A simple example illustrating how finite-context models are implemented. The rows of the table correspond to probability models at a given instant *n*. In this example, the particular model that is chosen for encoding a symbol depends on the last five encoded symbols (order-5 context)

## Abstract

The use of medical imaging has increased in the last years, especially with magnetic resonance imaging (MRI) and computed tomography (CT). Microarray imaging and images that can be extracted from RNA interference (RNAi) experiments also play an important role for large-scale gene sequence and gene expression analysis, allowing the study of gene function, regulation, and interaction across a large number of genes and even across an entire genome. These types of medical image modalities produce huge amounts of data that, for several reasons, need to be stored or transmitted at the highest possible fidelity between various hospitals, medical organizations, or research units.

### Keywords

- Image coding
- Lossless coding
- Progressive decoding
- Biomedical images
- Image coding standards
- Image decomposition
- Arithmetic coding
- Finite-context modeling

## 1. Introduction

Image compression is a very important research field. It is fundamental in many different areas, such as biomedical imaging, consumer electronics, and Internet, among others. The goal of an image compression method is to represent an arbitrary image using the smallest possible number of bits.

The use of medical imaging has increased in the last years, especially with magnetic resonance imaging (MRI) and computed tomography (CT) [1]. These types of medical image modalities produce huge amounts of data that, for several reasons, need to be stored or transmitted with the highest possible fidelity between various hospitals and medical organizations.

Also related to biomedical imaging, microarray images play an important role for large-scale gene sequence and gene expression analysis, allowing the study of gene function, regulation, and interaction across a large number of genes and even across an entire genome [2, 3]. The output of a microarray experiment is a pair of 16-bits-per-pixel images, usually with a very high resolution, sometimes exceeding 13,000×4,000 pixels. Consequently, over 200 MB can be required to store a single microarray image. Due to the development of these digital imaging technologies, some concerns appeared regarding efficient ways of storing and transmitting the images. In order to overcome these problems, sophisticated compression methods are required.

Another type of images that are addressed in this chapter are those that can be extracted from RNA interference (RNAi) experiments. Those experiments involve marking cells with various fluorescent dyes to capture three components of interest, namely, DNA, actin, and PH3 channels [4].

Typically, lossless compression algorithms are recommended for dealing with these types of images. In fact, lossless methods are generally required in applications where cost, legal issues, and value play a decisive role, such as in medical imaging or in image archiving [5]. On one hand, the use of lossless algorithms avoids problems of losing diagnostic information vital to identify life-threatening illnesses in the early stages. On the other hand, if a given image is lossless compressed, it is possible to recompress it in the future, using a more efficient algorithm and without losing any information.

Usually, the performance of image coding standards, such as JPEG, JBIG, JPEG-LS, and JPEG2000, falls short when they are applied to certain types of biomedical images. In order to overcome this lower-performance issue, specific compression algorithms are essential. In the class of lossless compression algorithms, there is a specific type that has progressive decoding capabilities. This type of compression is usually designated as "lossy-to-lossless compression." These algorithms are very flexible, because they allow stopping decoding at a certain point, according to the available resources or user requirements. However, the original image can also be obtained, without any loss, if the decoding process goes until the end.

In this chapter, we study the performance of several specific compression methods developed by the authors, as well as of image coding standards, when applied to medical images (computed radiography, computed tomography, magnetic resonance, and ultrasound), RNAi images, and microarray images. The addressed compression algorithms are based on image decomposition, finite-context models, and arithmetic coding. We can divide these algorithms into two categories. In the first one, the input image is split into several bitplanes and each bitplane is encoded individually. This approach is combined with several preprocessing techniques in order to improve the compression efficiency. In the second approach, the intensity levels of a given image are organized in a binary-tree structure, where each leaf node is associated with an image intensity.

We start by presenting the compression algorithms, namely, bitplane decomposition and binary-tree decomposition. We include a brief explanation about finite-context models and arithmetic coding, the entropy coding block used in both approaches. Then, we present a set of experiments that have been performed using the compression algorithms described in Sect. 4, as well as experimental results using the most important image coding standards (e.g., PNG, JBIG, JPEG-LS, and JPEG2000). At the end, we draw some conclusions.

This chapter intends to be a reference for comparison of new compression algorithms that may be developed in the future, for two main reasons. On one hand, the experimental results presented in Sect. 5.2 are state of the art regarding the compression of some of these types of images. On the other hand, several approaches and preprocessing techniques are presented, giving hints about new developments that can be done.

## 2. Compression methods

The compression algorithms that we address here can be classified into two different categories. One based on bitplane decomposition and the other one based on binary-tree decomposition. For the first category, we use a sophisticated bitplane decomposition approach that was successfully developed for the compression of microarray images [6]. Furthermore, we include some preprocessing techniques, such as segmentation and histogram reduction, in order to improve the compression results. An alternative bitplane decomposition approach based on *bit modeling by pixel value estimates* is also considered.

The other approach, based on binary-tree decomposition, was developed with success for the compression of medical images [7] and microarray images [8]. In this decomposition approach, the intensity levels of a given image are organized in a binary-tree structure, where each leaf node is associated with an image intensity.

In the following subsections, we address these two decomposition approaches and give a brief explanation regarding finite-context models and arithmetic coding. The implementation of the compression algorithms can be found in [9].

### 2.1. Finite-context models

Markov modeling is widely used in several fields, including image [7, 8, 10, 11, 12] and DNA [13, 14, 15] compression. Finite-context models rely on the Markov property, since an order-

The probability estimates

where

is the total number of events that has occurred so far in association with context

Initially, all counters are set to zero, i.e., the symbols are assumed to be equally probable. The counters are updated each time a symbol is encoded. However, it is possible to update the counters according to a specific rule. Since the context templates are causal, the decoder is able to reproduce the same probability estimates without needing additional side information.

Table 1 presents a simple example of how a finite context is typically implemented. In this example, we are dealing with an order-5 finite-context model, which means that the context uses the last five encoded symbols to assign the symbol probabilities. Each row of Table 1 represents a probability model that is used to encode the current symbol, using the last five encoded ones. For example, if the last symbols were "11000," i.e.,

Context, c5,n | C(0∣c5,n) | C(0∣c5,n) | C(c5,n) |

00000 | 23 | 41 | 64 |

∶ | ∶ | ∶ | ∶ |

00110 | 14 | 34 | 48 |

∶ | ∶ | ∶ | ∶ |

01100 | 25 | 12 | 37 |

∶ | ∶ | ∶ | ∶ |

11000 | 28 | 41 | 69 |

∶ | ∶ | ∶ | ∶ |

11111 | 8 | 2 | 10 |

One important aspect that must be considered is the size of the context. For an order-

#### 2.1.1. Mixtures of finite-context models

In order to attain better compression results, the previous approach can be used in a more robust scheme. The goal here is to use a model mixture with more than one finite-context model. In our case, we decided to use only two different models: the one used by [6] and another one based on *bit modeling by pixel value estimate*. This approach could be extended to use more models, but at a cost of some computation time. In what follows, we will describe how this mixture scheme works.

The per symbol information content average provided by the finite-context model of order-

bits per symbol (or bits per pixel, bpp, in the case of images). Hence, the

where

In order to compute the probability estimate for a certain symbol, it is necessary to combine the probability estimates given by (1) using (4). The weight assigned to model

i.e., by considering the probability that model

where

where

Calculating the logarithm, we get

which is related to the code length that would be required by model

where

and removing the logarithms, we can rewrite () as

and, finally, set the weights to

Usually, a compression algorithm can be divided into two parts, modeling and coding. The Markov models are responsible for providing a statistical model as reliable as possible to be used later in the coding stage. The coding stage is where the statistical model is used to compress the data. Arithmetic coding is usually the technique that is used. In the following section, we give a brief explanation on how arithmetic coding works.

#### 2.1.2. Arithmetic coding

Arithmetic coding is a compression technique developed by Rissanen [19] in the late 1970s. This method is a good alternative to Huffman coding [20], because it usually generates better compression results. In order to obtain better results, an appropriate probability estimator must be used in the arithmetic encoder (e.g., based on finite-context models, as described above).

This method represents a set of symbols using a single number in the interval [0,1). As the number of symbols of the message grows, the initial interval [0,1) will shrink and the number of bits necessary to represent the interval will increase. When we are processing the pixels of an image in a raster scan order, the probabilities of the intensities of the pixels are conditioned by the context determined by a combination of the already encoded neighboring pixels. The encoder and the decoder estimate this context model dynamically adapting it to the input data, during the encoding/decoding process. According to [16, 17, 18], this arithmetic encoding method generates output bitstreams with average bitrates almost identical to the entropy of the source model.

### 2.2. Bitplane decomposition

The technique to separate an image into different planes (bitplanes), known as bitplane decomposition, plays an important role in image compression. Usually, each pixel of a grayscale image is represented by 8 bits, or 16 bits as the case of some biomedical images used in the experimental results presented in this chapter. Suppose that the image has

The bitplane decomposition technique is very useful on image compression. On one hand, it allows some bi-level compression methods, such as JBIG, to be applied to typical grayscale images. The compression method is applied to each bitplane after the decomposition. On the other hand, it is possible to create sophisticated models that take advantage of this decomposition. For instance, it is possible to use information of the previous bitplanes (usually the MSBPs) to improve the compression performance of the LSBPs.

In [23], an embedded image-domain adaptive compression (EIDAC) scheme used with success on simple images (with a reduced number of active intensity values) is presented. EIDAC uses a context-adaptive bitplane coder, where each bitplane is encoded using a binary arithmetic coder. Later, in [24], a compression method inspired from EIDAC for microarrays image is proposed. The same authors improved their method using a more robust 3D context modeling [24]. We used this last approach in our experiments and also other alternative versions using some preprocessing techniques, namely, segmentation and bitplane reduction.

#### 2.2.1. Segmentation

In the literature, we can find several algorithms for image segmentation. In our approach, we decided to use a threshold-based method inspired from the work presented in [25]. The segmentation is attained by means of a dynamic thresholding scheme. By applying a threshold value, the pixels of the image can be split in two sets (background and foreground). For each threshold value, it is possible to obtain the number of pixels and the standard deviation of the intensities of these pixels in each set. The desired threshold is calculated according to (17). Using (17), it is always guaranteed that the weighted sum of the standard deviation of both the background and foreground is minimal,

where

#### 2.2.2. Bitplane reduction

Bitplane reduction is an interesting method that can further improve compression efficiency by eliminating redundancy in the pixel precision for simple images. Simple images are images where the number of different intensities that occur is very small, compared to the total number of possible intensities. For example, if we have only 24 different intensities out of 256 for an 8-bit image, we only need 5 bits to represent each pixel intensity. This means that, when we are encoding the image, we only need to encode five bitplanes instead of the original eight bitplanes.

Yoo et al. [26] have shown that it is possible to obtain compression gains using the simplest form of bitplane reduction, known as histogram compaction (HC). Later, [23] presented a more robust bitplane reduction method called scalable bitplane reduction (SBR). This approach finds the reduced bitplane codeword by growing a binary tree. The method splits each node of the binary tree into two nodes using a simple MINMAX metric to measure the distortion.

In order to better understand the SBR algorithm, we present a small example in Fig. 4. Initially, we associate all the active pixel values to the root node. In this small example, the image only has four active pixel values. Starting in the root node, the algorithm splits all the children sub-nodes using a MINMAX criterion. The split process in the root node starts by computing the value 32,767 from

Taking as example the intensities

where

IntensityValue | HC | SBR | ||

Codeword | Codeword size | Codeword | Codeword size | |

0 | 00 | 2 | 0 | 1 |

32,760 | 01 | 2 | 10 | 2 |

65,530 | 10 | 2 | 110 | 3 |

65,535 | 11 | 2 | 111 | 3 |

#### 2.2.3. Simple bitplane coding

In [27], Kikuchi et al. introduced the concept of bit modeling by the pixel value estimates. In their approach, instead of using the true bit values of each bitplane, they used the expectation values of the pixels to build up the contexts. This approach is known as *bit modeling by pixel value estimate*. They extended their work more recently to be applied to various types of images (color, grayscale, color-quantized, bi-level, and halftone) [28] and for HDR (high-dynamic-range) images [29].

Let us consider that a given pixel value at location (

Contrarily to Kikuchi's method, our approach uses only one type of context, denoted as *neighborhood context* in Kikuchi's work. The contexts are built by the estimates of partially decoded pixels based on the template depicted in Fig. 5. The pixel location of

where

In Kikuchi's method, the inter-bit correlation on a bitplane is not used. Instead, for modeling a target bit, the authors used the pixel value estimates of which more significant bits have been already available at the decoder. Their method is referred to as *bit modeling by pixel values*, where the pixel value estimates are used rather than of the unknown true values at the decoder. Contrarily to Kikuchi's method that uses a nine-pixel template, our approach uses a variable-size 15-pixel template (see Fig. 5). In our case, a greedy search routine is performed in each bitplane in order to obtain the context size that attains the best compression performance (lower bits per pixel as possible). According to Kikuchi's method, the decoded pixel values are spatially correlated to each other as significantly high as the true pixel values which will make sense to use a larger template. Furthermore, since the alphabet size is only two, the probability of having context dilution is low. Similar to Kikuchi's method, we are considering some noncausal locations with respect to the scanning order of the pixels (locations

Let us consider that we are coding an

at the

Initially, all the pixel estimate values are set to

The pixel value estimate is used in a coming chance of reference and will be the decoded pixel value, when the decoding is stopped (after all the bitplanes are processed). The most recent estimate of a given pixel is always made up of two parts: its significant bits are those already encoded/decoded true bits and the other less significant bits are 0 (zero) followed by a successive 1s (ones). The value of the less significant bits is equal to the expectation value of the unknown lower bits, if binary symbols of zero and one are assumed to occur in those bits with equal probability.

### 2.3. Binary-tree decomposition

Binary trees are also an important data structure that can be used in several algorithms. In the case of image compression, we can associate each leaf node of the binary tree to an image intensity. The binary tree can be viewed as a simple nonlinear generalization of lists; instead of having one way to continue to another element, there are two alternatives that lead to two different elements [30]. Every node (or vertex) in an arbitrary tree has at least two children (see Fig. 7). Each child is designated as left child or right child, according to the position in relation to the tree root.

One of the first methods where binary trees were used for image compression was proposed by Chen et al. [31] regarding the compression of color-quantized images. Chen's method uses a binary-tree structure of color indexes instead of a linear list structure. Using this binary-tree structure, Chen's method can progressively recover an image from two colors to all of the colors contained in the original image. Inspired by the work done by [31], a few years later, the authors of [7, 11, 12] developed a lossy-to-lossless method based on binary-tree decomposition and context-based arithmetic coding. In the last approach, the authors studied the performance of their method in several kinds of grayscale images, including medical images. As can been seen, this decomposition approach is very versatile because it can be applied in color-quantized images and also in grayscale images.

This binary-tree decomposition approach was intended to be used in images with a small number of intensities, usually with eight or less bits per pixel, due to a tight relation between the processing time and the number of different intensities of the image. In this work, we intend to study the performance of this approach to compress biomedical images, such as microarray images, RNAi, etc. Another interesting feature of this approach is its capability of progressive decoding, which means that the decoding process can be stopped at any moment according to a specific distortion metric, obtaining an image with some loss. Moreover, it is possible to obtain the original image without any loss if the full decoding process is performed. In the next two sections, we describe in more detail the compression algorithm that is based on a binary-tree decomposition and context-based arithmetic coding.

#### 2.3.1. Hierarchical organization of the intensity levels

This method is based on a hierarchical organization of the intensity levels of the image. This organization of the intensity levels is attained by means of a binary tree. Each node of the binary tree,

where

In order to better understand the construction of the binary tree, we present in Fig. 7 a small example for an image with only five active pixel values

In order for the decoder to be able to build the same tree, it is necessary to send the information of the active pixel values to the decoder using a 65,536-bit indicator. In order to encode this indicator, the encoder uses the following strategy. First, the maximum intensity value

#### 2.3.2. Encoding pixel locations

After each node is expanded, two new nodes are created, each one with a representative intensity (

The performance of the compression method is directly dependent on the encoding of these binary masks. The encoding efficiency of these binary masks can be controlled by a carefully chosen context modeling that will then drive the binary arithmetic encoder. The context is constructed based on the template depicted in Fig. 9. The number of context pixels can go up to 16 at most, and they are numbered according to their distance to the encoding pixel (represented in gray in Fig. 9). A particular context is represented using a sequence of bits,

where

and where

The value

There is also an alternative way to encode the binary mask. If the number of bits required to encode the mask and the context size is bigger than the total number of pixels associated with the node to be expanded, the encoder sent the binary mask as a binary string, without compression. In order for the decoder to differentiate between these two modes, a binary stream is needed to be encoded for each node that is expanded.

## 3. Evaluation of the compression methods

In this section, we present compression results obtained by the compression methods presented earlier in Sect. 4. We decided to present only the results of the best version of each method in order to avoid extending this section. We start with the description of the data sets used in this work. Then, we provide the obtained compression results. At the end of this section, we present a study of the rate distortion, comparing the two compression approaches described in Sect. 4 and two image coding standards, JBIG and JPEG2000.

### 3.1. Data sets

To evaluate the compression methods presented in this chapter, we used three types of images: microarray, medical, and RNAi. The output data obtained in a microarray experiment is a pair of 16-bits-per-pixel grayscale images, one from the so-called red channel and the other from the green channel (see Fig. 10). We use a total of 298 microarray images of nine different data sets as described in Table 3.

Regarding the medical images, we used images of four modalities: computed radiography (CR), computed tomography (CT), magnetic resonance (MR), and ultrasound (US). In Fig. 11, we can find five examples of medical images used in this work. The data set used in [33] was also considered as a medical reference data set. All the medical image data sets are depicted in Table 4, a total of 370 images.

Finally, we also used a recent and popular image type that can be extracted from RNA interference (RNAi) experiments. Those experiments involve marking of cells with various fluorescent dyes to capture three components of interest, namely, DNA, actin, and PH3 channels [4]. In Table 5, we can find the RNAi image data sets used in this work, a total of 34,560 images. The images were retrieved from the RNAi experiments without any kind of transformation (no normalization was applied). For us, it does not make sense to perform a normalization operation in order to reduce the image depth from 10 to 8 bits per pixel. The normalization is a lossy process, and hence, applying a lossless compression method to images that suffered some loss is not appropriate. In Fig. 12 we present some examples of RNAi images.

In Tables 3-5, we can see several important measures: image size, depth, entropy, intensity usage, and a sparsity measure called Gini index or GI [34]. The GI is a normalized measure that assumes values between zero and one. Values close to zero means that image has a lower sparsity histogram. On the other hand, values close to one represent an image that has a very sparse histogram.

Data sets | Year | Images | Approximate size (cols × rows) | Depth | Average entropy (bpp) | Average intensity usage (percentage) | Gini Index[0-1] |

ApoA1 [35] | 2001 | 32 | 16 | 11.038 | 39.507% | 0.494 | |

Arizona [36] | 2011 | 6 | 16 | 9.306 | 82.821% | 0.774 | |

IBB [37] | 2013 | 44 | 16 | 8.503 | 54.072% | 0.806 | |

ISREC [38] | 2001 | 14 | 16 | 10.435 | 33.345% | 0.710 | |

Omnibus-LM [39] | 2006 | 25 | 16 | 5.713 | 50.130% | 0.726 | |

Omnibus-HM [40] | 2006 | 25 | 16 | 7.906 | 98.076% | 0.892 | |

Stanford [41] | 2001 | 40 | 16 | 8.306 | 27.515% | 0.615 | |

Yeast [42] | 1998 | 109 | 16 | 6.614 | 5.391% | 0.518 | |

YuLou [43] | 2004 | 3 | 16 | 9.422 | 36.906% | 0.556 | |

Overall | 298 | 16 | 7.415 | 67.003% | 0.782 |

Data sets | Year | Images | Approximate size (cols × rows) | Depth | Average entropy (bpp) | Average intensity usage (percentage) | Gini Index[0-1] |

CR | 2003 | 12 | 10-16 | 9.905 | 11.487% | 0.240 | |

CT | 2003 | 12 | 12-16 | 8.032 | 2.879% | 0.389 | |

MR | 2003 | 12 | 16 | 6.837 | 1.636% | 0.624 | |

US | 2003 | 12 | 8 | 4.561 | 0.358% | 0.714 | |

MIAS [33] | 1995 | 322 | 8 | 4.544 | 0.357% | 0.681 | |

Overall | 370 | 8-16 | 5.169 | 1.596% | 0.631 |

Data sets | Year | Images | Approximate size (cols × rows) | Depth | Average entropy (bpp) | Average intensity usage (percentage) | Gini Index[0-1] | |

RNAi D1 | ∣ (DNA) | 2,304 | 7.408 | 4.140% | 0.274 | |||

∣ (PH3) | 2006 | 2,304 | 12 | 5.631 | 1.019% | 0.056 | ||

∣ (Actin) | 2,304 | 7.583 | 2.287% | 0.192 | ||||

RNAi D2 | ∣ (DNA) | 2,304 | 6.617 | 2.925% | 0.172 | |||

∣ (PH3) | 2006 | 2,304 | 12 | 5.572 | 0.661% | 0.050 | ||

∣ (Actin) | 2,304 | 6.565 | 1.334% | 0.099 | ||||

RNAi D3 | ∣ (DNA) | 2,304 | 7.210 | 4.195% | 0.257 | |||

∣ (PH3) | 2006 | 2,304 | 12 | 5.659 | 0.981% | 0.054 | ||

∣ (Actin) | 2,304 | 7.653 | 3.215% | 0.242 | ||||

RNAi D4 | ∣ (DNA) | 2,304 | 7.530 | 4.249% | 0.286 | |||

∣ (PH3) | 2006 | 2,304 | 12 | 5.662 | 1.080% | 0.057 | ||

∣ (Actin) | 2,304 | 7.907 | 3.070% | 0.244 | ||||

RNAi D5 | ∣ (DNA) | 2,304 | 7.281 | 4.046% | 0.256 | |||

∣ (PH3) | 2006 | 2,304 | 12 | 5.622 | 0.904% | 0.052 | ||

∣ (Actin) | 2,304 | 7.649 | 2.772% | 0.221 | ||||

RNAi all | ∣ (DNA) | 11,520 | 7.209 | 3.911% | 0.249 | |||

∣ (PH3) | 2006 | 11,520 | 12 | 5.629 | 0.929% | 0.054 | ||

∣ (Actin) | 11,520 | 7.471 | 2.536% | 0.199 | ||||

Overall | 2006 | 34,560 | 12 | 6.770 | 2.458% | 0.167 |

Taking into consideration Table 3, we can see that the image sizes of the microarray data sets are very large (ranging from *Omnibus-HM* data set.

Regarding the medical image data sets (see Table 4), we can see that the image sizes are lower when compared to the microarray images (except for the *mini-MIAS* data set). The bit depth varies from 8 to 16 bits and the average entropy from 4.5 bpp to 9.9 bpp. The average intensity usage is very low for these medical images. For the *CR* data set, only 11.5% of the total available intensities are actually used. For the other data sets, this measure is even lower *MR*, *US*, and *mini-MIAS* data sets have a GI higher than 0.6. On the other hand, for the *CR* and *CT* data sets, the GI values are lower (0.24 and 0.39, respectively).

Finally, we provide 5 RNAi image data sets in Table 5. We split each data set into three channels (DNA, PH3, and actin). All the images have size

### 3.2. Experimental results

In this subsection, we present the results obtained using several general purpose compression tools (e.g., gzip, bzip2, PPMd, and LZMA), image coding standards (e.g., PNG, JBIG, JPEG-LS, and JPEG2000), and the methods described in Sect. 4. In Tables 6-8, we can find the compression results for all data set described earlier in Sect. 5.1, using several general compression tools and image coding standards. Default parameters were used in all compression tools. Hence, we did not try to adjust some parameters in order to improve the compression results. Only the lossless mode was imposed.

JBIG results were obtained using version 2.0 of the JBIG-Kit package[1] -. The results for the JPEG-LS standard were obtained using version 2.2 of the SPMG JPEG-LS codec[1] -. JPEG2000 lossless compression was obtained using version 5.1 of JJ2000 codec with default parameters for lossless compression[1] -. For additional reference, we also provided compression results using the gzip, bzip2, ppmd, and lzma general-purpose compression tools.

Compression methods | ||||||||

Data sets | Gzip | Bzip2 | PPMd | LZMA | PNG | JBIG | JPEG-LS | JPEG2000 |

ApoA1 | 12.711 | 11.068 | 10.984 | 11.374 | 12.568 | 10.851 | 10.608 | 11.063 |

Arizona | 11.263 | 9.040 | 8.980 | 9.402 | 11.017 | 8.896 | 8.676 | 9.107 |

IBB | 10.453 | 9.081 | 8.495 | 8.985 | 10.090 | 9.344 | 9.904 | 10.516 |

ISREC | 12.464 | 10.922 | 10.730 | 11.126 | 12.476 | 10.925 | 11.145 | 11.366 |

Omnibus (LM) | 7.124 | 5.346 | 4.977 | 5.527 | 6.781 | 5.130 | 4.936 | 5.340 |

Omnibus (HM) | 9.558 | 7.523 | 7.219 | 7.787 | 9.160 | 7.198 | 6.952 | 7.587 |

Stanford | 9.972 | 7.961 | 7.809 | 8.273 | 9.776 | 7.906 | 7.684 | 8.060 |

Yeast | 7.672 | 6.075 | 5.794 | 6.389 | 8.303 | 6.888 | 8.580 | 9.079 |

YuLou | 11.434 | 9.394 | 9.285 | 9.708 | 11.428 | 9.298 | 8.974 | 9.515 |

Average | 9.044 | 7.189 | 6.859 | 7.388 | 8.729 | 7.051 | 6.996 | 7.511 |

Compression methods | ||||||||

Data sets | Gzip | Bzip2 | PPMd | LZMA | PNG | JBIG | JPEG-LS | JPEG2000 |

CR | 9.284 | 6.140 | 6.230 | 7.197 | 7.991 | 6.152 | 5.784 | 5.845 |

CT | 8.790 | 5.604 | 5.847 | 6.996 | 9.411 | 8.822 | 8.089 | 8.334 |

MR | 8.342 | 5.313 | 5.514 | 6.663 | 9.795 | 9.940 | 9.321 | 9.389 |

US | 3.496 | 2.742 | 2.642 | 2.820 | 3.061 | 2.908 | 2.750 | 3.138 |

mini-MIAS | 2.637 | 1.696 | 1.590 | 2.022 | 1.672 | 1.562 | 1.416 | 1.443 |

Average | 3.452 | 2.242 | 2.163 | 2.660 | 2.484 | 2.183 | 2.005 | 2.040 |

Compression methods | |||||||||

Data sets | Gzip | Bzip2 | PPMd | LZMA | PNG | JBIG | JPEG-LS | JPEG2000 | |

RNAi D1 | ∣ (DNA) | 9.120 | 6.846 | 6.832 | 7.314 | 9.857 | 6.802 | 6.311 | 6.533 |

∣ (PH3) | 7.461 | 5.933 | 5.804 | 6.008 | 11.552 | 5.994 | 5.806 | 5.830 | |

∣ (Actin) | 9.274 | 6.807 | 6.839 | 7.353 | 11.801 | 6.790 | 6.304 | 6.452 | |

RNAi D2 | ∣ (DNA) | 8.358 | 6.469 | 6.389 | 6.714 | 9.421 | 6.469 | 6.094 | 6.233 |

∣ (PH3) | 7.396 | 5.884 | 5.760 | 5.963 | 12.420 | 5.964 | 5.793 | 5.804 | |

∣ (Actin) | 8.309 | 6.378 | 6.315 | 6.638 | 12.449 | 6.422 | 6.032 | 6.122 | |

RNAi D3 | ∣ (DNA) | 8.948 | 6.735 | 6.637 | 7.153 | 9.665 | 6.704 | 6.238 | 6.436 |

∣ (PH3) | 7.490 | 5.952 | 5.819 | 6.030 | 11.756 | 6.042 | 5.812 | 5.842 | |

∣ (Actin) | 9.362 | 6.874 | 7.387 | 7.431 | 10.582 | 6.838 | 6.353 | 6.509 | |

RNAi D4 | ∣ (DNA) | 6.243 | 6.925 | 6.915 | 7.415 | 10.014 | 6.878 | 6.372 | 6.612 |

∣ (PH3) | 7.491 | 5.950 | 5.824 | 6.028 | 11.548 | 6.028 | 5.813 | 5.845 | |

∣ (Actin) | 9.586 | 7.014 | 7.070 | 7.629 | 10.955 | 6.977 | 6.460 | 6.630 | |

RNAi D5 | ∣ (DNA) | 9.007 | 6.765 | 6.745 | 7.198 | 9.775 | 6.729 | 6.259 | 6.459 |

∣ (PH3) | 7.450 | 5.928 | 5.798 | 6.001 | 11.745 | 5.994 | 5.804 | 5.825 | |

∣ (Actin) | 9.346 | 6.857 | 6.893 | 7.414 | 11.185 | 6.828 | 6.342 | 6.493 | |

RNAi all | ∣ (DNA) | 8.935 | 6.748 | 6.703 | 7.159 | 9.746 | 6.716 | 6.255 | 6.455 |

∣ (PH3) | 7.458 | 5.929 | 5.801 | 6.006 | 11.804 | 6.004 | 5.806 | 5.829 | |

∣ (Actin) | 9.176 | 6.786 | 6.900 | 7.293 | 11.394 | 6.771 | 6.298 | 6.441 | |

Average | 8.523 | 6.488 | 6.468 | 6.819 | 10.982 | 6.497 | 6.120 | 6.242 |

According to the results depicted in Tables 6-8, it seems that, globally, JPEG-LS is the best image coding standard. Among the general compression methods used, PPMd is the one that attained the best compression results.

Tables 9-11 provide lossless compression results using the methods described in Sect. 4. BPD (bitplane decomposition) corresponds to the results using method [6]; BFS (background-foreground separation) corresponds to using the previous approach with segmentation. The columns HC (histogram compaction) and SBR (scalable bitplane reduction) correspond to the approaches that use bitplane reduction in method [6]. The SBC-Mix (simple bitplane coding-mixture) corresponds to the approach described in Sect. 4.2.3 using a mixture of finite-context models. The last column, denoted as BTD (binary-tree decomposition), corresponds to the results obtained using the method described in Sect. 4.3.

If we look closely to Table 9, we can see that the method based on binary-tree decomposition is the one that attained the best compression results among all the others, for the case of microarray images. The best method is about 11% better than JPEG-LS. However, the method SBC-Mix is the best for three data sets: *ApoA1*, *ISREC*, and *Yulou*.

Regarding the experiments performed on medical images and taking into consideration the results presented in Table 10, we can conclude once again that the method based on binary-tree decomposition outperforms all the others and is about 9% better than JPEG-LS.

Compression methods | |||||||

Data sets | BPD | BFS | HC | SBR | SBC | SBC-Mix | BTD |

ApoA1 | 10.194 | 10.234 | 10.231 | 10.232 | 10.205 | 10.142 | 10.194 |

Arizona | 8.242 | 8.245 | 8.244 | 8.243 | 10.308 | 8.219 | 8.186 |

IBB | 7.974 | 7.982 | 7.978 | 7.978 | 8.537 | 7.966 | 7.943 |

ISREC | 10.159 | 10.193 | 10.195 | 10.199 | 10.260 | 10.148 | 10.198 |

Omnibus (LM) | 4.567 | 4.570 | 4.561 | 4.565 | 4.645 | 4.545 | 4.539 |

Omnibus (HM) | 6.471 | 6.479 | 6.473 | 6.472 | 6.581 | 6.443 | 6.400 |

Stanford | 7.379 | 7.349 | 7.350 | 7.349 | 7.403 | 7.305 | 7.303 |

Yeast | 5.453 | 5.395 | 5.527 | 5.466 | 5.492 | 5.326 | 5.318 |

YuLou | 8.619 | 8.641 | 8.626 | 8.626 | 8.669 | 8.591 | 8.592 |

Average | 6.284 | 6.288 | 6.286 | 6.285 | 6.437 | 6.257 | 6.235 |

Compression methods | |||||||

Data sets | BPD | BFS | HC | SBR | SBC | SBC-Mix | BTD |

CR | 5.338 | 5.207 | 5.291 | 5.214 | 5.764 | 5.137 | 5.136 |

CT | 4.836 | 4.840 | 4.857 | 4.875 | 8.302 | 4.762 | 4.809 |

MR | 4.654 | 4.702 | 4.835 | 4.833 | 9.382 | 4.594 | 4.783 |

US | 2.472 | 2.564 | 2.478 | 2.471 | 2.722 | 2.439 | 2.462 |

mini-MIAS | 1.390 | 1.378 | 1.391 | 1.391 | 1.430 | 1.358 | 1.352 |

Average | 1.877 | 1.853 | 1.874 | 1.865 | 2.016 | 1.826 | 1.822 |

Compression methods | ||||||||

Data sets | BPD | BFS | HC | SBR | SBC | SBC-Mix | BTD | |

RNAi D1 | ∣ (DNA) | 6.250 | 6.293 | 6.250 | 6.255 | 6.225 | 6.179 | 6.186 |

∣ (PH3) | 5.559 | 5.570 | 5.570 | 5.569 | 5.571 | 5.545 | 5.555 | |

∣ (Actin) | 6.255 | 6.311 | 6.244 | 6.258 | 6.220 | 6.185 | 6.205 | |

RNAi D2 | ∣ (DNA) | 5.959 | 5.986 | 5.968 | 5.968 | 5.960 | 5.918 | 5.930 |

∣ (PH3) | 5.537 | 5.547 | 5.539 | 5.540 | 5.545 | 5.518 | 5.525 | |

∣ (Actin) | 5.919 | 5.948 | 5.904 | 5.928 | 5.904 | 5.876 | 5.892 | |

RNAi D3 | ∣ (DNA) | 6.175 | 6.213 | 6.169 | 6.164 | 6.145 | 6.106 | 6.111 |

∣ (PH3) | 5.580 | 5.598 | 5.584 | 5.585 | 5.590 | 5.564 | 5.572 | |

∣ (Actin) | 6.313 | 6.366 | 6.299 | 6.306 | 6.277 | 6.236 | 6.247 | |

RNAi D4 | ∣ (DNA) | 6.309 | 6.355 | 6.312 | 6.317 | 6.289 | 6.239 | 6.244 |

∣ (PH3) | 5.577 | 5.593 | 5.582 | 5.584 | 5.587 | 5.561 | 5.570 | |

∣ (Actin) | 6.423 | 6.484 | 6.412 | 6.419 | 6.395 | 6.348 | 6.363 | |

RNAi D5 | ∣ (DNA) | 6.195 | 6.233 | 6.194 | 6.198 | 6.166 | 6.126 | 6.133 |

∣ (PH3) | 5.558 | 5.568 | 5.564 | 5.567 | 5.569 | 5.543 | 5.552 | |

∣ (Actin) | 6.299 | 6.355 | 6.286 | 6.298 | 6.263 | 6.224 | 6.240 | |

RNAi all | ∣ (DNA) | 6.178 | 6.216 | 6.179 | 6.182 | 6.157 | 6.113 | 6.121 |

∣ (PH3) | 5.562 | 5.575 | 5.568 | 5.569 | 5.573 | 5.546 | 5.555 | |

∣ (Actin) | 6.242 | 6.293 | 6.229 | 6.242 | 6.212 | 6.174 | 6.189 | |

Average | 5.994 | 6.028 | 5.992 | 5.998 | 5.980 | 5.945 | 5.955 |

Finally, in Table 11, we present the results for the RNAi images, and in this case, method SBC-Mix is the one that attained the best compression results, almost 3% better than JPEG-LS. The difference between the SBC-Mix and BTD is very low (close to 0.01 bpp). In our opinion, it seems that the methods described in Sect. 4 attain better compression results when compared to JPEG-LS for images with higher GI values (sparse images). This is the reason why the binary-tree decomposition approach attains between 9% to 11% better results than JPEG-LS for the microarray and medical images. On the other hand, the RNAi images have a lower GI (around 0.17), therefore the lower performance of our methods when compared to JPEG-LS.

In Figs. 15 and 16, we can see the encoding and decoding speed in kilobytes per second for the compression methods described in Sect. 4. In the encoding phase, it seems that the method based on binary-tree decomposition is the fastest one among all the others for all data sets. In the decoding phase, the fastest method is different from data set to data set. For the microarray and medical image data sets, the methods based on bitplane decomposition are faster than the method based on binary-tree decomposition. On the other hand, for the RNAi images, the method based on binary-tree decomposition is faster than all the others. The SBC-Mix method is the slowest one in all data sets. The lower decoding speed for the SBC-Mix is due to the mixture procedure. For each symbol that is processed, the model weights need to be updated, which is a very time-consuming task.

### 3.3. Rate distortion study

The two decomposition approaches described in Sect. 4 have progressive decoding capabilities (also known as lossy to lossless). Due to that, it is important to understand what is the error induced in the image during the decoding phase when only part of the data is decoded. This characteristic allows also to obtain the original image without any loss.

In Figs. 15 and 16, we present the rate distortion curves of two images, "array1" from the *YouLou* data set and "cr_17218" from the medical *CR* data set, according to two metrics:

Regarding Fig. 15, we can notice that the method based on binary-tree decomposition outperforms all the other methods in terms of

In Fig. 16, we can see in the first plot, related to the

## 4. Conclusions

The use of biomedical imaging has increased in the last years. These biomedical images include microarray, medical, and RNAi images. In this chapter, we have studied two image decomposition approaches to be applied to this type of images in order to compress them efficiently.

We presented a set of comprehensive results regarding the lossless compression of biomedical images using general coding methods (e.g., gzip), image coding standards (e.g., JPEG2000), and the two image decomposition approaches that rely on finite-context models and arithmetic coding.

From the obtained experimental results, we conclude that JPEG-LS gives the best compression results, among the image coding standards, but lacks lossy-to-lossless capability, which may be a decisive functionality if remote transmission over possibly slow links is a requirement.

We developed compression algorithms based on two decomposition approaches: bitplane decomposition and binary-tree decomposition. In the bitplane decomposition, we also used segmentation, bitplane Reduction, and an approach based on bit modeling by the pixel value estimates. All the developed methods allow lossy-to-lossless compression and are based on finite-context models and arithmetic coding. According to the obtained results, the approach based on binary-tree decomposition seems to be the one that attains the best compression results. The only exceptions are the RNAi images, for which the best method is the one based on a mixture of finite-context models.

In terms of coding speed, the compression algorithm based on a binary-tree decomposition seems to be the fastest one among all the others in the encoding phase. In the decoding stage, the approaches based on bitplane decomposition seem to be faster for the microarray and medical image data sets. For the RNAi images, the method based on binary-tree decomposition seems to be the fastest one.

A rate distortion study was also performed and, according to the obtained results, it seems that the method based on binary-tree decomposition attains the best results for the majority of the cases. The results obtained by the developed compression algorithms have been compared with general-purpose compression methods and also with image coding standards. According to the results, we can conclude that these compression methods have better compression performance in all the image test sets used.

## Acknowledgments

This work was developed at IEETA (Institute of Electronics and Telematics Engineering of Aveiro).It was partially supported by FEDER, through the Operational Program Competitiveness Factors (COMPETE), and by National Funds through FCT (Foundation for Science and Technology) in the context of a PhD Grant (FCT reference SFRH/BD/85855/2012) and projects PEst-OE/EEI/UI0127/2014 and Incentivo/EEI/UI0127/2014.

## Notes

- http://www.cl.cam.ac.uk/ mgk25/jbigkit
- The original website of this codec, http://spmg.ece.ubc.ca, is currently unavailable. However, it can be obtained from http://sweet.ua.pt/luismatos/codecs/jpeg_ls_v2.2.tar.gz
- The original website of this codec, http://jj2000.epfl.ch, is currently unavailable. Nevertheless, this codec can be obtained from http://sweet.ua.pt/luismatos/codecs/jj2000_5.1-src.zip