Coding and Compression

Pre-processing. It splits the image into smaller parts (Tiling), makes the values of the input samples symmetric around zero and decorrelates the color components by using the ICT (Irreversible Color Transform) and the RCT (Reversible Color Transform).
DWT. It is the core of JPEG2000. Instead of the DCT (Discrete Cosine Transform) used in JPEG baseline, JPEG2000 uses the Discrete Wavelet Transform, chosen by the JPEG2000 group among a lot of others transforms. The way how it works can be easily explained in one dimension; the 2D version is immediately deducible.

In one-dimensional DWT an analysis block is used. It is made up of two digital filters ( e ): a high pass and a low pass one. In JPEG2000, both the filters always have odd components and practically (5,3) and (9,7) are used: 5 is the number of the low-pass components, whereas 3 is the number of the high-pass ones. The difference between the numbers of the two components must be always a power of two. The output of either filter is subject to a downsampling that halve the number of samples for each filter (if summed, they are still equal to the input number): these samples are called wavelet coefficients.

In order to reconstruct the bit stream, an upsampling that restores the lost samples by putting zeros between one sample and the other is used; then another couple of filters is used.

In 2 dimensions the DWT is applied to each row and then to each column of the image. As a result, the image is divided in subbands marked by their position:

The image can be still seen only in the 1LL subband, which can be transformed once more and divided into other 4 subbands: this is called 2^nd level DWT. This process can be repeated as long as a satisfactory compression level is reached.

Quantizer. Depending on the sample value, it chooses a number of bits to represent the sample value itself. JPEG2000 introduces something new: the deadzone; the central point among the values taken by the samples is represented by a number of bits equal to zero.

What does ‘image compression’ mean?

Compression allows to reduce the number of bits used to represent the coded image, a number that is smaller than how we would use to store the image in the original format. This number is variable and it depends on the compression algorithm we use.

JPEG2000 supports two different kinds of compression: lossy and lossless.

Why do we need a lower bit rate?

A lower bit rate implies a lower occupation of the transmission channel and of the storage supports. Therefore it is advisable for a standard to assure enough lower bit-rates depending on the user’s requirements. In this case JPEG2000 reaches much better results than traditional JPEG.

What does ‘quantization’ mean?

Quantization allows to pass from a continuous to a discrete set; therefore it is the main step in image digitalization. Typically one byte is used for each chromatic component.

What does ‘ranging’ mean?

It consists of the necessity of having a range of integers to be codified. Both the encoder and the decoder must use a number of bits enough large to represent the quantization factor for each subband.

HVS: what is it? Which is its purpose in image compression?

The abbreviation HVS is the acronym of Human Visual System. Its analysis and study represent a basic step in the realization of image compression standards (both for still pictures and video sequences of any kind), in order to improve the quality perceived in image vision as much as possible. Human eye sensibility changes with respect to the different image components (brightness level, spatial frequency, color, contrast, temporal frequency, etc…). In the case of gray levels perception, human eyes do not have a linear behavior but a behavior similar to a cube root. Therefore, if we applied a linear quantization, the perceived quality would be worse than in the case of a quantization made with a cube root function. From a mathematical point of view, it is as though we had two filters whose transfer functions are one the inverse of the other: putting them in a cascade structure, we obtain an uniform perception of gray levels. Similar considerations can be made for every aspect of an image.

Spatial redundancy

In an image it is highly likely that pixels close to each other have a similar color. By exploiting this property, called spatial redundancy, it is possible to reduce the image size considerably, using a smaller number of bits.

Which is the meaning of lossy and lossless compression?

A lossy compression involves a loss of information compared to the original signal; therefore it is not possible to reconstruct the original signal from one that has been compressed by using a lossy compression algorithm. Vice versa, a lossless compression does not involve a loss of information and allows a perfect reconstruction of the original signal after the decompression.

Why is a lossy compression acceptable?

A lossy compression is acceptable mainly for three reasons:

Ø The human visual system is often able to tolerate loss in image quality without compromising the ability to perceive the contents of the scene.

Ø It happens very often that the digital input of the lossy compression algorithm is, in its turn, an imperfect representation of the real image. It should be considered, indeed, that the image is reconstructed starting from a finite number of samples, which can take only discrete values.

Ø A lossless compression would never be able to grant the high levels of compression that can be obtained by using a lossy compression and that are necessary for applications in storage and/or distribution of images.

When is a lossless compression preferable?

Ø In medical applications, when it is difficult to choose an acceptable error level for image representation.

Ø In those cases when we deal with highly structured images (typically non-natural ones) like texts or graphics, which are usually more easily handleable by the means of lossless compression techniques.

Ø In all those applications when images are frequently modified or re-compressed: using a lossy coding, there would be too many errors in the image that would make the quality absolutely unacceptable.

How can the compression of an image be measured?

Image compression is measured by the means of an index called Compression Ratio: =; and represent the number of pixels along the horizontal and vertical dimension of the image, B is the number of bits required to represent each pixel, while ||c|| is the final bit-stream obtained after the compression.

Does the peak signal to noise ratio represent a reliable index of image quality?

The PSNR can give an approximate index of image quality, but by itself it cannot make a comparison between the quality of two different images. It is possible, indeed, that an image with a lower PSNR might be perceived as an image of better quality compared to one with a higher signal to noise ratio.

What is the ‘concept of irrelevance’?

The concept of irrelevance is applied to that information which is unnecessary for the exact reconstruction of an image. There are two different kinds of irrelevance:

Ø Visual irrelevance, if the image contains details that cannot be perceived by human eye.

Ø Irrelevance due to the fact that, for specific applications (for instance medical and military ones), only some specific regions of the image may be of interest rather than the image in its wholeness. Therefore, those parts that do not belong to the region of interest are classified as irrelevant.

What is a code-block?

A code block is a rectangular assemblage (Tile) of coefficients belonging to the same subband. Each Tile can be identified by four parameters: and locate the starting point of each rectangle, whereas and identify its size.

What does DPCM mean?

Differential Pulse Code Modulation. It is a predictive coding used in order to transmit the value of a sample calculated by the means of previous samples instead of the real value.

How is DPCM used in JPEG2000?

DPCM is typically used before applying entropy coding, after the wavelet transform, in order to optimize the performance of the compression algorithm.

What is the embedded bit-stream?

Filtering the image through subsequent quality layers creates the embedded bit-stream; the first one (Q₀) codifies the different code-blocks by using a prefixed length l₀ and minimizing the distortion. The following layers contain additional contributions related to every code-block, minimizing further the distortion.

Each quality layer contains the contributions of each code-block.

In order to have an embedded bit-stream that allows scalability in distortion, it is necessary to introduce enough information to identify the contributions of each quality layer related to the code-blocks.

Which is the difference between an embedded and a non-embedded bit-stream?

The most important advantage of using an embedded bit-stream rather than a non-embedded one is that it is possible to choose the aimed compression level after the compression of the source: this is called an a-posteriori approach.

It is therefore possible to carry out scalability in distortion.

What is the chromatic space?

The chromatic space is a vectorial space used to represent the colors. The most important chromatic spaces are:

RGB (Red Green Blue): primary colors are Red, Green and Blue;

· CYMB (Cyan, Yellow, Magenta, Black);

YC_RC_B: the chromatic components are obtained using one luminance and two chrominances.

What is the meaning of ‘higher quality image’?

There are different approaches to image quality evaluation and they are based on objective and subjective parameters.

The quality of a compressed image is evaluated by analyzing the difference between the original and the compressed one.

One of the most widely used parameters for the evaluation of image quality is the MSE.

MSE: what is it? Is it reliable? Is there an alternative?

MSE means ‘Mean Square Error’. It represents the classical error estimate given by the equation:

where M and N are the image dimensions.

It is a widely used criterion but is often not enough representative.

JND and Hosaka Plots may be used as an alternative to MSE.

Noise resilience

Many new applications of JPEG2000 standard demand that compress data should be transmitted over channels with different error characteristics. For example, ‘wireless’ communications are subject to ‘random’ channel errors, while Internet communications are subject to a loss of data due to congestioning traffic.

As we know, JPEG2000 carries out a codification that is independent from the code-blocks; this is an advantage in error resilience, as each error in the ‘bit-stream’ that corresponds to a code-block will remain inside it.

How much does the PSNR improve if we codify an image using JPEG2000 instead of JPEG?

The following graphic, related to the same image coded with JPEG baseline and JPEG2000, shows that despite of using the same number of bits per pixel, the difference between the two standards is remarkable. For example, for 1.125 bpp, JPEG2000 performs better with a difference of 6 dB; this confirms the higher visual quality of the new standard.

Figure 1: JPEG baseline (0.125 bpp)

Figure 2: JPEG 2000 (0.125 bpp)

What is Canvas?

The Canvas is a reference greed (that is a coordinate system) that JPEG2000 uses to describe the action of different components (typically the color components) on the image.

This greed allows to deal more easily with components management, above all if we notice that these components can have different size. A particular localization inside the Canvas is given to each sample of each component.

Why do we make a distinction between bit-stream and code stream?

The two expressions are not equivalent but they are related to different objects.

Ø The bit-stream is the sequence of bits derived from the codification of a set of symbols; it is typically used to refer to the sequence of bits that comes from the codification of each code-block.

Ø The code-stream is, on the other hand, a set made up of several bit-streams to which the necessary information for the image decodification and reconstruction is associated. This additional information may refer, for example, to the locations of particular bit-streams or to information regarding transformation, quantization and coding methods.