## Abstract

The need for compression of images becomes apparent when one counts the number of bits needed to represent the information content within each image. This Chapter provides an introduction to the realm of data compression and to approaches for image compression.

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—The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point.Shannon, Claude

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- 1.
Shannon used the older spelling ‘Markoff’ for the famous Russian mathematician Andrei Andreyevich Markov.

- 2.
Each new pixel (to the right of the first) represents the difference of its original value and the value of the previous pixel.

- 3.
The prefix condition, which represents a necessary and sufficient condition for the creation of variable length codes, states that no code can be the beginning of another code for the same alphabet.

- 4.
This is a typical symbolism for digital images. Every image consists of a sequence of two-dimensional samples. The parameters

*x*and*y*represent these two dimensions: \(0 \le x< N_1, 0 \le y < N_2\), \(N_1\) and \(N_2\) being the image dimensions in the horizontal and vertical directions respectively. - 5.
It should be noted that in this example the transform has been applied once on the total image area, which is not the usual case. Usually, these transforms are being applied in a block-by-block basis, typically of \(8\times 8\) pixels.

- 6.
The green channel was selected because it is easy to follow; the upper left triangular region of the image, which is yelowish is expected to exhibit higher values in this channel than those in the rest of the block, which shows up redish.

- 7.
Again, as in the case of the example given for DCT, the transform has been applied once on the total image area, which is not the usual case; in the usual case the transform is being applied in a block-by-block basis, typically of \(8\times 8\) pixels.

- 8.
A family of special piecewise constant functions assuming only the two values, \(\pm 1\).

- 9.
‘Famous’ due to its acceptance and usage in image compression applications, like in the Joint Photographic Experts Group 2000 (JPEG2000) image compression standard.

- 10.
It should be noted that the definition of what is ‘favorable’ purely depends on the context.

- 11.
For simplicity,

*geometric*is a distribution sharply concentrated around the mean value with only small values around it being probable (small variance). - 12.
This MATLAB function can be executed either for a scalar (single) or for a vector input and will respond with the appropriate results.

- 13.
This MATLAB function can be executed either for a scalar (single) or for a vector input and will respond with the appropriate results.

- 14.
This MATLAB function can be executed only for scalar inputs.

- 15.
The term is used metaphorically to denote the ability of a method to adapt to the input data and the exploitation of statistical, spectral or any other inherent redundancies.

- 16.
There is a misunderstanding on this case, as JPEG does not support lossless compression. This mode does not have anything in common with the classical JPEG algorithm and can only support compression rates of about 2:1, using prediction in a causal neighborhood of pixels. It is essentially an entirely different method.

- 17.
The Wang database is accessible at http://wang.ist.psu.edu/docs/related/.

- 18.
These are Tables K.3 and K.4 in the JPEG standard.

- 19.
It should be stressed that the negative values are represented in the twos-complement format. Twos complement representation requires to flip all bits of the absolute value binary representation and then add 1.

- 20.
It is worth noting that there are cases in which it is possible high RMSE values to correspond to visually acceptable quality.

- 21.
In the context of signal processing, a filter bank is a set of band-pass filters, each of which separates an input signal into one single frequency sub-band of the overall spectrum of the original signal. A graphic equalizer is usually referenced as one illustrative example of a filter bank, in which various components of the input signal are being attenuated separately and all components are recombined to form the modified signal at the output. Apparently this process includes two steps, a decomposition or an analysis and a reconstruction or a synthesis step.

- 22.
It is reminded that an

*adaptive binary arithmetic coder*accepts the binary symbols of an input sequence, along with a corresponding probabilistic model, and outputs a codestream with a length of at most two bits greater than the combined ideal lengths of the code of the input symbols. By updating the probability estimate of symbols adaptivity is enabled (Pennebaker et al. 1988). - 23.
It is noted that this section is provided here purely to illustrate the potential in adopting the JPEG2000 coding strategy and to highlight an example of successful engineering. The system that is described was implemented in a

*pre-HTML5 era*, so the need for all those complementing technologies for client-server communication was imperative. The value of this section is purely illustrative and educative. - 24.
Remember this was an implementation in an early pre-HTML5 era.

- 25.
Transmission times are reported for a noiseless transmission channel in a guaranteed constant 1 Mbps bandwidth.

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Pavlidis, G. (2017). Data Coding and Image Compression. In: Mixed Raster Content. Signals and Communication Technology. Springer, Singapore. https://doi.org/10.1007/978-981-10-2830-4_2

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