A new lossless compression scheme based on Hu!man coding scheme for image compression

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1 Signal Processing: Image Communication 16 (2000) 367}372 A new lossless compression scheme based on Hu!man coding scheme for image compression Yu-Chen Hu, Chin-Chen Chang* Department of Computer Science and Information Engineering, National Chung Cheng University, Chiayi 621, Taiwan, ROC Received 25 August 1998 Abstract A novel lossless image-compression scheme is proposed in this paper. A two-stage structure is embedded in this scheme. A linear predictor is used to decorrelate the raw image data in the "rst stage. Then in the second stage, an e!ective scheme based on the Hu!man coding method is developed to encode the residual image. This newly proposed scheme could reduce the cost for the Hu!man coding table while achieving high compression ratio. With this algorithm, a compression ratio higher than that of the Lossless JPEG method for images can be obtained. In other words, the newly proposed algorithm provides a good means for lossless image compression Elsevier Science B.V. All rights reserved. Keywords: Lossless image compression; Hu!man coding; JPEG 1. Introduction * Corresponding author. Tel.: /6011; fax: address: ccc@cs.ccu.edu.tw (C.-C. Chang). Lossless image compression [9}12] is very important in many "elds such as biomedical image analysis, medical images, art images, security and defense, remote sensing, and so on. During the past few years, several schemes have been developed for lossless image compression. Usually, a twostage coding technique is embedded in these schemes. In the "rst stage, a linear predictor such as di!erential pulse code modulation (DPCM) [2,8] or some linear predicting functions, is used to decorrelate the raw image data. In the second stage, a standard coding technique, such as Hu!man coding [3,5], arithmetic coding [6,14] or Lempel Ziv coding, is used to encode the residual magnitudes. Such a two-stage scheme is useful because the high correlation between neighboring pixels in most images can be decorrelated, which results in a signi"- cant entropy reduction. To date, the Lossless JPEG [12] is one of the most e!ective compression schemes for gray scale images. The Lossless JPEG has two basic versions: DPCM with arithmetic coding and DPCM with Hu!man coding. The DPCM with arithmetic coding version typically provides slightly better compression results than the DPCM with Hu!man coding version, but it requires a higher computation cost. Some modi"ed methods [1,4,7] have been proposed to improve the linear predictor in the "rst stage. Besides, the Hu!man coding of most-likely magnitude (MLM) [13] had been developed to improve the coding performance in the /00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S ( 9 9 )

2 368 Y.-C. Hu, C.-C. Chang / Signal Processing: Image Communication 16 (2000) 367}372 second stage. However, achieving a better compression ratio for lossless image compression is still an important issue. In this paper, we shall propose a new coding technique based on the MLM scheme for lossless image compression. The rest of this paper is organized as follows. In Section 2, we shall review the Hu!man coding scheme and the MLM coding scheme. In Section 3, the newly proposed method shall be proposed. In Section 4, some experimental results shall show that our newly proposed algorithm indeed produces good image quality. The discussion and conclusion shall be given in Section Previous works The Hu!man coding technique is a commonly used scheme for data compression because it is very simple and e!ective. It requires the statistical information about the distribution of the data to be encoded. Besides, an identical coding table is used in both the encoder and the decoder. The details of the Hu!man coding technique are described in the following. Hu4man Coding Algorithm Step 1. Build a Hu!man tree by sorting the histogram and successively combine the two bins of the lowest value until only one bin remains. Step 2. Encode the Hu!man tree and save the Hu!man tree with the coded value. Step 3. Encode the residual image. In Step 1, the Hu!man tree of the given data is built by successively combining the two bins with the lowest value. The Hu!man coding table is then constructed to record the mapping of the data of each bin and the corresponding encoding bit pattern. The performance of the Hu!man coding technique relies on the distribution of the input data. If the input data has a large dynamic range, the Hu!man coding technique requires a higher computation cost. The Hu!man coding of the most-likely magnitude (MLM) is thus proposed to solve this problem since the fundamental goal of MLM is to encode the residual image generated for the linear predictor. It exploits two properties of image residual distribution: 1. The distribution of these residual magnitudes is symmetrical to the origin. 2. The number of the small residual magnitudes is much larger than that of the large residual magnitudes. By exploiting Property 1, the MLM scheme cuts down the number of code calculations by half by creating Hu!man codes only for the residual magnitudes and then adds a sign bit. By exploiting Property 2, the MLM scheme uses the Hu!man coding technique only for the most-likely (or small) residual magnitudes and employs another special pre"x code to identify the least-likely (or large) residual values. The details of the MLM scheme are described in the following. The MLM Method Step 1. Compute the residual histogram H, such that H(x)"the number of pixels having residual magnitudes. Step 2. Compute the symmetry histogram S such that S(y)"H(y)#H(!y), where y*0. Step 3. Find the range threshold R such that S(j))P N( S(j), where N is the total number of the pixels in the input residual image and P is the desired proportion of the most-likely magnitudes. Note that 0)P)1 and 0)R. Step 4. Construct the modi"ed symmetry histogram S with R#1 entries, where S (y)" if0)y)r!1, S(y) N! S(j) ify"r. Note that S (R) stores the total number of the least-likely occurrences. Step 5. Construct the Hu!man tree and generate the Hu!man table C for histogram S using the traditional Hu!man coding tech- nique. Step 6. Encode each residual magnitude x by the following three rules. Rule 1. If 0)x)R!1 then output conc(c(x), 0).

3 Y.-C. Hu, C.-C. Chang / Signal Processing: Image Communication 16 (2000) 367} Rule 2. If!(R!1))x(0 then output conc(c( x ), 1). Rule 3. If x*r or x)!r then output conc(c(r), x). Here the function conc(x, z) concatenates the bit representations of x and z. Here C(x) denotes the encoded bit pattern for input x stored in the Hu!man coding table. In Step 1, the residual histogram H is computed and to be transformed into the symmetry histogram S in Step 2. The range of symmetry histogram S is about half of the original residual histogram H. However, the information about the sign value should be kept for residual re-production in the decoder. That is, to record the sign value of each residual, one sign bit should be added in the encoding step. Given a prede"ned proportion threshold P, where 0)P)1, the range threshold R to classify the most-likely magnitude and the least-likely magnitude is decided in Step 3. If P is set to 0.9, it means that we want to encode 90% of the residual magnitudes using the Hu!man coding technique with slight modi"cation. Thus, a switch value should be added in front of these least-likely magnitudes to accommodate the encoding technique. So, these least-likely magnitudes are collected together in Step 4. At present, there are R#1 entries in the modi"ed symmetry histogram. After completing the computation of histogram, the standard Hu!man coding technique is applied to the construction of the Hu!man tree and the Hu!man coding table for in Step 5. The actual encoding process for input residual x is done in Step 6. If x is a most-likely magnitude, the coding output is composed of the encoded bit pattern for C( x ) followed by a sign bit. Otherwise, if x is a least-likely magnitude, the coding output is composed of the encoded bit pattern for C(R) followed by the binary representation of x. The parameter P provides a means of trading o! speed for compression performance. The authors claim that P"0.99 is a good compromise for a wide variety of images. A large value of P generally leads to more computation cost though a Hu!- man table with more entries can be stored. On the other hand, if the parameter P is small, the MLM scheme has some loss in encoding these least-likely magnitudes. 3. The newly proposed scheme In this section, we shall propose an enhanced Hu!man coding algorithm based on the MLM method. We call it the enhanced MLM method (EMLM). In the MLM method described in Section 2, the output of the most-likely magnitude x is composed of the bit pattern of C( x ) followed by a sign bit. In such an arrangement, there is one unused entry for the residual magnitude x when x equals zero. In other words, this entry is not used in the MLM scheme. To make use of this entry, two solutions are found. First, we can omit the sign bit when x equals zero because Hu!man code is a pre- "x code, or alternatively we can utilize this entry by storing the switch value of these least-likely magnitudes and thus lower the required bit length for the switch value. Therefore, the required bit length of these least-likely magnitudes can be reduced and the range of modi"ed symmetry histogram can be decreased by one. From the description of the MLM method in Section 2, we know that the output of each leastlikely residual magnitude x is composed of the switch value, namely C(R), followed by the binary representation of x. The output length of the leastlikely magnitude may be longer than that when the standard Hu!man coding scheme is used instead. To reduce the required bit length and the size of Hu!man table, we integrate the folding technique into the EMLM method. The details of the EMLM method are described in the following. Our EMLM Method Step 1. Compute the residual histogram H, such that H(x)"the number of the pixels having residual magnitude x. Step 2. Compute the symmetry histogram S such that S(y)"H(y)#H(!y), where y*0. Step 3. Given the prede"ned range threshold R. The symmetry histogram with the folding technique is constructed according to the following sub-steps.

4 370 Y.-C. Hu, C.-C. Chang / Signal Processing: Image Communication 16 (2000) 367}372 Step 3.1. If 0)y)R!1, then (y)" S(y). Step 3.2. If R)y and y"q R#r, then the following three sub-steps are executed sequentially to build the histogram. Step (R)" (R)#S(y). Step (q)" (q)#s(y). Step (r)" (r)#s(y). Now, there are R#1 entries in the modi- "ed symmetry histogram. Step 4. Rearrange the modi"ed symmetry histogram with the unused zero entry by computing S " (0)# (R) S (y) if y"0, if 1)y)R!1. At present, the modi"ed symmetry histogram has only R entries. Step 5. Construct the Hu!man tree and generate the Hu!man table C for histogram using the traditional Hu!man coding technique. Step 6. Encode each residual magnitude x by the following four rules. Rule 1. If 0)x)R!1, then output conc(c(x), 0). Rule 2. If!(R!1))x, then output conc(c( x ), 1). Rule 3. If x*r and x"q R#r, where q and r are among the integers, then output conc(conc(conc(c(0), 1),0), conc(c(q), C(r))). Rule 4. If x)!r and x "q R#r, where q and r are among the integers, then output conc(conc(conc(c(0),1),1), conc(c(q), C(r))). Here, C(x) denotes the encoded bit pattern for x in the Hu!man table, and the function conc(x, z) concatenates the bits of x and z together. In this method, the residual histogram H and the symmetry histogram S are computed in Steps 1 and 2, respectively. Given a prede"ned range threshold R, the whole magnitudes can be divided into a group of the most-likely magnitudes and of the least-likely magnitudes. Then the modi"ed symmetry histogram can be constructed using the folding technique in Step 3. Now, the modi"ed symmetry histogram has R#1entries.InStep4, the modi"ed symmetry histogram is rearranged by making use of the unused entry for magnitude zero. Now, the modi"ed symmetry histogram has R entries. The Hu!man tree and the Hu!man coding table for are constructed in Step 5. In Step 6, the actual encoding process for input residual x is described. If x is a most-likely magnitude, the coding output is composed of the encoded bit pattern for C( x ) followed by a sign bit. If x is a least-likely magnitude, the coding output is composed of the encoded bit pattern for the switch element followed the sign bit and the encoded bit pattern for the quotient and the remainder of x. Here the encoded bit pattern of the switch element makes use of the unused zero entry to reduce the required bit length. In our EMLM, we use the wasted entry in MLM originally meant for the zero magnitude. The required bit length of the switch element to justify the most-likely magnitudes and least-likely magnitudes is much smaller that that in MLM. Besides, a folding technique for histogram rearrangement is developed. The goal of the folding technique is to reduce the bit length for each least-likely magnitude. By rearranging the histogram, the entropy of these residual magnitudes can be further lowered down. A better compression ratio and a smaller Hu!man table can thus be obtained. 4. Experimental results All our experiments are done upon SUN SPARC10. In these simulations, we apply the DPCM method to compute the residual image in the "rst stage and then use the EMLM scheme to encode the residuals in the second stage. In this section, we shall compare the performances of the DPCM/Hu!man, DPCM/MLM scheme and our DPCM/EMLM scheme using still images with 256 gray levels. The criterion used to compare them is compression ratio (with respect to the number of bits), which is de"ned as follows: original file size Compression Ratio (CR)" compressed file size.

5 Y.-C. Hu, C.-C. Chang / Signal Processing: Image Communication 16 (2000) 367} Table 1 The performance comparison of the compression ratios using images Images Methods Hu!man Compress Gzip DPCM/ Hu!man DPCM/ MLM DPCM/ EMLM Airplane Boat Family Lenna Ti!any Toys Table 2 The total number of nonzero entries needed to be processed in di!erent methods Methods Hu!man DPCM/ DPCM/ DPCM/ Images Hu!man MLM EMLM Airplane Boat amily Lenna Ti!any Toys The experimental results of six images are listed in Table 1. The Hu!man method shown in Table 1 uses the Hu!man coding technique to encode each image without decorating the raw image data. Compress and Gzip, on the other hand, are two popular compression tools. Also, the DPCM/Hu!man method is one of the two basic versions of the JPEG standard. Furthermore, the proportion threshold P of the MLM method is set to This indicates that we want to encode 99% of the most-likely magnitudes using the traditional Hu!man coding technique and encode 1% of the least-likely magnitudes by transmitting the switch value along with the binary representation of the residual to the decoder. It is shown that the best compression ratios for these images are obtained using the DPCM/ EMLM method. Note that the storage cost of the Hu!man table is included when computing the compression ratios. The total number of nonzero entries from these four di!erent methods using six images are shown in Table 2. Here, the proportion threshold P of the MLM method is set to 0.99, and the range threshold R of the EMLM method is set to 12. The nonzero entries of the Hu!man method show the raw distribution of the input raw image. The nonzero entries of the DPCM/Hu!man method include the positive and the negative residual magnitudes after the DPCM scheme is used to decorate the raw image. The DPCM/MLM method uses the concept of symmetry origin to reduce the nonzero entries. Thus, the number of nonzero entries from DPCM/MLM is about half of that in DPCM/Hu!man. The DPCM/EMLM employs the folding technique to lower the nonzero entries further. The DPCM/EMLM method has the fewest nonzero entries as shown in Table 2. In general, large numbers of nonzero entries imply bigger storage spaces and higher computation complexities are demanded. The problem can be very serious if the images to be encoded are with high residual dynamic ranges. For example, if each pixel in the image is stored using 12 bits, the dynamic range of the residual image will be bigger than when 8 bits are used to represent each pixel. The issue about the selection of a good range threshold for the EMLM method is very important. Since these least-likely magnitudes are encoded by the folding technique instead of being encoded by the switch value along with their residual magnitudes, the bit patterns for the quotient and the remainder should be as small as possible. Thus, the selection of the range threshold tends to be a small value. However, the range threshold should be larger than the square root of the nonzero entries of

6 372 Y.-C. Hu, C.-C. Chang / Signal Processing: Image Communication 16 (2000) 367}372 the symmetry histogram. Therefore, the range threshold is set to 12 in our EMLM method to meet the two conditions halfway. 5. Discussion and conclusion The compression ratio is an important criterion in choosing a compression scheme for lossless image compression. Currently, the Lossless JPEG is the most common scheme for lossless image compression. It has two basic versions: the DPCM/ arithmetic coding and the DPCM/Hu!man coding. The DPCM/Hu!man coding scheme is more popular than the DPCM/Arithmetic scheme because it needs less computation cost. The Hu!- man coding technique is simple and e!ective, but it needs a lot of computation time if the dynamic range is very large. Therefore, the MLM scheme is proposed to resolve this problem. The DPCM/MLM scheme reduces computation time by reducing code computation while reaching high compression performance. This is because it saves a much smaller code tree with the encoded value. The DPCM/MLM scheme turns out to be an algorithm with signi"cant improvements in computation time and compression performance compared with the DPCM/Hu!man method of Lossless JPEG. These advantages of the DPCM/ MLM scheme become most apparent for images with high residual dynamic ranges. In this paper, an EMLM scheme based on the MLM scheme has been proposed to further improve the compression ratio in lossless image compression. By employing the folding technique and making use of the originally unused entry for zero magnitude in the MLM scheme, our EMLM scheme can achieve a higher compression ratio while keeping all the advantages of the MLM scheme. In other words, our method inherits the advantages when encoding images for high residual dynamic ranges. From the experimental results, it is shown that our newly proposed method outperforms the DPCM/Hu!man method and the DPCM/MLM method in compression ratio. Moreover, this DPCM/EMLM method reduces the number of generated codes because the number of the nonzero entries used is smaller than those of the DPCM/ Hu!man method and of the DPCM/MLM scheme. In other words, the newly proposed method indeed serves well as an e$cient and e!ective scheme for lossless image compression. References [1] M.F. Carreto-Castro, J.M. Ramirez, J.L. Ballesteros, D. Baez-Lopez, Comparsion of lossless compression techniques, in: IEEE Proceedings of the 36th Midwest Symposium on Circuits and System, Vol. 2, 1993, pp. 1268}1270. [2] C.C. Cutler, Di!erential quantization of communication signals, U.S. Patent 2,605,361, [3] R.G. Gallager, Variations on a theme by Hu!man, IEEE Trans. Inform. Theory 24 (1978) 668}674. [4] M. Giridhar, A. Nasir, D.S. Samuel, A two-stage scheme for lossless compression of images, in: IEEE Symposium on Circuits and System, Vol. 2, 1995, pp. 1102}1105. [5] D.A. Hu!man, A method for the construction of minimum redundancy codes, Proc. IRE 40 (1952) 1098}1101. [6] G.G. Langdon, An introduction to arithmetic coding, IBM J. Res. Develop. 28 (1984) 135}149. [7] Y.W. Nijim, S.D. Stearns, Di!erentiation applied to lossless compression of medical images, IEEE Trans. Med. Imaging 15 (4) (1996) 555}559. [8] J.B. O'Neil, Entropy coding in speech and television di!erential PCM systems, IEEE Trans. Inform. Theory IT-17 (1971) 758}761. [9] M. Rabbani, P.W. Jones, Digital image compression techniques, Tutorial Texts in Optical Engineering, Vol. TT7, SPIE Optical Eng. Press. [10] K. Sayood, K. Anderson, A di!erential lossless image compression scheme, IEEE Trans. Signal Process. 40 (1) (1992) 236}241. [11] S.D. Stearns, L. Tan, N. Magotra, Lossless compression of waveform data for e$cient storage and transmission, IEEE Trans. Geosci. Remote Sensing 31 (3) (1993) 645}654. [12] G.K. Wallace, The JPEG still picture compression standard, Comm. ACM 34 (4) (1991) 30}40. [13] J.C. Wehnes, H.T. Pai, A.C. Bovik, Fast lossless image compression, in: Proceedings of the IEEE Southwest Symposium on Image Analysis and Interpretation, 1996, pp. 145}148. [14] I.H. Witten, R.M. Neal, J.G. Cleary, Arithmetic coding for data compression, Comm. ACM 30 (1987) 520}540.