A fast and efficient method for compressing fmri data sets

Transcription

1 A fast and efficient method for compressing fmri data sets Fabian J. Theis 1,2 and Toshihisa Tanaka 1 1 Department of Electrical and Electronic Engineering, Tokyo University of Agriculture and Technology, Tokyo , Japan 2 Institute of Biophysics, University of Regensburg, Regensburg, Germany Abstract. We present a new lossless compression method named FTTcoder, which compresses images and 3d sequences collected during a typical functional MRI experiment. The large data sets involved in this popular medical application necessitate novel compression algorithms to take into account the structure of the recorded data as well as the experimental conditions, which include the 4d recordings, the used stimulus protocol and marked regions of interest (ROI). We propose to use simple temporal transformations and entropy coding with context modeling to encode the 4d scans after preprocessing with the ROI masking. Experiments confirm the superior performance of FTTcoder in contrast to previously proposed algorithms both in terms of speed and compression. 1 Introduction Modern imaging techniques become increasingly important and common in medical diagnostics; such techniques include X-ray computerized tomography (CT) and magnetic resonance imaging (MRI). Although much of this paper can be applied to more general (medical) image series, we will in the following focus on the latter. MRI visualizes three-dimensional objects by measuring magnetic properties within small volume elements called voxels. Even single scans of CT and MRI are already large in size (up to several megabytes). More recently, it has become popular to also study time series of MRI scans, during which the subject performs various functional tasks, hence the term functional MRI (fmri). These data sets measure up to several hundred megabytes, and efficient storage methods have to be applied. As even small deviations within adjacent scans may already contain important information (which can for example be revealed by blind source separation [1]), lossy image and time series compression is out of question, and lossless (or at most near lossless) techniques have to be used. Moreover efficiency not so much in compression rate but also in speed is essential, as fast and preferably sequential, single-pass processing enables close to real-time data preprocessing and analysis. The field of medical image and volume compression is still rather young, with popular algorithms being direct generalizations of image compression techniques such as for example 3d context modeling [2]. Image sequence coding methods

2 have also recently gained some attention, for instance by compressing sequences using integer wavelet transform with subsequent entropy coding [3]. However specific application to fmri are rare, and to our knowledge only some simple algorithms exist at present: SmallTime [4] performs a very fast but rather inefficient fmri compression by simply taking difference images between adjacent recordings and then storing the typically 8-bit difference image in contrast to the 16-bit recorded image. Adamson employs the efficient LOCO-I (low complexity lossless compression for images) algorithm [5] to compress MRI and fmri recordings in sequence [6]. This enables high data throughput with acceptable compression efficiency, and we will generalize his proposal using JPEG-LS( Joint Photographic Experts Group -lossless) with added preprocessing in the following. Lossy fmri compression has been proposed in [7] but has not been adopted by the community, most probably due to the above mentioned sensitivity of MRI recordings to small deviations. Our algorithm processes the time series, interpreted as recordings from multiple sources by also taking into account additional recordings such as a regionof-interest (ROI) or mask selection. This information, stemming from multiple sources, is packed into a single data stream allowing for efficient and fast storage and recovery, outperforming present fmri compression algorithms considerably both in speed and compression ratio. 2 Lossless image compression using JPEG-LS Lossless compression should be used for a variety of applications, particularly those involving medical imaging such as CT and (f)mri. For these applications, ISO/IEC provided a lossless algorithm in the JPEG international standard, but, unlike the baseline JPEG (lossy), this lossless version is poor in compression performance. Instead, a new lossless image compression standard called JPEG- LS is provided [8]. This compression algorithm draws heavily from the LOCO-I method developed by Weinberger at al. [5] and aims at improving compression performance with low complexity. JPEG-LS is an image compression algorithm involving prediction and entropy coding. The coder effectively uses four neighboring pixels. Let x[n 1, n 2 ] be a value of the current pixel. The neighborhood consists of the four samples: x[n 1, n 2 1], x[n 1 1, n 2 ], x[n 1 1, n 2 1], and x[n 1 1, n 2 + 1], respectively denoted by x a, x b, x c, and x d, which are exploited for modeling contexts of an entropy coder as well as determining a mode. JPEG-LS switches between two modes: normal and run modes, depending on the neighborhood. At the current pixel, if one of gradients defined as 1 = x d x b, 2 = x b x c, and 3 = x c x a are non-zero, then the JPEG-LS coder is in normal mode. In normal mode, prediction of x denoted by µ x is obtained by a non-linear function of the four neighbor pixels. Then, the residual that is actually coded in JPEG-LS is given by e x = s x (x µ x ) β x, where s x represents the sign and β x the bias compensating term, which is needed to make the probability distribution of residuals unbiased symmetric. The JPEG-LS coder represents the mapped residuals by using an adaptive Golomb code [9] with context modeling.

3 xc x a xb x xd x Current pixel xa Memory x b xc xd xa x b Normal Mode x c e x Predictor Residual Coder x a x ref xa Run Mode x b xc xd Gradient Calculator Mode Switch Controller Bitstream Run Coder Fig.1. Schematic JPEG-LS compression This residual coding is context adaptive, where the context used for the current sample is identified by a context quantization function of the three gradients. Then, context-dependent Golomb and bias parameters are estimated sample by sample. If all the gradients are identical, then the JPEG-LS coder moves to the run mode. The assumption here is that x and possibly a large number of consecutive samples are all likely to have the same value as x a. The number of samples which are all the same as x a in a scanning direction is called the run-length. This run-length is coded by using Golomb code again. But, here a so-called MELCODE [5], which is specialized for encoding the run-length, is utilized in JPEG-LS. This compression scheme is visualized in figure 1; for details, see [5] for example. 3 Compressing information from multiple sensors The goal of the proposed compression algorithm is to fuse the data set, acquired by multiple scans over time, into a single easy to store file of decreased size as quickly as possible. Furthermore, additional information such as masked voxels or stimulus components corresponding to the protocol used in the fmri recordings can be merged with the data stream. 3.1 fmri compression The MRI measurements x(t) {0, 1,..., α} w h d are taken for time points t = 1,...,T with a temporal resolution in the range of seconds and size T 100. The measured data for each time point is a three-dimensional data structure; each scan is of size w h d and voxels take integer values between 0 and α. Our compression algorithm is simple in concept just apply the efficient and fast context-based JPEG-LS onto each slice image, but after some intelligent preprocessing. A desirable image property enabling high compression rates is simple structure together with low α. This property can be achieved in our case of biomedical time series by temporal operations. Different methods and filters are possible such as integer wavelet transformation using for example a simple

4 Haar wavelet, or discrete cosine transform. For fmri it is advisable to employ a structurally simple, preferably even linear transformation that uses a rather small time window this would increase coding and decoding speed and keep the memory consumption low albeit at the possible loss of some compression efficiency. Hence we decided to fully encode the first scan x(1), and then only encode the difference images x(t) := x(t) x(t 1) (after possible translation to have a zero-valued minimum). Another property of fmri data is that typical scans are taken of restricted regions that do not fully fill out the whole scan volume for example fmri is a common tool for brain imaging, and the non-brain volume or at least voxels outside of the head can be easily identified and masked out, see figure 2(a). After preprocessing by motion alignment [10] the scans are temporally aligned. Hence we can assume to have a single time-independent mask y {0, 1} w h d. The mask is assumed to be given by the user, and to be binary with ones indicating the ROI. Context-based coding of the full volumes is not as efficient, so the additional ROI data can be used to enhance compression: only encode the non-masked voxels of each line, and during reconstruction recover the full line by adding zeros (or mask values) at the masked voxels. Then of course the mask y has to be stored in the compressed file. For this we use run-length encoding, which works well due to the often simple structure of the mask. Given a sufficiently large number of scans additional masking turns out to be of significantly higher efficiency. The resulting algorithm is single-run, as there is no need of returning to previous scans if at most two scans are held in memory at the same time (which is acceptable, typical sizes are up to 2MB per scan). So memory efficiency is also provided, and the algorithm is fast as confirmed by the experiments later. We call the resulting compression algorithm FTTcoder (fmri temporal transformation coder), and refer to algorithms 1 and 2 for details. 3.2 Fusing the stimulus A peculiarity of functional MRI data sets is the presence of a stimulus protocol. It describes the functional task used in the experiment. Typically either a simple block design or a so-called event-based protocol is used. The former describes a periodic on-off stimulus, whereas the latter consists of activities interspersed with a varying period of non-activity, which can also depend on the subject s interaction. In this section, we will fuse the additional before-hand knowledge of the stimulus with the data to achieve more efficient compression ratios; for simplicity, we will use a block design protocol. In addition to the observed data set x(t) assume that the binary stimulus protocol σ(t) {0, 1} is given for t = 1,...,T. A simple model for incorporating the stimulus with the MRI data can be built by assuming that an underlying stimulus-independent activity is additively overlayed by the stimulus-related brain activity at time instants t where σ(t) = 1. If x (0),x (1) R w h d denote the stimulus-independent and stimulus-dependent data component respectively, then according to the model the data time series x(t) can be written as x(t) = x (0) + σ(t)x (1) + e(t), (1)

5 Data: scan sizes w h d, scan range [0, α] N 0, T fmri scans x(1),...,x(t) [0, α] w h d, optional common mask y {0, 1} w h d Result: compressed bit stream b 1 store sizes h, w, d, T and range α in b 2 if mask is used then run-length encode and store binary y in b for t 1,..., T do 3 if t = 1 then z x(1) else z x(t) x(t 1) 4 if necessary translate z and determine and store new range [0, α ] by calculating minima and maxima of z for j 1,..., h, k 1,..., d do 5 determine masked current and previous lines: l {z(i, j, k) i with y(i, j, k) = 1} pl {z(i, j 1, k) resp. z(i, j, k 1) i with y(i, j, k) = 1} 6 context encode line l using context (l,pl) to stream b by JPEG-LS end end Algorithm 1: compression algorithm FTTcoder Data: fmri-compressed bit stream b Result: fmri scans x(1),...,x(t) [0, α] w h d 1 read scan sizes h w d, number of scans T and maximal range α from b 2 run-length decode optional binary mask y {0,1} w h d from b for t 1,..., T do for j 1,..., h, k 1,..., d do 3 determine masked previous line: pl {z(i, j 1, k) resp. z(i, j, k 1) i with y(i, j, k) = 1} 4 context decode line l using context pl from stream b by JPEG-LS 5 recover unmasked line: z({i y(i, j, k) = 1}, j, k) l end 6 if necessary translate z 7 if t = 1 then x(1) z else x(t) x(t 1) + z end Algorithm 2: decompression algorithm inverting FTTcoder where e(t) denotes the model error at time instant t. The model is fulfilled well if e(t) is small for all t, and compression can be improved in this case. Please note that of course more advanced models (including convolutions induced by the BOLD effect, using blind separation for additional component identification etc.) are possible and used in the analysis of fmri data sets, but our goal is to keep the algorithm simple, fast and efficient; especially the single-run property must not be destroyed, so more complex models might be difficult to include. Let σ(t) := σ(t) σ(t 1), t = 2,...,T, and denote time instants t with σ(t) 0 as stimulus jumps. The compression performance can be increased by reducing the range and the deviation of the difference image x(t) to be compressed. However at stimulus jumps, the differences can be expected to be larger than at other time instants. By using model (1), we get x(t) = x (0) +σ(t)x (1) +e(t) x (0) σ(t 1)x (1) e(t 1) = σ(t)x (1) + e(t)

6 with e(t) := e(t) e(t 1). So x(t) is larger at stimulus jumps. For compression, we now propose to estimate the stimulus component x (1) by the normalized difference image ˆx (1) := σ(t 0 ) x(t 0 ) (which equals x (1) ± e(t) and hence approximately x (1) ) at the first jump t 0. Subsequently, instead of encoding x(t), we compress z(t) := x(t) σ(t)ˆx (1) = x(t) σ(t) σ(t 0 ) x(t 0 ), (2) which equals σ(t) ( x (1) ˆx (1)) + e(t) and can therefore expected to be small. The stimulus can be included in the compressed stream using run-length encoding. For decompression, this can easily be inverted by restoring the stimulus approximation ˆx (1) at the first jump, and then reconstructing subsequent x(t) from the decompressed frame z(t) by 4 Results x(t) := x(t 1) + z(t) + σ(t)ˆx (1). (3) We demonstrate the performance of FTTcoder (freely available for download at when applied to real data sets. For this we use two data sets. The first one is a two-dimensional slice with w = h = 128, α = 2048 and T = 98 scans. The data set has been masked with a rather large mask, see figure 2(a) for the (masked) first scan of the series. Physically, the data set is represented by a large file containing a concatenation of all scans. The second data set is given by 240 analyze files with scan size The data has not been masked, and the largest possible α = is used for the 16bit range. For illustration, FTTcoder performance is compared with two previously proposed fmri compression schemes as well as three general algorithms namely direct file copy, zip and the efficient bzip2. For the latter three algorithms, the plain data itself is directly compressed, not the temporally preprocessed one. The two fmri compression utilities are iterative LOCO-I compression of images [6], which due to lack of code we emulate by using FTTcoder without temporal differences and masking, and direct storage by mapping 16bit values into 8bit values if possible (SmallTime, [4]). We compare both compressed file size and speed. The experiments have been made on a Pentium M 2.0GHz using cygwin. The results are shown in table 1. Clearly FTTcoder outperforms the other algorithms both in compression ratio as well as in speed the latter is at first a bit astonishing when comparing against file copy, but this is due to the fact that the much smaller compressed file takes less time to be stored on hard disc than the file copy of the larger one. Apparently due to its implementation, SmallTime is considerably slower than the other algorithms and also less efficient, although we note that the given data set contains a large number of non-roi voxels, which SmallTime does not compress efficiently. SmallTime was unable to compress the second data set, which we believe is due to the fact that almost all differences were 16bit. Direct application of LOCO-I performs comparably well in terms of speed as FTTcoder, but

7 Table 1. Algorithm performance; speed was measured for combined compression/decompression using the mean over 100 runs (first data set) respectively 10 runs (second data set). data set original size algorithm compression speed 2d data set 3137 kb FTTcoder (mask) 245 kb (7.8%) 0.182s 2d data set 3137 kb FTTcoder (no mask) 261 kb (8.3%) 0.185s 2d data set 3137 kb FTTcoder (no diffs) 350 kb (11.2%) 0.166s 2d data set 3137 kb FTTcoder (no mask&diffs) 366 kb (11.7%) 0.162s 2d data set 3137 kb SmallTime 1589 kb (50.7%) 1.3s 2d data set 3137 kb bzip2 394 kb (12.5%) 0.514s 2d data set 3137 kb zip 562 kb (17.9%) 0.255s 2d data set 3137 kb file copy 3137 kb (100%) 0.192s analyze data set 22.6 MB FTTcoder (no mask) 14.1 MB (62.4%) 5.16s analyze data set 22.6 MB FTTcoder (no mask&diffs) 17.0 MB (75.2%) 5.9s analyze data set 22.6 MB SmallTime runtime error - analyze data set 22.6 MB tar and bzip MB (69.0%) 14.6s analyze data set 22.6 MB tar and gzip 18.0 MB (79.7%) 12.5s analyze data set 22.6 MB file copy 22.6 MB (100%) 14.3s the compression rate is considerably lower. In practical applications, FTTcoder is about 3 times as fast as traditional zip algorithms, and considerably more efficient. We finally compare the proposed extended stimulus-based compression algorithm from section 3.2 with plain FTTcoder. The algorithm performance depends greatly on how well the model (1) is fulfilled i.e. how large the error terms e(t) are. In the following, we construct a toy data set by choosing x (0) to be the brain slice from figure 2(a) with an additive stimulus component x (1) constructed by setting pixels randomly to ±1 within a fixed rectangle in the brain part, see figure 2(b). Within the brain, white Gaussian noise is added with varying SNR in [ 1.5dB, db]. A stimulus of 6 off, 6 on periods of total length T = 98 was used. We compare the file sizes of the compressed data sets. Figure 2(c) shows the ratios. Clearly, the stimulus based algorithm considerably outperforms the normal one in the no- and low-noise cases, but the performance decreases with increasing noise. Similar performance is achieved starting at SNRs of around 8dB. We conclude that depending on how well the stimulus model is fulfilled, fusing the additional stimulus component with the data may increase compression ratios. In the future, we will study more advanced temporal preprocessing. 5 Conclusion We have proposed a novel lossless compression scheme for sequences of medical images, focusing on fmri recordings. The algorithm is based on JPEG-LS and turns to be more efficient in both speed and compression rate than more generic compression algorithms. It is well known that even for data sets with a small number of two-dimensional slices and a high slice distance there is a significant gain in compression ratio by compressing the 3d data spatially in contrast to compressing the two-dimensional slices separately [2], and we currently work on generalizing JPEG-LS to 3d contexts and employing this for fmri compression.

8 1.1 1 compression ratio (a) slice&mask (b) toy data set noise level (c) compression performance Fig. 2. In (a) a single fmri slice together with a selected ROI/mask differentiating brain from non-brain voxels is presented. (b) shows the toy stimulus component x (1) together with a slice image at active stimulus and noise level 100. Figure (c) compares the fmri compression based on stimulus-fusion versus normal compression by plotting the ratio of the stimulus-compressed file size and the non-stimulus-compressed size. References 1. Keck, I., Theis, F., Gruber, P., Lang, E., Specht, K., Puntonet, C.: 3D spatial analysis of fmri data - a comparison of ICA and GLM analysis on a word perception task. In: Proc. IJCNN 2004, Budapest, Hungary (2004) Klappenecker, A., May, F., Beth, T.: Lossless compression of 3D MRI and CT data. In Laine, A., Unser, M., Aldroubi, A., eds.: Proc. SPIE, Wavelet Applications in Signal and Imaging Processing VI. Volume (1998) M.Wu, Forchhammer, S.: Medical image sequence coding. In: Proc. DSAGM, Copenhagen, Denmark (2004) Cohen, M.: A data compression method for image time series. Human Brain Mapping 12 (2001) Weinberger, M., Seroussi, G., Sapiro, G.: The LOCO-I lossless image compression algorithm: Principles and standardization into JPEG-LS. IEEE Transactions on Image Processing 9 (2000) Adamson, C.: Lossless compression of magnetic resonance imaging data. Master s thesis, Monash University, Melbourne, Australia (2002) 7. Taswell, C.: Wavelet transform compression of functional magnetic resonance image sequences. In: Proc. SIP, Las Vegas, USA (1998) Taubman, D.S., Marcellin, M.W.: JPEG 2000 Image Compression Fundamentals, Standards and Practice. Kluwer Academic Publishers, Massachusetts (2002) 9. Golomb, S.: Run-length encodings. IEEE Transactions on Information Theory 12 (1966) Woods, R., Cherry, S., Mazziotta, J.: Rapid automated algorithm for aligning and reslicing pet images. Journal of Computer Assisted Tomography 16 (1992)