On the Utilization of Reversible Colour Transforms for Lossless 2-D Data Compression
On the Utilization of Reversible Colour Transforms for Lossless 2-D Data Compression
Waleed, Muhammad;Um, Tai-Won;Khan, Aftab;Khan, Ali
applied sciences Article On the Utilization of Reversible Colour Transforms for Lossless 2-D Data Compression 1 1, 2 2 Muhammad Waleed , Tai-Won Um * , Aftab Khan and Ali Khan Department of Information and Communication Engineering, Chosun University, Gwangju 61452, Korea; email@example.com Department of Computer Systems Engineering, University of Engineering and Technology (UET), Peshawar 25120, Pakistan; firstname.lastname@example.org (A.K.); email@example.com (A.K.) * Correspondence: firstname.lastname@example.org Received: 25 December 2019; Accepted: 21 January 2020; Published: 31 January 2020 Abstract: Reversible Colour Transforms (RCTs) in conjunction with Bi-level Burrows–Wheeler Compression Algorithm (BBWCA) allows for high-level lossless image compression as demonstrated in this study. The RCTs transformation results in exceedingly coordinated image information among the neighbouring pixels as compared to the RGB colour space. This aids the Burrows–Wheeler Transform (BWT) based compression scheme and achieves compression ratios of high degree at the subsequent steps of the program. Validation has been done by comparing the proposed scheme across a range of benchmarks schemes and the performance of the proposed scheme is above par the other schemes. The proposed compression outperforms the techniques exclusively developed for 2-D electrocardiogram (EEG), RASTER map and Color Filter Array (CFA) image compression. The proposed system shows no dependency over parameters like image size, its type or the medium in which it is captured. A comprehensive analysis of the proposed scheme concludes that it achieves a signiﬁcant increase in compression and depicts comparable complexity similar to the various benchmark schemes. Keywords: Burrows–Wheeler Compression Algorithm (BWCA); colour image compression; raster map; Colour Filter Array (CFA); EEG compression; Reversible Colour Transform (RCT) 1. Introduction Lossless image compression is addressed as one of the most challenging tasks among the research community. Lossless coding methods are employed to retain the quality of an image in visual quality critical applications. Among the ﬁelds like remote sensing, business documentation, digital radiography, medical industry and satellite imagery, work is being done to achieve signiﬁcant error-free compression [1–3]. Lossless image compression is also utilized in techniques such as information hiding and RASTER map compression [4,5]. Satellite and medical imagery are often high-resolution and at times require the original data to be protected from any loss. Different types of images such as grayscale, multi-spectral images, hyperspectral images, electrocardiogram (EEG), and Magnetic Resonance Imaging (MRI) have been processed to achieve lossless compression [6–13]. Lossless compression is majorly achieved due to the redundancy of intensity levels in the image thus making it a tedious task to bring down the size without compromising its quality [14–16]. By transforming the image data earlier to compression, higher image data redundancy can be achieved which in turn leads to higher compression . Highly correlated image pixels regions offer greater information redundancy. This inter-pixel redundancy is utilized for effective lossless compression. Appl. Sci. 2020, 10, 937; doi:10.3390/app10030937 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 937 2 of 17 This research work is oriented around the enhancement of lossless compression of colour images based on the Bi-level Burrows–Wheeler Compression Algorithm (BBWCA) utilizing colour space transformations. The BBWCA developed by Khan et al. in  utilized Reversible Colour Transform (RCT) focusing on its extensive possibilities of applications given in . In the proposed technique, various colour space transformations were analyzed for their increase in inter-pixel redundancy leading to higher compression. Section 2 presents the various techniques employed for lossless compression with a major focus on the literature related to the Burrows–Wheeler Compression Algorithm (BWCA). RCTs are presented in Section 3. Experimental results in comparison with benchmark schemes and different image types are given in Section 4. Conclusions, contributions and future directions for the study are presented in Section 5. 2. Burrows–Wheeler Compression Algorithm (BWCA) The classical Burrows–Wheeler Compression Algorithm (BWCA) was presented by J. Abel in  and is shown in Figure 1. It comprised of four stages which are explained brieﬂy below and are depicted in Figure 2. A comprehensive examination of the scheme and its various stages is presented in the works of [14,15,18–21]. Currently, various variants of the algorithm have been developed by modifying either the complete scheme or its various stages [18,22,23]. Figure 1. The classical Burrows–Wheeler Compression Algorithm (BWCA) . Figure 2. (a) Generated permutations from string (b) Permutations sorted with the ﬁnal column containing the output string (c) M-Term Pursuit (MTF) encoding of string followed by Run-Length Encoding (RLE) in (d). Appl. Sci. 2020, 10, 937 3 of 17 The source data is input to the ﬁrst stage of the Burrows–Wheeler Transform (BWT). The BWT is a lossless transform invented by M. Burrows and D. J. Wheeler for text compression . By increasing the recurrence of image pixels, redundancy is enhanced which in turns leads to better compression. In the case of BWCA, it is the BWT that generates runs of symbols which is encoded by the MTF transform in such a way that it increases the occurrence of zeros. The later stages of RLE and entropy coding the various stages of BWCA are detailed as follows in order to elucidate their effectiveness for lossless transformation and compression. The BWCA is based on the Burrows–Wheeler Transform (BWT) with secondary stages taking advantage of it . The BWT is a reversible transform and creates chains of symbol repetitions by lexicographically sorting the permutations of the input string or pixels. The output contains long runs of the symbols which allows for the later stages of MTF and RLE to easily compress these runs effectively . Figure 2 shows the transformation of the example string ‘bananana’ for clarity. In the ﬁrst step of BWT, the strings permutations are created by circular left shifting one element at a time. Figure 2a shows all the permutations. Figure 2b depicts the lexicographical sorting of the matrix in Figure 2a. The permutations are sorted in ascending alphabetical order where order is checked for each column of the string. For the example string, the original string position is now 5 and the last column of the sorted matrix contains the string ‘nnnbaaaa’ having repetitions of alphabets ‘n’ and ‘a’. Large size of input data will create better runs however, it will induce computation cost as memory and computations increase exponentially. The string ‘nnnbaaaa’ is given to the MTF encoder stage in the reverse order to put it in ascending order. The MTF stage encodes a symbol from the list of all symbols. Its position is then moved to the front of the list so that if the same symbol is being repeated, it will get an index of “zero” from the list. Through this method, more runs of zeros are generated for repeating symbols. As shown in Figure 2c, the string is encoded in conjunction with a list of all symbols. Currently the list is ‘abn’. The position of alphabet ‘a’ is index number ‘0’. The four ‘a’ runs are encoded as ‘0000’. When the alphabet ‘b’ is reached, its position in the list is checked, which is 2 and the run is encoded as ‘00001’. the symbol ‘b’ is moved to the front of the list such that if it is repeated, more ‘0’ indexes will be produced. The MTF encoder through this principle aims at increasing the runs of zeros and ones. The RLE stage then encodes these runs. RLE-0 is usually employed and other symbol runs are not encoded by RLE as the runs of zeros are usually more. Different algorithms are presented for this RLE. Abel  presented the RLE-BIT and RLE-EXP algorithms while Burrows and Wheeler  used the Zero Run Transform (RLE-0). The proposed model uses the RLE-0 scheme. Once RLE has been done, the output is encoded by an entropy coder such as Huffman or arithmetic encoder. An entropy coder compresses data by assigning fewer bits to frequently occurring symbols. This way symbols with elevated occurrence are decreased effectively. Both Huffman and arithmetic encoder models considers the probability of each data symbol or pixel. Huffman encoders provide better speed while arithmetic coders provide better compression. This research work has employed the arithmetic coder. Details of the arithmetic encoder are given in the Appendix A. The BWCA is not complex in logic and offers high compression due to its main component, the Burrows–Wheeler Transform (BWT) which increases the redundancy of the data, in turn leading to higher compression. To further enhance the compression ability of the scheme, various modiﬁcations to the BWCA have been made including the works of Arnavut et al. in , Deorowicz et al. in , J. Abel in [17,18,22], Schindler in , Balkenhol and Shtarkov in , Arnavut et al. in  and Khan et al. in  to name a few. This research work augments the compression performance of the Bi-level Burrows–Wheeler Compression Algorithm (BBWCA) devised by Khan et al. in . The scheme offers higher image compression as compared to other benchmarks schemes by transforming the colour intensity levels over the dimensional space using a Reversible Colour Transform (RCT). The RCT increases the Appl. Sci. 2020, 10, 937 4 of 17 inter-pixel redundancy aiding in the compressibility of the data at the later stages of the algorithm. Figure 3 shows an overview of the BBWCA. Prior to the input to RCT, true colour image data is DC shifted resulting in a grey-level range from 128 to +127. In comparison to the original image, the resulting image becomes more skewed aiding in much better compression during the later stages. Once the data is DC shifted; converted to YUV colour space from RGB, it is further processed during the BWT stage. To achieve better compression results Khan et al. in  employed the JPEG 2000 RCT . Figure 3. Overview of the Bi-level Burrows–Wheeler Compression Algorithm (BBWCA) . In contrast to the conventional BWCA approach, BBWCA transﬁgures the image through row-wise and column-wise application of BWT over image data resulting in a highly correlated 2-D image. Redundancy among the pixels is enhanced by the utilization of the KMTF transform [28,30], in conjunction with RLE-0 and arithmetic encoding. By utilizing the RCT, BBWCA is able to achieve higher inter-pixel redundancy and in turn higher compression. There exist various RCTs, time and frequency based, which can be employed and explored for higher compression. The main aim of this research work is to investigate these RCTs and demonstrate their effect on the compression performance offered by the BBWCA. The following section presents a brief history and various types of RCTs designed and devised by the research community in the ﬁeld of image processing. 3. Reversible Colour Transforms (RCTs) Prior conversion of data from imaging domain to another can produce better compression results by the BWCA. Guo et al. utilized Discrete Wavelet Transform (DWT), a lossless wavelet transform in conjunction with Huffman encoder, and achieved higher compression. DWT for lossless compression has been utilized in methodologies proposed by Srikanth and Meher in  and Mozammel et al. in . Telagarapu et al. in  employed Discrete Cosine Transform (DCT) and Discrete Wavelet Transform (DWT) showing that the DWT outperformed the DCT in terms of lossless compression. Strutz et al. in  used automatic selection of colour transforms for efﬁcient coding, either lossy or lossless. It was 2 percent more efﬁcient in the case of lossless coding. In Strutz’s earlier work in , various RCTs have been developed for efﬁcient compression. By not ﬁxing the RCT for image regions, better compression was achieved. Colour space transformation refers to the conversion of image information from one colour domain to another. Multiple linear along with the non-linear transformations of the true colour image is utilized for the deﬁnition of a colour image in terms of luminance and colour [16,36–39]. RCTs have been utilised to boost compression of the RGB images by increasing data redundancy in other colour spaces [39,40]. Totally reversible integer colour transform transforms the imaging data into various colour spaces [29,41] and has been utilized for lossless image compression by JPEG 2000 [29,42–44]. Appl. Sci. 2020, 10, 937 5 of 17 This colour conversion helps in achieving high compression by reducing an image’s entropy prior to encoding [38,39]. Researchers have proposed various RCTs in the ﬁelds including the methodologies presented in [16,29,37,38,42,44–47]. This research focuses on the evaluation and investigation of the new and efﬁcient RCT proposed by Starosolski in . RDgDb, LDgDb, RDgEb, LDgEb, YA2UA2VA2, YA6UA6VA6, YA7UA7VA7 were listed as some of the few lossless transforms utilized and a comparative analysis has been drawn with RCTs of JPEG-2000, JPEG-XR and JPEG-LS etc. Each transform offers it advantages e.g., RDgDb has a requirement of subtraction of two integers per image pixel thus having lesser computational complexity  while LDgDb lossless colour transform offers a better estimation of the analog transformation in human vision system thus outperforming the YCoCg-R transform. Table 1 lists various RCTs used in image processing. Forward and Reverse RCT equations are listed as well. Table 1. List of RCTs and their forward and reverse equations. Transform Forward RCT Reverse RCT Reference 2 3 2 32 3 2 3 2 32 3 R 1 0 0 R R 1 0 0 R 4 5 4 54 5 4 5 4 54 5 RDgDb D = 1 1 0 G G = 1 1 0 D Roman Starosolski g g D 0 1 1 B B 1 1 1 D et al.  b b 2 3 2 32 3 2 3 2 32 3 L 1/2 1/2 0 R R 1 1/2 0 L 4 5 4 54 5 4 5 4 54 5 LDgEb D = 1 1 0 G G = 1 1/2 0 D Roman Starosolski g g E 1/2 1/2 1 B B 1 0 1 E et al.  b b 2 3 2 32 3 2 3 2 32 3 L 1/2 1/2 0 R R 1 1/2 0 L 4 5 4 54 5 4 5 4 54 5 LDgDb D = 1 1 0 G G = 1 1/2 0 D Roman Starosolski g g D 0 1 1 B B 1 1/2 1 D et al.  b b 2 3 2 32 3 Y 0 1 0 R A2 4 5 4 54 5 YA2UA2VA2 U = 0 1 1 G Roman Starosolski A2 V 1 1 0 B et al.  A2 2 3 2 32 3 Y 1 0 0 R A6 4 5 4 54 5 YA6UA6VA6 U = 1 0 1 G Roman Starosolski A6 V 1 1 0 B et al.  A6 2 3 2 32 3 Y 0 0 1 R A7 4 5 4 54 5 YA7UA7VA7 U = 0 1 1 G Roman Starosolski A7 V 1 0 1 B et al.  A7 mYA2UA2VA2 Modular Arithmetic Variant of YA2UA2VA2 R = R R = R N N mRDgDb mDg = (R G)smod2 G = (R mDg)smod2 Roman Starosolski N N et al.  mDb = (G B)smod2 B = (G mDb)smod2 mDg = (R G)smod2 R = (mL + [mDg/2])mod2 N N mLDgEb mL = (R [mDg/2])mod2 G = (R mDg)mod2 Roman Starosolski N N et al.  mEb = (B mL)smod2 B = (mEb + mL)mod2 mLDgDb Modular Arithmetic Variant of LDgDb h i h i R+2G+B U+V Y = G = Y 4 4 RCT [14,30] U = R G R = U + G V = B G B = V + G mRCT Modular Arithmetic Variant of RCT  2 3 2 3 2 32 3 Y R 1 1 1 Y 4 5 4 5 4 54 5 YCoCg-R C = G = 1 1 0 C JPEG XR Standard g g C B 1 1 1 C  O O 2 32 3 1/4 1/2 1/4 R 4 54 5 1/4 1/2 1/4 G 1/2 0 1/2 B Appl. Sci. 2020, 10, 937 6 of 17 4. Experimental Setup And Results All the simulations and experimentations of the proposed method were performed over machines equipped with an Intel Core i7 processor, Quad Core CPU running @ 2.9 GHz, 8 GB RAM, 2 GB virtual memory executing programs and applications over the Microsoft Windows operating system. Algorithms were developed in MATLAB and were evaluated using the following image datasets. Kodak colour image dataset University of Loughborough Uncompressed Colour Image Database (UCID) RASTER Maps Images Colour Filter Array (CFA) Images EEG Images Simulations were performed by applying the proposed technique to these image sets and comparing against benchmark schemes. Six different colour transforms that include RDgDb, LDgDb, LDgEb, YA2UA2VA2, YA6UA6VA6, and YA7UA7VA7  were utilized. Simulation results are presented as follows. 4.1. Lossless Compression Of Colour Images The BBWCA was tested on sets of colour images to guess the efﬁciency of the RCTs. Systems comprised of BMF, FLIC, STUFF IT, FP8, GRALIC, and BMF were used as benchmark schemes for the evaluation purposes. Image compression benchmark website  features the afore-mentioned techniques serving as the core of the most recent compression schemes. The proposed scheme was gauged against the formerly crafted YUV RCT based BBWCA . The Kodak test images  shown in Figure 4 were used for initial testing. All the pictures have a resolution of 768 512 3 pixels totaling 1152 KB each. Figure 4. Kodak Test Images. Compression results for these images are listed in the tabular form in Table 2. The YA6UA6VA6 RCT based BBWCA compressed the archive of images to 4199 KB in opposition to the raw data set of 11,520 KB. Subsequently, the size of 4231 KB was achieved using GRALIC method. The YA6UA6VA6 RCT based BBWCA peaks the benchmark compression schemes as it compresses the well homogenized gray levels resulting from the RCT. Appl. Sci. 2020, 10, 937 7 of 17 Table 2. Comparative Analysis of Compression sizes achieved in KB with existing benchmark schemes (for Kodak images). All Kodak images have a resolution of 768 512 3 totaling 1152 KB. BPPs for separate Y, U, V channels for the BBWCA(YUV) transform are shown for individual channel efﬁciency. Existing Compression Schemes Image FLIC ZPAQ FP8 PAQ8 STUFF IT kodim02 422 461 352 341 391 kodim05 517 731 575 549 499 kodim07 390 501 350 332 370 kodim08 540 723 595 564 507 kodim11 425 556 424 404 432 kodim13 561 711 585 562 552 kodim20 335 411 325 313 421 kodim21 455 543 432 435 488 kodim22 480 688 523 503 482 kodim23 401 620 417 394 404 TOTAL 4526 5945 4578 4397 4546 Existing Compression Schemes Proposed Compression Scheme Image BMF GRALIC BBWCA (RDgDb) BBWCA (LDgDb) BBWCA (LDgEb) kodim02 438 397 700 685 539 kodim05 512 480 742 742 847 kodim07 395 333 654 653 715 kodim08 524 498 681 681 878 kodim11 438 401 585 586 703 kodim13 565 534 707 708 821 kodim20 339 308 506 533 612 kodim21 465 474 524 525 690 kodim22 485 439 675 681 788 kodim23 411 367 635 633 659 TOTAL 4572 4231 6409 6427 7252 Proposed Compression Scheme Image BBWCA (YUV) BBWCA (YUV) BPP for Y | U | V Channel BBWCA (YA2UA2VA2) BBWCA (YA7UA7VA7) BBWCA (YA6UA6VA6) kodim02 372 2.73 | 0.58 | 0.56 506 691 294 kodim05 398 3.02 | 1.79 | 1.09 580 785 441 kodim07 387 2.77 | 1.45 | 1.25 442 709 374 kodim08 437 3.06 | 1.67 | 1.70 628 700 537 kodim11 375 2.79 | 1.56 | 1.23 451 598 497 kodim13 472 3.11 | 1.46 | 0.96 483 806 458 kodim20 492 2.42 | 1.27 | 0.74 334 709 205 kodim21 419 3.17 | 1.48 | 1.53 480 560 545 kodim22 478 3.00 | 1.43 | 1.18 527 726 428 kodim23 493 2.56 | 1.38 | 1.20 460 710 420 TOTAL 4323 4891 6994 4199 Proposed scheme tested over natural images acquired from Uncompressed Colour Image Database (UCID) of the University of Loughborough as shown in Figure 5. In this case, the proposed system employed the existing distinguished top performer YA6UA6VA6 RCT for BBWCA. PNG, FP8, BMF, FLIC, GRALIC and BBWCA (YUV) were utilized as benchmark schemes. Compression results are presented in Table 3. BBWCA surpasses the benchmark schemes by showing a signiﬁcant reduction of the total archive of size 14,450 KB to a compressed size of 5519 KB with a Compression Ratio (CR) of 2.62. It is considered as a signiﬁcant enhancement in the pre-existed RCT based BBWCA scheme equipped with the ability of reduction of the 7829 KB archive data with CR as 1.85. The proposed lossless compression technique was also extended to other 2-D data. Section 4.2 exhibits the compression capability of the proposed scheme to RASTER maps. Section 4.3 and 4.4 represent the lossless compression for the CFA and EEG images respectively. Appl. Sci. 2020, 10, 937 8 of 17 Figure 5. University of Loughborough - UCID Outdoor Images. Table 3. Comparison of compression sizes for UCID outdoor images. Compression Scheme UCID Image No Original Size BMP PNG FP8 BMF FLIC GRALIC BBWCA (YUV) BBWCA (YA6UA6VA6) 1 578 468 299 522 278 264 378 207 2 578 486 297 256 276 259 395 210 3 578 606 363 296 325 307 410 410 4 578 536 339 297 316 298 342 237 5 578 576 382 328 346 332 374 241 6 578 560 367 317 335 321 370 217 7 578 625 416 336 352 337 334 224 8 578 544 343 284 307 291 308 201 9 578 456 265 224 239 227 300 174 10 578 547 357 303 319 304 368 235 11 578 565 335 271 299 280 334 171 12 578 502 329 284 309 288 335 227 13 578 546 339 280 302 286 309 194 14 578 568 359 298 319 304 327 242 15 578 346 217 191 204 195 177 122 16 578 499 309 256 276 260 275 152 17 578 581 384 320 338 324 240 206 18 578 435 279 244 259 247 352 154 19 578 592 376 306 328 314 209 263 20 578 583 372 310 339 316 222 270 21 578 554 332 266 299 278 200 240 22 578 591 413 367 379 369 212 259 23 578 595 395 332 359 336 431 230 24 578 615 424 362 376 364 382 258 25 578 597 388 335 352 339 245 175 Total 14450 13573 8679 7585 7831 7440 7829 5519 4.2. Lossless Compression of RASTER Maps RASTER maps are electronic maps embedded in navigation devices and comprises chieﬂy of redundant yet indispensable data. The data takes up a lot of storage space and being climacteric, its loss is highly pernicious. Lossless image compression is employed to decrease the data size while maintaining full quality of these images. Techniques proposed by Mao et al. in  and Akimov et al. in  prove to be pivotal in this aspect. Akimov et al. scheme works on splitting the colour map into separate layers that are compressed singly. Block-Line Separated Encoding (BLiSE) technique by Mao et al. outperforms JPEG-LS, Portable Networks Graphic (PNG) and Graphics Interchange Format (GIF) compression. However, BBWCA Appl. Sci. 2020, 10, 937 9 of 17 utilizing the YUV RCT gave higher compression results than BLiSE and different schemes used as benchmarks. The YA6UA6VA6 RCT BBWCA shows promising results in comparison to the YUV RCT based BBWCA. RASTER maps used to draw a comparative analysis of compression achieved with the benchmark schemes in comparison with the proposed BBWCA (YA6UA6VA6) algorithm is shown in Figure 6. Figure 7 demonstrates the results of the system for the respective RASTER maps. Figure 6. Set of RASTER maps used for compression. Figure 7. Comparative analysis of RASTER maps of different compression sizes. Original image size is 843 KB. Values on y-axis are displayed on the logarithmic scale for a better visual representation. 4.3. Lossless Compression of CFA Images CFA images undergo lossless compression to lower down the computational complexity in later stages. Lee et al.’s in  proposed a scheme working on the lossless compression of HDR imagery. Initially, the images are de-interlaced into individual channels followed by the weighted template matching based prediction is performed compressing the high-resolution images at a very Appl. Sci. 2020, 10, 937 10 of 17 low computational cost. Chung et al. in  proposed a method to use context matching accompanied with reduction techniques in the spectral domain for colour difference estimation. CFA images utilized for the evaluation of the designed system are given in Figure 8. Bits Per Pixel (BPPs) achieved by the schemes in the discussion are represented in Figure 9. JPEG-LS, JPEG2000, LCMI, CMBP and HPCM techniques  are used as a benchmark scheme to draw a comparison with the proposed system. YUV RCT based BBWCA of Khan et al. in  is also compared with the proposed method to determine its compression efﬁciency and performance. The proposed YA6UA6VA6 RCT based BBWCA performs the compression reducing the data with an average BPP of 3.35. In comparison to the YUV BBWCA, HPCM and CMBP schemes, it shows an increase of 18, 20, and 26 percent increase in the effective lossless compression. The transform mainly offers the advantage of its ability to produce a redundant number of grey scale values lying among the U and V channels. Once the redundant data is transformed by BWT, it generates data with correlation matrix among the 2-D image which is efﬁciently compressed by the later stages. Figure 8. Colour Filter Array (CFA) images utilized for the evaluation of the proposed system. Figure 9. Bits Per Pixel (BPPs) employed by the benchmark and BBWCA for lossless compression of CFA images. Appl. Sci. 2020, 10, 937 11 of 17 4.4. Lossless Compression of 2-D EEG Data Inspection and storage of EEG produce large data containing patient’s signiﬁcant information. Compression algorithms reduce the storage capacity of the data and provide effective methods for preventing overwriting due to the small capacity of EEG machines. Antonio et al. in  proposed a method based on securing the derivation of EEG instead of the unprocessed data. Lossless compression was achieved by the amalgamation of various techniques including Huffman encoding and vector quantization. Dauwels et al. in  devised a methodology utilizing wavelets to achieve compression over multi-channel EEG. The effective compression is achieved by the utilization of wavelet-based volumetric coding, energy-based lossless compression of the resulting wavelet bands followed by coding based on the tensor. The latter scheme recorded with the best compression results. Lin et al. in  proposed a multi-channel EEG compression scheme based on the Independent Component Analysis (ICA) followed by Set Partitioning in Hierarchical Trees (SPIHT). Prior to the application of ICA, the algorithm utilizes Principal Component Analysis (PCA) followed by the application of SPIHT compressing any residues adding effective compression in comparison to the independent components. Xu et al. in  proposed a compression method based on No List Set Partitioning in Hierarchical Trees (NLSPIHT) algorithm for 1.5 D EEG. It considers the I-D DWT instead of SPIHT based 2-D compression requiring lower computational complexity and power. Daou et al. in  proposed another scheme utilizing the same techniques as mentioned above i.e., DWT and SPIHT. Experimentation was conducted utilizing the EEG data samples from the University of Bonn . Figure 10 depicts the compression results achieved for the EEG data in terms of size and ratios. Five datasets labelled (A-E) constitute the data where each label presents an ailment. The proposed YA6UA6VA6 RCT based BBWCA shows a signiﬁcant improvement in compression for the source signals. Figure 10. Comparative analysis of compression size and ratios for lossless compression of EEG data (size in KB). Appl. Sci. 2020, 10, 937 12 of 17 4.5. Scalability to High Resolution Imagery The proposed scheme provides a single platform for the compression of various forms of 2-D data and images including imagery acquired from large resolution sensors. 4K images, HDR images, MRI/CT image volumes are our future work directions. Since there is too much redundancy in these datasets, one method that we are currently investigating involves applying MTF transform encoding prior to BBWCA. Figure 11 shows the amount of data that can be reduced to long runs of zeros by applying MTF transform on the pristine image before encoding it with BBWCA. Figure 11a shows the original image and its MTF encoded version is given in Figure 11b. Figure 11c shows the histogram of both images. Large number of zeros in the MTF encoded version is beneﬁcial in data reduction prior to BWT and its successive stages. Figure 11. (a) Original image  (b) MTF encoded image (c) Histogram of original image and its MTF encoded version. 5. Conclusions The utilization of the reversible transforms in conjunction with BWCA algorithm resulted in greater compression as compared to the benchmark schemes. The YA6UA6VA6 RCT achieved the best compression results in general for various size of images and creates a single base for the compression of a variety of data including CFA, EEG and RASTER data. The proposed compression outperforms Appl. Sci. 2020, 10, 937 13 of 17 the techniques exclusively developed for 2-D EEG, RASTER map and Color Filter Array (CFA) image compression. In the future, the BBWCA is also aimed for the application of DCT and DWT to achieve lossy compression over the data as well. In comparison to the existing schemes, for CFA images the proposed system gives 3.35 Bits Per Pixel (BPPs) compared to 3.95. Similarly, for EEG data, the proposed system achieved a compression ratio of 3.12 and gave promising results for RASTER image compression. Author Contributions: Conceptualization, M.W., T.-W.U. and A.K. (Aftab Khan); methodology, M.W., T.-W.U. and A.K. (Aftab Khan); software, M.W. and A.K. (Ali Khan); validation, T.-W.U. and A.K. (Aftab Khan); formal analysis, T.-W.U. and A.K. (Aftab Khan); investigation, M.W. and A.K. (Ali Khan); resources, T.-W.U. and A.K. (Aftab Khan); data curation, M.W. and A.K. (Ali Khan); writing—original draft preparation, M.W. and A.K. (Aftab Khan); writing—review and editing, M.W., T.-W.U. and A.K. (Aftab Khan); visualization, M.W., T.-W.U. and A.K. (Aftab Khan); project administration, T.-W.U.; funding acquisition, T.-W.U. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by research fund from Chosun University and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.2018R1A2B2003774). Conﬂicts of Interest: The authors declare no conﬂict of interest. Appendix A. The Arithmetic Coder The Huffman Coding trades a symbol with a particular code depending on the Huffman tree generated from the prior probabilities of the symbols in the message. On the other hand, the Arithmetic coding evades this idea by replacing a set of input symbols with a single ﬂoating point output number. The length of the message and its complexity leads to higher number of bits required for generating the output. The range of an arithmetic coding output, say x here, is a single number such that 0 x < 1. x is uniquely decodable taking considering probabilities of the symbols. This interval keeps on reducing for lengthier messages which in turn leads to the utilization of a higher numbers of bits to represent the range. The process continues for the subsequent symbols and frequently occurring symbols narrow the range lesser as compared to less frequent symbols. Figure A1 shows the encoding example for the Arithmetic encoder for better comprehension. The example of a ﬁxed model arithmetic code from the works of Witten et al. , is presented below. Figure A1. Symbols ranges in Arithmetic encoder for the message ‘eaii!’. Let suppose we have a set of alphabets i.e., [a,e,i,o,u,!] where ! is used as end of message indicator. The probabilities of these alphabets are given in Table A1. Now, a message “eaii!” is encoded as follows. Table A1. Symbols probabilities and ranges for ﬁxed model Arithmetic coder example. Symbol Probability Range a 0.2 [0.0, 0.2) e 0.3 [0.2, 0.5) i 0.1 [0.5, 0.6) o 0.2 [0.6, 0.8) u 0.1 [0.8, 0.9) ! 0.1 [0.9, 1.0) Appl. Sci. 2020, 10, 937 14 of 17 The alphabet “e” is the ﬁrst symbol to be encoded as it has the highest probability. The ﬁrst symbol is the important part of the arithmetic encoder as it deﬁnes the highest interval which will be subsequently narrowed. As seen in Figure A1, in the ﬁrst step, the message is arranged on the vertical probability interval. The range of “e” starts from 0.2 and ends at 0.5. The second alphabet in the message is “a”. The range of e [0.2,0.5) needs to be narrowed. The equations below let us calculate that new range. range = upper_bound lower_bound (A1) Range of a “symbol” is calculated then as lower_limit : lower_limit + range ”symbol”_probability (A2) So, for “a” the overall range is 0.5 0.2 = 0.3. The new range for symbol “a” is calculated as 0.2 : 0.2 + 0.3 0.2 giving us [0.2, 0.26] Similarly, the ranges for all the symbols are calculated as shown in Table A1. Now, let’s see how we can decode a message for the Arithmetic coder. Say the decoder only has the information about the last range of the message i.e., [0.23354, 0.2336). This range according to the probability given in Table A2 shows the ﬁrst character has to be “e”. Once the ﬁrst character is decoded it imitates the probability ranges for the rest of the symbols. Table A2. Calculated ranges for all the symbols. Initially [0, 1) After Symbol e [0.2, 0.5) a [0.2, 0.26) i [0.23, 0.236) i [0.2333, 0.2336) ! [0.23354, 0.2336) Initially the range was [0, 1) and after seeing “e” it became [0.2, 0.5). From here, as per the probabilities, the range is narrowed for symbol “a” giving us [0.2, 0.26). The process is repeated till all the symbols ranges are generated at the decoder end. Then from the last range, in the reverse order, the message is identiﬁed. References 1. Ouni, T.; Lassoued, A.; Abid, M. Lossless Image Compression Using Gradient Based Space Filling Curves (G-SFC). Signal Image Video Process. 2015, 9, 277–293. [CrossRef] 2. Anusuya, V.; Raghavan, V.S.; Kavitha, G. Lossless Compression on MRI Images Using SWT. J. Digit. Imag. 2014, 27, 594–600. [CrossRef] [PubMed] 3. Srinivasan, K.; Dauwels, J.; Reddy, M.R. A Two-Dimensional Approach for Lossless EEG Compression. Biomed. Signal Process. Control. 2011, 6, 387–394. [CrossRef] 4. Mao, Q.; Zhou, B.; Zou, Q.; Li, Q. 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On the Utilization of Reversible Colour Transforms for Lossless 2-D Data Compression