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JPEG-based Variable Block-Size Image Compression using CIE La*b*Color Space.

1. Introduction

Since its inception in 1993, JPEG has become a widely used image compression format in internet and multimedia applications [1-3]. As a lot of applications and work-flows have already been developed around it, its widespread usage remains unabated [4] despite the emergence of image compression standards such as JPEG 2000, and HEIC (HEVC still image compression standard). The basic block diagram of JPEG is shown in Fig. 1. As noted in Fig. 1, plenty of research effort has been made on improving the performance of almost all the major blocks of JPEG. In [5], trapezoid and triangular blocks are used instead of the regular 8x8 blocks in JPEG. Shape Adaptive DCT (SA-DCT) has also been used in conventional image/video coding algorithms since the 1990s. In [6], image is segmented using splitting and merging technique and SA-DCT is used to transform the segmented arbitrary shaped image regions. SA-DCT along with overlapped transform is also used in [7] where smooth regions of the image are downsampled and then transformed. Superpixel Driven Graph Transform (SDGT) is used in [8] instead of SA-DCT. In [8], image is segmented using the graph technique and each segment is transformed using SDGT. Authors in [9] have proposed the use of fuzzy logic for approximating the 2D-DCT used for transforming image sub-blocks in JPEG. Discrete Hartley Transform (DHT) is proposed in [10] as an alternative to DCT for image sub-block transformation. As DHT is used, another efficient quantization approach and scanning order (in lieu of zigzag scanning order used in JPEG) is proposed in [10]. To enhance the quantization performance, Just Noticeable Difference (JND) based quantization techniques for JPEG have been proposed in [11-13]. A lot of research has been invested to find efficient JND profiles as Human Visual System (HVS) characteristics can be accurately and efficiently modeled using JND. As a result, many DCT based JND models have been proposed in the past [14 - 21]. A new quantization table is generated for JPEG based on a generic psychovisual error threshold in [22]. In [23], the existing transform and quantization operations of JPEG are modified so that only bit-shifts are required for quantization operation. This leads to lower power consumption while encoding/decoding images. In [24], local prediction based adaptive scanning is used instead of conventional zigzag scanning in DCT based compression techniques such as JPEG and H.264/AVC-intra. The Run Length Coding (RLC) technique of JPEG is further optimized in [25] so that the redundancies in the JPEG's RLC technique are further reduced. In [26], four modifications are proposed to improve the efficiency of JPEG's binary arithmetic QM coding. Adaptive Golomb Coding is used as entropy coding in [27] to improve the compression efficiency of JPEG. A saliency based approach to improve the performance of JPEG is proposed in [28] and a block-based image quality metric is used to improve the rate-quality performance of JPEG in [29]. Apart from these techniques, many other optimization techniques have been proposed for the performance improvement of JPEG. Interested reader is referred to [4] for more details.

On the other hand, to the best of the authors' knowledge, not a lot of research is focused on the color space conversion technique used in JPEG. Color space conversion plays an important role in any compression scheme as it represents the color image in a color space to minimize the psycho-visual redundancies. YCbCr has been conventionally used for coding of images and videos as it is a luminance-chrominance color space.

In this paper, we use a perceptually uniform and linear CIE La*b* color space for compression of images. Though CIE La*b* color space has been used in past for compression of images/videos, it has been majorly used for pre-processing while the actual encoding/decoding is done in YCbCr color space [34, 35]. As linear and perceptually uniform CIE La*b* color space is used, we use a linear CSF for generation of the quantization matrices. Linear CSF used in this work is derived from the CSF proposed by Klein et al. [36]. Besides, a novel histogram based approach of dividing the image into variable block sizes is also proposed. Variable block size algorithms have come up in the past years and have been incorporated in [30 - 33]. They are mostly top-down quad-tree decomposition based approaches. These algorithms start from a large, uniform and fixed block called Macro Block (MB) typically of size 16x16 or 32x32. Each MB is then divided into smaller non-uniform blocks according to the image structure. In this process, though the size of image sub-blocks is variable, their positions are fixed. Our algorithm is more compatible to the image structure as both size and position of the image blocks are variable.

To describe the proposed techniques in detail, rest of the paper is organized as follows. In Section 2, we discuss the important techniques and concepts leading to the proposed techniques. Section 3 presents the proposed techniques while experimental results are presented and discussed in Section 4. We conclude this paper in Section 5.

2. Fundamental Methods and Materials

In this section, we review some of the fundamental techniques that are utilized in the proposed work.

2.1 CIE La*b* Color Space

Images are captured and/or displayed by most of the digital devices in RGB color space since it is an orthogonal color space and hence is hardware friendly. RGB color space being highly correlated is not suitable for compression. Generally, YCbCr color space is used by most of the standard image and video compression systems as it is a highly decorrelated luminance-chrominance color space. However, in this paper, we have used CIE La*b* color space instead of YCbCr due to its inherent advantages which are discussed as follows:

1) Like YCbCr, La*b* is also a luminance-chrominance color space. Therefore, it gives high decorrelation and hence chrominance subsampling [37] can be effectively implemented.

2) La*b* color space is a color-opponent color space with dimensions L for lightness and a* and b* for the color-opponent dimensions, based on nonlinearly-compressed CIE XYZ color space coordinates. The three planes L, a* and b* have pixel intensity ranges; L [0, 100], a* [-100, 100] and b* [-100, 100]. Hence, L plane can be represented using 7 bits/pixel as against the 8 bits/pixel required by Y plane of YCbCr. This itself contributes to a saving of 1 bit/pixel. The values that a pixel can take in a* and b* plane are centered on 0. Hence, there is no need of level shifting for a* and b* planes unlike YCbCr where level shifting is required for all the three planes.

3) CIE La*b* is a linear and perceptually uniform color space [35]. Hence quantization can be implemented effectively without perceptual loss of visual quality.

4) Another advantage of CIE La*b* is that it is a device independent color space [38].

Since authentic direct color conversion formulae for converting from RGB to CIE La*b*

color space are not available, we first convert to CIE XYZ and then to La*b*. For conversion to CIE XYZ color space, gamma-corrected RGB values are converted to linear RGB values before applying the following conversion formula [39];

[mathematical expression not reproducible] (1)

where R, G, B are the linear RGB values.

Conversion from CIE XYZ to CIE La*b* [34, 40, 41] is accomplished using (2), (3) and (4) as discussed below:

L = 11+ x f([Y/[Y.sub.n]]) -6 (2)

a* = 500 x [f([X/[X.sub.n]])- f([Y/[Y.sub.n]])] (3)

a* = 200 x [f([Y/[Y.sub.n]])- f([Z/[Z.sub.n]])] (4)

where X, Y, Z are calculated using (1) above and [X.sub.n], [Y.sub.n], [Z.sub.n] are the tri-stimulus values of the adapting white [34, 40, 41]. The function f used in (2), (3) and (4) is given as;

[mathematical expression not reproducible] (5)

2.2 Exponential Golomb Coding

Exponential Golomb codes were first proposed for the representation of non-negative integers with exponentially decaying probability distribution [42]. Exponential Golomb codes offer a host of advantages over other DCT based entropy coding techniques. Firstly, they don't require look-up tables to provide supplemental information to the encoder. And secondly, theoretically they can be used to encode a data source with an infinite number of possible symbols unlike Huffman coding [27]. Given a non-negative integer n, the zeroth order exponential Golomb code of n is represented as [G.sup.0.sub.exp](n) and can be computed using following steps:

Step 1: Calculate prefix m from the symbol to be coded n using,

[mathematical expression not reproducible] (6)

and form the unary code of m (unary code of a number x is the number of x zeros followed by 1).

Step 2: Determine the binary representation of

(n + 1) - [2.sup.m] (7)

truncated to m least significant bits.

Step 3: Concatenated binary representation of results of steps 1 and 2 is the exponential Golomb code of n.

2.3 Binary Arithmetic (QM) Coding

Binary arithmetic coding is used as an entropy coding technique in many image/video compression algorithms. QM-coder is a version of binary arithmetic coding used with some implementations of the JPEG standard [3, 43]. The process of binary arithmetic QM coding is presented here in brief.

In binary arithmetic QM coding, instead of coding symbols 0 and 1 directly, they are mapped to most probable symbol (MPS) or least probable symbol (LPS). The basic idea behind this mapping is that when a completely black image (black denoted by 0) has a small white patch (white denoted by 1) in it, 0 is mapped to MPS while 1 is mapped to LPS. Whenever a symbol is input, QM coder first decides whether it is MPS or LPS based on the symbols input in the past with the help of probability estimation tables and then encodes it.

Suppose we have an interval A and the LPS probability estimate is Qe, MPS probability estimate will obviously be 1 - Qe since there are only two possible input symbols. The interval A is then divided into two parts by the QM coder such that LPS sub-interval is AxQe and the MPS sub-interval is Ax(1 - Qe). The positioning and sizes of the sub-intervals according to convention is depicted in Fig. 2. The input symbol sequence of MPS and LPS is encoded into a code stream (pointer) C by QM coder. Ideally, C can point anywhere in the current interval (either MPS or LPS) but for ease of operation, QM coder points C at the bottom of the current interval.

3. Proposed Work

Flowchart of the proposed JPEG-based image compression scheme is shown in Fig. 3. The blocks that are modified as compared to the basic JPEG block diagram in Fig. 1 are highlighted. The structure of Fig. 3 is deliberately kept similar to that in Fig. 1 in order to highlight the modifications and bring out the differences.

In this scheme, classification and subsequent grouping of adjacent 8x8 blocks based on their similarity in terms of mean and variance is incorporated in CIE La*b* color space. This results in the formation of blocks of 10 different sizes. Most of the HVS based quantization techniques for DCT are proposed for perceptually non-uniform and non-linear YCbCr color space and fixed block sizes. We propose modifications to make them suitable for the perceptually linear and uniform CIE La*b* color space and for the image-dependent non-uniform image sub-blocks, as it is more pragmatic. Important and highlighted (modified) blocks/steps of the algorithm presented in Fig. 3 are explained below in detail.

3.1 Conversion to CIE La*b* Color Space

Input RGB image is first converted to CIE La*b* using (1) to (5) given in section 2.1.

Since human eye is less sensitive to chrominance information than luminance, chrominance subsampling is applied on a* and b* planes. Subsampling ratio of 4:2:0 is used [37]. In other words, a* and b* planes are subsampled by a factor of 2 both horizontally and vertically.

As mentioned in section 2.1, since the intensity values of a* and b* color planes are already centered on zero, there is no need of level shifting for a* and b* planes. However, a value of 50 is subtracted from each pixel intensity value of the L plane in order to center the mean of L plane on zero, in the proposed algorithm. Rest of the steps of the proposed algorithm are same for level shifted L plane and subsampled chrominance planes a* and b*.

3.2 Division into variable non-uniform blocks

The proposed algorithm uses blocks of different rectangular shapes and sizes as against the uniform 8x8 sized blocks used by most of the image compression algorithms. Sixteen different block sizes are used in the proposed work, smallest being 8x8 and the largest, 32x32. The process of division into these non-uniform blocks is described below with the help of Fig. 4.

3.2.1 8x8 blocks

Initially, the input image is divided into the smallest size 8x8 non-overlapping uniform blocks just like in JPEG and as shown in Fig. 4 (a).

3.2.2 Indexed histogram based merging

Mean and variance of each 8x8 block is calculated. Only first two statistical moments i.e. mean and variance are used to describe the characteristics of the 8x8 blocks. Thus, for each sub-plane (L, a* or b*) of an input image I of size PxQ, we have two planes each of size P/8xQ/8; namely mean plane containing all mean values of 8x8 blocks of the image shown in Fig. 4 (b) and variance plane having all variances of 8x8 blocks of the image shown in Fig. 4 (c). Mean and variance values of 8x8 image sub-blocks are displayed in Fig. 4 (b) and Fig. 4 (c). These values are normalized to the range 0 to 255 only for display purpose so that the information contained in them is clearly visible. However, for the indexed histogram formation and subsequent grouping, original mean and variance values are used. This data is stored in a 3 dimensional matrix (A) with two P/8xQ/8 planes. Thus, A(1:P/8,1:Q/8,1) denotes the mean plane and A(1:P/8,1:Q/8,2) denotes the variance plane.

A histogram table is formed using these values and statistical bin size thresholds are chosen as discussed further. A sample indexed histogram table structure is shown in Fig. 5.
Fig. 5. Indexed Histogram Table H with arbitrary data.

Index No.   Mean    Variance   Population

1           15.64   0.79       50
2           13.72   0.35       217
3           19.42   1.31       41
*           *       *          *
*           *       *          *
*           *       *          *


The histogram formation process is described as below;

1) Initialize an indexed histogram table structure H with k = 1 row and four columns, namely index, mean, variance and population. The number of rows k goes on increasing if the next block [mean, variance] is different than the earlier registered one [mean, variance] using thresholds [Th.sub.mn] and [Th.sub.var].

2) Initialize the first entry in the table by the mean and variance of the first 8x8 image block that is the top left corner block in the image and population and index to 1.

H(1,2:3) = A(1,1,1:2), H(1,4) = 1 and H(1,1) = 1.

3) Scan the matrix A along the rows and read the next mean and variance values stored in A. Compare every new pair of mean and variance stored in matrix A with all the earlier recorded mean and variance pairs in H and increment the population of that row of H which satisfies the condition given in (8), else record mean and variance values at that location of A in the next new row of H and initialize its population to 1.

(|A(m,n,1) - H(k,2)| < [Th.sub.mn])AND(|A(m,n,2) - H(k,3)|<[Th.sub.var]) (8)

The thresholds [Th.sub.mn] and [Th.sub.var] are obtained by calculating standard deviation of the mean and variance values in the mean plane and variance plane, respectively.

Note that, here, [Th.sub.mn] is used to threshold the difference between mean values and similarly, [Th.sub.var] is used to threshold the difference between variance values. Hence, standard deviation of the respective values is used as the threshold. In addition to the formation of the histogram table H, a new index matrix Idx of size P/8xQ/8 is formed simultaneously. Idx contains the indices of all the block locations in matrix A, as indicated by histogram table H. These indices in Idx are further used for grouping of the adjacent blocks. If indices of the immediate neighbouring blocks (N4) are same, then blocks are merged to form bigger blocks.

3.2.3 Merging of adjacent blocks

Adjacent blocks are grouped based on the index matrix Idx to form blocks of sizes 32x32, 32x24, 24x32, 24x24, 32x16, 16x32, and so on, till 8x8. These block sizes are allotted indices from 15 to 0, respectively. Note that all possible combinations of block sizes starting from 32x32 till 8x8 in steps of 8 are used.

The highest block size chosen is 32x32. This is because sharpness of human vision is highest in the fovea region covering approximately 2[degrees] of visual angle. This corresponds to a radius of approximately 40 pixels for [LAMBDA] = 1.5 min/pixel for normal viewing distance. Thus a 32x32 image sub-block is faithfully covered by the visual angle [21]. Smallest block size considered is 8x8 in accordance with the JPEG standard. In the above procedure, vertical block sizes such as blocks 32x24, 24x16, 32x16, etc. are considered before horizontal blocks 24x32 or 16x32 as natural images contain more vertical structures than horizontal ones [44]. Experimental analysis shows that natural images contain very less number of block sizes greater than 32x32. Thus, considering block sizes greater than 32x32 will further increase the blocking overhead and the compression performance may deteriorate.

Further, a block size index array, BIdx, is constructed using Idx and the block indices 0 to 15. BIdx indicates the spatial position and its block index in the image and is used, further, for forming the compressed bitstream. The same BIdx will be used at the decoder to decode the image. Section of a sample index matrix Idx is shown in Fig. 6 (a). Adjacent blocks having same indices are merged to form bigger non-uniform blocks which are shown using different colors. Fig. 6 (b) shows a sample block size index array BIdx formed using the sample index matrix of Fig. 6 (a). The algorithm used for merging is diagrammatically explained using Fig. 7 and summarized in the form of a pseudo code in Algorithm 1.

It is clearly evident from Algorithm 1 that priority is given for the formation of bigger block sizes such as 32x32, 32x24, 24x24 etc. Thus, the order in which blocks are checked for merging based on Idx is: 32x32, 32x24, 24x24, 32x 16, 16x32, 24x 16, 16x24, 16x 16, 32x8, 8x32, 24x8, 8x24, 16x8, 8x16 and 8x8. In this way, the complete image is encoded and a block index array named BIdx is formed. This block index array is compressed using exponential Golomb coding and sent to the decoder as overhead information along with the compressed image blocks.
Fig. 6. Sample Idx and BIdx matrices

(a) Section of a sample index matrix Idx

32   32   32   32   40   40   39   39   39   36   36    5
32   32   32   32   40   40   19   43   43   36   36    5
32   32   32   32   40   39   24   15   43   43   17    5
32   32   32   32   40   39   40   38   15   13   14    5
18   18   18   32   32   32   33   33   36   38   39    5
18   18   18   32   32   32   33   33   12   17   10   10
18   18   18   18   17   16   33   33   37   37   37   37

(b) Corresponding block size index array BIdx

16    8    3    4   ...  ...   6

Algorithm 1: Grouping of adjacent 8x8 image sub-blocks

Input:  Index matrix Idx, Input image plane (L, a* or b*) of size PxQ
Output: Block size index array BIdx, compressed codestream
begin
    1. Initially set index variables to zero, i = 0, j = 0 and p = 1.
    Following steps are executed until all the 8x8 blocks of the image
    are encoded:
    2. If all values of matrix Idx(i:i+3,j:j+3) are same
       Code 32x32 block of the image plane;
       BIdx(p) = 16;
       j = j + 3;
    3. Else check all values of matrix Idx(i:i+3,j:j+2), if same
       Code 32x24 block of the image plane;
       BIdx(p) = 15;
       j = j + 2;
    4. Else check values of matrix Idx(i:i+2,j:j+3), if same
       Code 24x32 block of the image plane;
       BIdx(p) = 14;
       j = j + 3;
    5. Else check values of matrix Idx(i:i+2,j:j+2), if same
       Code 24x24 block of the image plane;
       BIdx(p) = 13;
       j = j + 2;
    6. ...
    7. ...
   10. Code 8x8 block of the image plane;
       BIdx(p) = 0;
       j = j + 1;
   11. p = p + 1; Increment i appropriately;
end


Thus, after similarity based merging of adjacent 8x8 blocks, the input image gets divided into non-uniform blocks as seen in Fig. 4 (d). The above splitting and merging algorithm is implemented on the level shifted L plane and subsampled a* and b* planes separately as all the three planes are compressed separately. Fig. 4 (d) shows the results of division of L plane only. Colored image is shown instead of the gray scale luminance image for better understanding. From Fig. 4 (d), it can be clearly seen that smooth regions have larger blocks whereas edges and textured regions get divided into smaller blocks, more specifically, 8x8 blocks. Horizontal and vertical edges get divided into preferably 8x32, 8x24, 8x16 and 32x8, 24x8, 16x8 sized blocks, respectively.

However, division into non-uniform blocks may not be properly justified from Fig. 4 (d) as regions appearing uniform are shown to be divided into smaller blocks. Hence, a simple color image shown in Fig. 8 (a) is used to illustrate the result of division into non-uniform blocks. Fig. 8 (b) shows the results of splitting and merging algorithm implemented on level shifted L plane of Fig. 8 (a) justifying the division into non-uniform blocks. However, there are still some discrepancies as red and pink colored stripes are getting merged with gray colored boundary region in single 32x32 blocks. Similarly, yellow, green and aqua colors are also getting merged. This happens as luminance values of red, pink and gray colors and that of yellow, green and aqua colors are similar as seen from Fig. 8 (c). This is taken care of by implementation of this algorithm on subsampled chrominance planes. Result of division of chrominance b* plane into non-uniform blocks is shown in Fig. 8 (d). In Fig. 8 (d), all the color components are getting divided faithfully. This also justifies the need of implementation of the splitting and merging algorithm on the three image planes separately. As compared to the hierarchical variable block size algorithm of H.264 or HEVC, our algorithm is more compatible to the image structure as both size and position of the image blocks are variable.

The MxN image sub-blocks thus formed, are transformed into MxN spectral domain coefficients using two-dimensional Discrete Cosine Transform (2D-DCT).

3.4 CSF based Quantization

Since non-uniform block sizes are used, we need quantization matrices of sizes suitable for the 16 different block sizes. CSF based quantization method proposed in [36] is modified and used for the generation of quantization matrices. Though [36] describes quantization matrix generation for 8x8 block size only, this method is heuristically extended for the generation of quantization matrices for all other non-uniform block sizes. As CIE La*b* space is used in the proposed scheme, a linear CSF is used for the generation of quantization matrices. As the range of intensity values is different for L plane and a* and b* planes, different set of quantization matrices are generated for L plane and a* and b* planes. CSF for quantization matrix generation as given in [36] is

CSF(f) = 100[square root of f] exp(-0.13f) (8)

where f is spatial frequency in cycles/degrees, given by

f(u,v) = 30[square root of [u.sup.2] + [v.sup.2]]/8x[LAMBDA] (9)

where u,v are co-ordinates of the DCT block. Pixel size [LAMBDA] is assumed to be 1.5 min/pixel [36].

Above equation gives frequency matrix for 8x8 block size only. Hence, the factor of 8 in the denominator of this equation. This equation can be extended to find frequency for any MxN size block using (10);

f(u,v) = 30[square root of [u.sup.2] + [v.sup.2]]/Nn x [LAMBDA] (10)

where Nn is given by (11) as follows

Nn = [square root of MxN] (11)

Since we are using CIE La*b* color space for compression, a linear CSF given by (12) below is used in this paper for quantization matrix generation.

CSF (f) = c(f - [f.sub.max]) (12)

where f is given by (10) and [f.sub.max] is the maximum frequency in an MxN image sub-block and is calculated using the formula shown below

[f.sub.max] = 30[square root of [M.sub.2] + [N.sub.2]]/[Nn x [LAMBDA]] (13)

In (12), c is a constant which governs the performance and quality of reconstructed image in terms of CR and PSNR as shown in Fig. 9.

As seen from Fig. 9, CR is highest for c values around zero, though PSNR is lowest for these values. An appropriate value of c can be chosen so that the tradeoff between CR and PSNR is optimum. Linear CSF given in (12) gives good results as we are using CIE La*b* space which is linear and perceptually uniform. This is evident from the results shown in Table 1.

Table 1 shows the CR and bpp (bits per pixel) achieved by linear CSF as well as non-linear CSF for fixed PSNR values. It can be clearly seen that for the same PSNR, CR achieved using the linear CSF proposed by us is better than the CR achieved by non-linear CSF proposed in [36]. Though the gain in CR is marginal in case of some images like Baboon, Yacht, Barbara, etc., compression performance of linear CSF based quantization is never poorer than that of non-linear CSF based quantization.

The role of orientation tuning is also considered by using the Orientation Tuning Function (OTF) as given by (14);

[mathematical expression not reproducible] (14)

If the value of OTF drops below 0.5, OTF value is considered to be 0.5.

Using (9) to (14), the thresholds for DCT basis functions in an MxN matrix are given as;

[mathematical expression not reproducible] (15)

Norm(u,v) are the normalization functions used in 2D-DCT. One thing to note from the above equation is that value of the threshold for u = 0 and v = 0 is [infinity] as value of CSF is zero for u = 0 and v = 0. Thus, quantization values for the DC coefficient are decided (or calculated) separately.

Finally, quantization matrix is given by

Quant (u, v) = min(T(u,v) x range,coeff_max(u,v)) (16)

where coeff_max is the matrix containing maximum values that DCT coefficients can take for a given range of spatial frequencies. Stimulus required to generate the matrix [36] is;

[x.sub.max](x,y) = [range/2] x (1 +sgn(cos((2x + 1)u[pi]/2M)cos((2y + 1)v[pi]/2N))) (17)

where sgn = +1 or -1 depending on the argument.

Above equation can be modified if the range of intensity values is from -X to +X, as in the case of a* and b* planes. The modified equation is

[x.sub.max](x,y) = [range/2] x sgn (cos((2x + 1)u[pi]/2M)cos((2y + 1)v[pi]/2N)) (18)

Quantization value for the DC coefficient is chosen depending upon the maximum value attained by the DC coefficient of the DCT sub-block that is calculated using (17) and (18). As quoted in [45], a human being can distinguish at the most 100 distinct luminance values ([L.sub.max] = 100). Hence, we quantize the maximum DC coefficient value for L plane of each block size in 100 levels. For example, for an 8x8 L block, maximum DC coefficient value is 800. If this value is quantized in 100 levels, we end up at a DC quantization value of 8. For 8x8 chrominance a* and b* blocks also, the DC coefficient quantization value is taken as 8. However, the range of a* and b* planes is -100 to +100 which is double than that of L plane. Hence, the same DC coefficient quantization value (i.e. 8 in case of 8x8 blocks) results in 200 quantization levels for a* and b* planes. Similarly, DC quantization steps for some of the 16 block sizes are calculated and presented in Table 2.

Thus, 32 quantization matrices are generated; 16 for L plane and 16 for a* and b* planes, using the approach discussed above. However, the quantization matrices for a* and b* planes are double of the quantization matrices for L plane. Therefore, only 16 quantization matrices for L plane need to be stored.

3.5 Entropy Coding

Non-zero discrete cosine transformed quantized coefficients of each image sub-block are zigzag ordered before entropy coding. In this paper, we have used modified binary arithmetic QM coding as entropy coding technique with our proposed algorithm. Modification of QM coding used with JPEG to encode variable block size DCT coefficient matrix is straightforward. Exponential Golomb coding is used for encoding the non-uniform block size index array, BIdx.

The decoding process is essentially opposite and exact reverse of the complete encoding process shown in Fig. 3.

4. Experimental Results and Discussion

Performance of the proposed algorithm is experimentally evaluated using 25 standard test images of different types and sizes available online at the referenced websites [46, 47]. Peak Signal to Noise Ratio (PSNR) is used to evaluate the reconstructed image quality whereas compression performance is tested using bits per pixel (bpp) and CR. If length of the compressed bitstream is denoted by no_bits, bits per pixel denoted by r is given as

r = [no_bits][ x q 3]bpp

where p and q are the number of rows and columns of the color image, respectively.

Value of CR can be obtained from r using the relation in (20);

CR = [8/r] (20)

All experiments are done on a desktop computer with an Intel Core i7-4770 processor running at 3.40 GHz with 32 GB DDR4 RAM. MATLAB programming environment is used for Windows 8 OS.

As discussed earlier, 16 quantization matrices are generated and can be stored at the encoder and the decoder for the CSF based quantization approach. Alternatively, with some increase in the computational complexity, quantization matrices can be generated using the CSF based quantization approach or by interpolation from a single 8x8 quantization matrix for a given value of CSF constant c. Each value of CSF constant c results in a unique set of quantization matrices. This fact is used to control the compression quality i.e. the achieved bpp, CR and PSNR values in the proposed work. A lower absolute value of c will result in higher compression (CR) and lower reconstructed image quality (PSNR) whereas a higher absolute value of c yields lower CR and higher PSNR. A value of c = -0.5 is chosen arbitrarily for comparison and benchmarking so that the JPEG, JPEG-XR and the proposed algorithm yield exactly same PSNR on the set of test images. However, the final choice of c depends on the end-user or the application for which the proposed compression scheme is being used. In JPEG and JPEG-XR, compression quality is controlled by quantization factor or quality factor (QF). QF is a constant by which the standard JPEG quantization matrices are multiplied in order to achieve required CR and PSNR. QF can be varied from 1 to 100 in JPEG whereas it can take values from 1 to 255 in JPEG-XR. Higher value of QF results in high compression and lower reconstructed image quality and vice versa in JPEG [42]. However, higher value of QF yields higher reconstructed image quality and hence, less compression in JPEG-XR.

Representative original images from the set of 25 standard test images are shown in Fig. 10 (a) - (e). Images reconstructed using the proposed technique along with bpp and PSNR are shown in Fig. 11 (a) - (e). It is clear that images with high neighboring interpixel intensity variations like Baboon yield poor bpp and PSNR. On the other hand, images like Woman-baby, which contain low neighboring interpixel intensity variations, yield high bpp and PSNR.

Results obtained using the proposed scheme are compared with JPEG using 25 standard test images and also with other related published research works [48-50].

4.1 Comparison with JPEG and JPEG-XR using standard test images

We compare the results obtained using the proposed scheme with JPEG and JPEG-XR [51]. For comparison with JPEG, we use MATLAB's built-in implementation of the JPEG encoder. Different parameters need to be set for the JPEG encoder in order to obtain optimum performance on different images in different situations. For JPEG, compression mode is set to 'lossy' and bit depth is set at its default value of 8. For comparison with JPEG-XR, we use an open source implementation by [52]. Since JPEG and the proposed scheme use non-overlapping blocks, we turn overlapping off in JPEG-XR for fair comparison of results. For the same reason, chrominance sub-sampling is also set to 4:2:0 in JPEG-XR. Image quality or quantization parameter is varied, in both JPEG and JPEG-XR, to get PSNR equal to that achieved by the proposed scheme for different images.

The benchmarking of the 25 standard test images along with the average results is shown in Table 3. We vary the quality setting parameter from 11 - 54 for JPEG and from 1 - 85 for JPEG-XR so as to get same PSNR values for the two algorithms.

As seen from Table 3, proposed scheme surpasses the bpp and CR achieved by JPEG for all the standard test images. It can be deduced that the proposed scheme outperforms JPEG by a large margin in case of images having large smooth regions i.e. low variance images such as Color bars, Chalk, Jupiter-moon, etc. Due to the use of variable block sizes, low variance images get divided into larger blocks instead of the conventional 8 x 8 blocks in JPEG. The proposed scheme performs only marginally better than JPEG for images having high variance (i.e. more texture) such as Peppers, Football, Cornfield, etc. This is because high variance images do not benefit from the use of larger block sizes as proposed in the variable block size algorithm. Due to absence of smooth (low variance) regions, these images get divided into smaller block sizes only. Proposed scheme achieves, on an average, 45.01% more CR than

JPEG.

Though the proposed scheme achieves 3.63% better CR than JPEG-XR, it performs better than JPEG-XR only for a few images such as Baboon, Color-bars, RGB full color cube, Cornfield, Jupiter-moon and Barbara. In case of images such as Football, Goldhill, Jupiter-moon close-up, etc. JPEG-XR performs marginally better than the proposed scheme. JPEG-XR outperforms the proposed scheme for rest of the images as evident from Table 3.

Performance of the proposed scheme is also compared with JPEG and JPEG-XR using rate-distortion curves shown in Fig. 12. Bits per pixel versus PSNR graphs are shown for eight standard test images. Graphs are plotted by keeping the PSNR achieved using JPEG and JPEG-XR same as that achieved by the proposed algorithm. Thus, the algorithm having lower bpp value at the same PSNR can be considered to have better compression performance. As seen from Fig. 12, performance of the proposed scheme is better than JPEG for all the images at lower bitrates (i.e. lower bpp values). For the Woman-baby image, performance of the proposed scheme is better than JPEG at all the bitrates. The proposed scheme performs only marginally better than JPEG-XR at lower bit rates for images such as Baboon and Jupiter-moon. Performance of JPEG-XR is marginally better than the proposed scheme for images such as Woman-baby, Cornfield and Yacht. In general, it can be said that performance of JPEG-XR is comparable to the proposed scheme at lower bit rates.

It is well-known that JPEG2000 achieves approximately 50% more CR than JPEG [53] while being 3 to 5 times more complex [54, 55]. Therefore, it is expected that the proposed scheme may not out-perform JPEG2000. Furthermore, JPEG2000 is a wavelet based compression scheme unlike the proposed scheme and JPEG-XR, which uses a transform similar to DCT and a block-based coding structure.

Due to the use of variable non-uniform blocks, the proposed scheme requires more computations than JPEG. The histogram formation step in the proposed algorithm is the most computationally intensive step. An image plane of size PxQ is first divided into 8x8 non-overlapping blocks and then a histogram is formed using thresholds [Th.sub.mn] and [Th.sub.var]. In the worst case scenario, if none of the histogram rows get merged with another; formation of histogram itself may require ([[PxQ])/[32]] x ([[PxQ])/[32]+1] comparisons. However, this scenario is not expected to occur as we are using image dependent thresholds [Th.sub.mn] and [Th.sub.var]. After experimentations, it has been observed that the number of histogram rows varies between 2 and 1167 for the standard test images and synthetic images used for benchmarking. Considering the number of histogram rows as 1167, the proposed algorithm requires approximately 1.5 times more computations as compared to JPEG.

4.2 Comparison with contemporary techniques

Proposed scheme is also compared with other contemporary techniques such as those proposed in [48-50] respectively, as shown in Table 4. [49] has achieved compression using machine learning techniques. Image is compressed using selected colors during encoding and missing colors are predicted during decoding. Better compression quality is achieved by minimizing the prediction error during decoding. Data-hiding and compression tasks are incorporated seamlessly by [48]. Blocks are embedded with secret data and based on the current bit being embedded, the block is encoded using either SMVQ or image inpainting. [50] incorporates HVS based models for thresholding and quantization of coefficients in a wavelet based compression scheme.

For fair comparison of results, PSNR of the proposed scheme is kept equal to the highest PSNR achieved by one of the above-mentioned schemes. However, in case of Baboon image, PSNR of 25.2 dB cannot be achieved using the proposed method. Hence, minimum PSNR achievable for Baboon i.e. 29.05 dB, is used. Similarly for images; Peppers, Airplane, Tiffany and Sailboat, minimum PSNR achievable by the proposed scheme is used. As seen from Table 4, our method performs better than almost all the techniques proposed in [48-50]. Note that the bpp values quoted in [50] are converted to CR using the relation in (26). For comparison with results in [50], two images, namely, Parrots and Statue, from the LIVE database [56] are used.

5. Conclusion

In this paper, images are compressed using a linear and perceptually uniform CIE La*b* color space in the JPEG image compression framework instead of the conventional YCbCr color space. A linear CSF suitable for the linear and perceptually uniform CIE La*b* color space is proposed. Image dependent variable size sub-blocks generated using the proposed novel histogram based merging technique are also used instead of the conventional 8x8 image sub-blocks in JPEG. In other words, images are compressed in the proposed scheme using the adaptively separated non-uniform blocks in the CIE La*b* color space using the JPEG image compression framework but our own quantization matrices generated using the proposed linear CSF function.

The proposed scheme achieves, on an average, 45.01% more CR than JPEG for the same reconstructed image quality in terms of PSNR. It outperforms JPEG in terms of reconstructed image quality at lower bitrates. The proposed scheme also yields reconstructed image quality better than JPEG at all the bitrates for simple, low variance images. The proposed scheme achieves on an average, 3.63% better CR than JPEG-XR. Performance of JPEG-XR is comparable to the proposed scheme at lower bit rates only. It is also demonstrated that the use of linear CSF for the perceptually uniform and linear CIE La*b* color space yields on an average 8% more compression than the non-linear CSF used with YCbCr color space.

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Samruddhi Y. Kahu received B.Eng. degree in Electronics and Communication Engineering from Nagpur University, Nagpur, India in 2011 and M.Tech. degree in Communication Systems Engineering from Visvsesvaraya National Institute of Technology, Nagpur, India in 2014 where she is currently pursuing her Ph.D. Her research interests include image and video compression and processing.

Kishor M. Bhurchandi received his B.Eng. and M.Eng. degrees in electronics engineering in 1990 and 1992. He further obtained his Ph.D. degree from Visvesvaraya Regional College of Engineering, Nagpur University, Nagpur, India in 2002, where he is currently working as a Professor. He is principal investigator of two funded major research projects in the field of signal processing and embedded systems. He has more than 45 publications to his credit. He has also worked on industrial problems on heavy and in motion weighing systems. He is the co-author of a popular book titled Advanced Microprocessors and Peripherals published by McGraw Hill, India. His research interests include color image processing and analysis, computer vision, digital signal processing and embedded systems.

Samruddhi Y. Kahu (1) and Kishor M. Bhurchandi (1)

(1) Department of Electronics and Communication Engg., Visvesvaraya National Institute of Technology

Nagpur, Maharashtra 440010 - India

[e-mail: samruddhikahu@gmail.com]

* Corresponding author: Samruddhi Y. Kahu

Received January 12, 2018; revised March 26, 2018; accepted April 16, 2018; published October 31, 2018

http://doi.org/10.3837/tiis.2018.10.023
Table 1. Effect of linear and non-linear CSF based quantization on
compression performance

Image Name     PSNR (dB)   Non-linear CSF [40]    Linear CSF
                           CR       bpp          CR       bpp

Baboon         30.2         13.95   0.57          14.41   0.55
Goldhill       34.52        34.21   0.23          36.72   0.22
Jupiter-moon   35.07        47.32   0.17          54.23   0.15
Lena           35.13        39.46   0.20          42.12   0.19
Woman-baby     41.4        107.98   0.07         119.63   0.07
Yacht          35.27        26.49   0.30          27.53   0.29
Barbara        34.42        28.84   0.28          30.42   0.26
Average        35.14        42.61   0.19          46.44   0.17

Table 2. DC coefficient quantization levels for CSF based quantization

Block Size     DC coefficient
             quantization levels

 8x8                 8
 8x16               11
16x8                11
16x16               16
16x24               20
24x16               20
24x24               24
24x32               28
32x24               28
32x32               32

Table 3. Comparison with image compression standards using 25 standard
test images

Image Name (*)        PSNR       JPEG       JPEG-XR         Proposed
                                                            Scheme
                      (dB)     r     CR     r        CR      r     CR
                             (bpp)         (bpp)           (bpp)

Baboon                29.6   0.26   30.24  0.26     30.35  0.22    36.6
Boats                 33.9   0.13   59.9   0.1      77.9   0.11    69.7
Cable-car             32.9   0.15   51.9   0.12     65.5   0.13    59.5
Caster-stand          33.3   0.13   60.6   0.11     71.7   0.12    64.8
Color-bars            43.6   0.13   63.7   0.11     71.2   0.03   310.4
RGB full color cube   37.9   0.09   81.1   0.06    144     0.04   186.9
Cornfield             32.3   0.17   46.4   0.17     48.1   0.16    48.6
Chalk                 34.9   0.12   68.2   0.06    137.8   0.08   104.5
Flower                34.4   0.12   69.3   0.09     90.1   0.10    76.2
Flowers               30.96  0.25   31.8   0.13     59.4   0.19    42.9
Goldhill              33.0   0.13   60.6   0.11     76     0.11    75.6
Jupiter-moon          33.4   0.13   63.9   0.11     75.9   0.11    73.9
close-up
Jupiter-moon          34.5   0.11   75.9   0.09     85.4   0.07   115.6
Lena                  33.6   0.12   64.02  0.09     85.4   0.09    81.3
Monarch               33.7   0.13   60.4   0.11     72.1   0.12    67.1
Peppers               32.6   0.12   65.4   0.09     85.4   0.12    65.6
Football              31.8   0.18   43.8   0.17     48.1   0.17    46.8
Strawberries-coffee   34.6   0.12   66.03  0.07    120.4   0.08    95.9
Stream                34.5   0.09   81.8   0.07    106.7   0.09    91.6
Woman-baby            39.3   0.08   99.5   0.04    190.4   0.05   168.6
Yacht                 32.9   0.15   53.5   0.12     65.5   0.14    57.9
Airplane              33.6   0.13   62.9   0.09     85.4   0.11    73.6
Sailboat              31.7   0.17   46.9   0.14     59.2   0.15    53.7
Tiffany               33.4   0.11   71.6   0.06    128.2   0.09    86.5
Barbara               32.4   0.18   45.4   0.16     50.7   0.14    57.3
Average of 25 std.    33.95  0.14   60.99  0.11     85.23  0.11    88.44
test images

(*) The images are taken from Image processing place [46] and Classic
Image processing library [47].

Table 4. Comparison with other proposed compression schemes

Image      Proposed Scheme   Zhang et al.'s scheme   Qin et al.'s scheme
Name       CR       PSNR     CR     PSNR             CR      PSNR
                    (dB)            (dB)                     (dB)

Lena       105.8    33.0     23.7   33.0             20.66   29.85
Peppers    137.2    31.37    22.6   31.0             21.23   30.35
Baboon     136.9    29.05    11.4   25.2             -       -
Airplane   174.7    31.98    -       -               21.37   29.31
Tiffany    157.94   32.5     -       -               21.74   30.54
Sailboat   141.8    30.76    -       -               20.12   28.42
Monarch     63.81   33.86    -       -               -       -
Parrots     77.19   36.51
Statue      89.62   33.67

Image           Sreelekha et al.'s scheme
Name          PLC-1                PLC-2
            CR      PSNR (dB)   CR      PSNR (dB)

Lena                -                   -
Peppers     -       -           -       -
Baboon      -       -           -       -
Airplane    -       -           -       -
Tiffany     -       -           -       -
Sailboat    -       -           -       -
Monarch     40      33.23       40      33.86
Parrots     57.14   33.75       57.14   36.51
Statue      36.36   32.80       36.36   33.66
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