - specifies the image dimensions, the number of components and the component subsampling

$ sos - start of scan marker

$ eoi - end of image marker

$ Extracting the huffman tables and the compressed data.

$ Decoding the compressed data using the huffman tables. Following the binary sequence, descending the tree until a leaf is reached. The value of the leaf is the decoded symbol.

$ Unpacking the tuples into a stream of 0s and values.

$ Arranging the values back into a 8x8 chunk using a zig-zag pattern.

$ Multiplying each value by the corresponding value in the quantization table stored in the jpeg file.

$

$ * Since the quantization process is lossy, dequantization does not produce the same values as before quantization.

$ Merging of the streams of 8x8 chunks into their respective channel, with dimensions of the original subsampled channels.

$ Duplicaiton of pixels in the chroma channels to match the dimensions of the luminance channel.

$ Transformation from the YCbCr color space to the RGB.

$

$ R = Y + 1.402 * (Cr - 128)

$ G = Y - 0.344 * (Cb - 128) - 0.714 * (Cr - 128)

$ B = Y + 1.772 * (Cb - 128)

$ Joining the R, G, B channels to form the final image.

$ The dot product of the input vector and a wave's coefficients measures the similarity between the two vectors, resulting in a frequency component.

$ Splitting the cosine wave into 8 equal parts, and sampling the wave at each part's midpoint

$

$ yn = cos((2n + 1)π/16)

$

$ n ∈ [0, 7]

$

$ Sampling of all 8 cosine waves at each of the 8 points gives us 64 coefficients.

$

$ * Doesn't include normalization

$ The DCT coefficient matrix is made by stacking the cosine waves' sampled points.

$ The matrix is multiplied by the input vector, so each frequency vector is multiplied by the input vector.

$

$ * Matrix multiplication is a more general form of the dot product

$

$

$ For a 8x8 chunk of the image, we pass the rows through the DCT matrix, and then the columns.

$ It is equivalent to multiplying the coefficient matrix to the input chunk, and them multiplying the result by the transpose of the coefficient matrix.