COLOR IMAGE COMPRESSION BASED ON ABSOLUTE MOMENT BLOCK TRUNCATION CODING USING DELTA ENCODING AND HUFFMAN CODING

1.   Introduction

Image compression is the art or science of effectively coding digital images to reduce the number of bits required to represent the image [1].It reduces the size of the image, so that the compressed image could be sent over the computer network from one place to another in short amount of time. In addition, the compressed image helps to store more number of images on the storage devices [2].Image compression plays an important role in many applications, including image database, image communications, remote sensing , etc [3].There are different techniques for compressing images. They are classified into two classes called lossless and lossy compression techniques. In lossless compression the reconstructed image is identical to the original image in every sense, whereas in lossy compression  the reconstructed image is similar to the original image but not identical to it, because some image information is lost during compression process[4].

Block truncation coding (BTC) is a simple and fast lossy compression technique for digitized grayscale images originally introduced by Delp and Mitchell [5]. The main idea of BTC is to perform moment preserving (MP) quantization for blocks of pixels so that the quality of the image will remain acceptable and at the same time decrease the storage space [6]. Since BTC has the advantages including  the encoding and decoding are extremely simple and fast[7] , requires a smaller  computational  load and much less memory, produce sharper edges which is important for a human visual system [8] , low computational cost, relatively high quality [9] , easy to implement compared to other algorithm such as vector quantization , and transform coding[10]. It was widely used in many compression applications such as high definition television(HDTV) , Internet video , digital cameras and printers [11],and software-based multimedia systems[8].On contrast the main drawbacks of original BTC are producing fixed length compression [11] , achieving low compression ratio[9], and it doesn’t perform well as well as transform coding  such as JPEG[8].

A simple and fast variation of BTC, called absolute moment BTC (AMBTC) was presented by Lema and Mitchell.It preserves the higher and lower mean of a block [12]. AMBTC has some advantages over  BTC such as  providing better image quality [13],and the coding and decoding processes are very fast [6].

Delta encoding is a simple coding technique.The key feature is that the delta encoded signal has a lower amplitude than the original signal.On the other words, delta encoding has increased the probability that each sample's value will be near zero, and decreased the probability that it will be far from zero.So it used in image compression to improve the perforamnce of entrpy coding techniques such as huffman coding[14].

It’s well known that the huffman’s algorithm is generating minimum redundancy codes compared to other algorithms. Huffman coding technique generating binary tree by calculating the probability value for each pixel in the image and sorting the pixels from the lowest probability value  to the highest probability value.Then allocates zero to the left node and one to the right node starting from the root of the tree[4].

In this paper we propose an image compression scheme based on  AMBTC by combining  delta encoding and huffman coding to produce variable  compression ratio of AMBTC.The remainder of this article is organized as follows: Section 2 introduces image compression and techniques. Section 3 discusses the proposed image compression technique. Section 4 includes the experimental results. Section 5 concludes this paper.

2.   Image compression

A digital image is a discrete two-dimensional function , f(x,y) of  picture  elements   (pixels). A digital image can be presented as an X*Y matrix where X refers to the number of image rows , and Y refers to the number of image columns [15].

There are three types of digital image:

Binary image (bi-level) where each pixel assumes one of only two discrete values: 1 or 0, where 1 is white , 0 is black [16].

Grayscale image (monochrome) which contains the values ranging from 0 and  255, where 0 is black , 255 is white and values in between are shades of grey [4].

Color image where each pixel of the image is represented by three color components, usually red, green and blue, shortly RGB. Each R, G and B is also in the range of 0 and 255 and each pixel is represented in three bytes. On the other hand a grayscale image is  represented  only  by  one  byte,  so that  the  storage  space  of  color image is three times  the size of grayscale image [4].

Image compression is the application of data compression on digital images[17].The main principle of image compression is based on  the fact that neighboring pixels are highly correlated ,where the adjacent pixels have the same color or very similar colors. This correlation is called spatial redundancy [18]. The purpose for image compression is to reduce the amount of data required for representing sampled digital images, and therefore reduce the cost for storage and transmission[3].

In the field of image compression , data redundancy plays an important role, where data with  redundancy  can  be  compressed; in contrast, data without  any  redundancy  can  not  be  compressed. So image compression techniques aims to reduce or remove the redundant data from the image [4].There  are  three  types  of  redundancies  , as follows :

Coding Redundancy: which is presented when less than optimum (the  smallest   length) code words are used to represent image data [17]. There are several lossless techniques for constructs such a code i.e. Huffman coding, Arithmetic coding[3]

Interpixel Redundancy: which is concerned with the correlations between the pixels of an image[19]. There are several lossless techniques to reduce the interpixel redundancy , such as predictive coding and run-length coding [4].

Psychovisual Redundancy: which is due to data that is ignored by the human visual system(HVS )[17]. Quantization is the famous lossy method used to reduce psychovisual redundant  from the image [4].

Figure 1 shows general model for image compression system. The first stage consists of  coding information into a 1-dimensional bit stream sequence. Then the encoded sequence is transmitted via transmission channels to decoder block , where the sequence of data is decoded. Decompression in the receiver realizes inverse operations i.e., channel decoder, and source decoder. The output image may or may not be an exact replica of original mage(lossless or lossy)[4].

 

 

 

 

 

 

Figure 1: A general compression system model.

Before handling the proposed image compression scheme let us review briefly image compression techniques.

2.1. Image Compression Techniques

In general image compression methods are classified into two classes :lossless methods such as (Run length encoding - Huffman coding - Delta coding - Dictionary methods, etc),and lossy methods such as (Quantization-BTC-AMBTC-Transform coding , etc). In this section we will focus on BTC, AMBTC, Delta encoding and Huffman coding as follows:

2.1.1. BTC algorithm

The BTC algorithm involves the following steps in the coding phase[12]:

1. The input image is divided into M*N (typically 4 * 4) of non-overlapping blocks of pixels.

2. For each block the statistical moments: mean  ,and  standard deviation  are calculated using the following equations [12]:

 

			(1)
					(1)
			(2)
					(2)

 

 

 

 

 

 

 

Where xi represents the i^th pixel value of the image block, and n is the total number of pixels in that block. The two values    and  are termed as quantizes of BTC.

3. A two-level bit plane is obtained by comparing each pixel value xi with the threshold value (). If x_i <  then the pixel is represented by ‘0’, otherwise by ‘1’. The compressed data contains the bit plane along with  and  .

In the decoding phase, an image block is reconstructed by replacing ‘1’ s in the bit plane with H and the ‘0’s with L, which are given by [12]:

 

	(3)
			(3)

 

 

 

 

 

	(4)
			(4)

 

 

 

 

Where p and q are the number of  0’s and 1’s in the compressed bit plane respectively.

2.1.2. AMBTC algorithm

The AMBTC algorithm involves the following steps in the coding phase [10]:

1. The input image is divided into M*N non-overlapping blocks (typically4 * 4 ).The average gray level of the block is calculated using the following equations [10]:

 

			(5)
	(5)

 

 

 

 

Where xi represents the ith pixels in the block.

2. The Pixels in the image block are classified into two ranges of values. The upper range is the gray levels which are greater than the block average gray level  , and the lower range is the gray levels which are smaller than the block average gray level  .

3. The mean of higher range X_H and the lower range X_L are calculated as [10]:

 

	(6)
			(6)
	(7)

 

 

 

 

 

 

 

(7)

 

Where k is the number of pixels whose gray level is greater than   .

4. A two-level bit plane is obtained by comparing each pixel value xi with the threshold value (). If x_i < then the pixel is represented by ‘0’, otherwise by ‘1’.The compressed data contains the binary block along with X_H and X_L.

After  generation of X_H , X_L , and bit plane, each block needs 32 bits (8 for X_H , 8 X_L , 16 for bit plane) to specify the bock data. Whereas the original block requires 128 bits (4*4*8).So that the compression ratio = 128/ 32 = 4.

On the other hand in the decoding phase, an image block is reconstructed by replacing the `1’ s with X_H and the ’0’ s by X_L .The coding and decoding processes of AMBTC are faster than original BTC, because square root and multiplication operations are omitted [6].

2.1.3. Delta encoding

Delta encoding is a lossless data compression method. The principle is to represent data as the difference between successive samples rather than original samples [14].Because delta encoding increases samples  probability , and  lowers the amplitude of signal. It’s a common strategy for compression signals particularly, when  followed by huffman or run-length encoding [14].Table 1 shows an example of image compression based on delta encoding. The first value in the encoded stream is the same as the first pixel value in the original stream. Thereafter, each value in the encoded stream represents the difference between the current and last pixel value in the original stream.

Table 1: Example of Delta Encoding.

It is noted in the previous example, the number of  elements in the encoded stream are equal to the number of elements in  the original stream .So delta encoding doesn’t compress data alone but  used as a pre-compression step to reduce interpixel redundancy from the image.

2.1.4. Huffman coding

Huffman coding is a commonly used method for lossless data compression. Since its development, in 1952, by D. Huffman[18].Huffman coding is an entropy encoding algorithm used for lossless data compression.The principle is to use a lower number of bits to encode the data that occur more frequently. Codes are stored in a code table which may be constructed for each image or a set of images. In all cases the code table and encoded data must be transmitted to enable decoding [20].

Huffman method is based on the three conditions. The conditions are [21] :

1. The codes corresponding to the higher probability symbols could not be longer than the code words associated with the lower probability symbols.

2. The two lowest probability symbols had to have code words of the same length.

3. The two lowest probability symbols have codes that are identical except for the last bit.

Example: Supposing an image with five pixels [P1, P2, P3, P4, P5], with probabilities of occurrence P(P1) = 0.15, P(P2) = 0.04, P(P3) = 0.26, P(P4) = 0.05, and P(P5) = 0.50. Huffman encoder uses the following steps for generating variable size codes:

1.Calculating the entropy rate as follows:

 

(8)

 

 

= 1 . 817684 bits

 

 

(8)

 

  

 

Where P(Pi)  is the occurrence probability of each Pi.

2.Sorting the image pixels in descending order of their probabilities as shown in the   following table:

Table 2

3. Building the binary tree as follows:

 

 

 

 

 

 

 

 

 

Fig 2: Huffman code for the five pixel.

4. Generating a huffman code dictionary for each pixel , as shown in table 3.

 

 

Table 3.

5. Replacing each pixel in the original image with the respective code in the dictionary.Where the pixels more frequent are coded with smaller number of bits.

6.Calculating the average length of the number of bits used to represent each pixel , which  is defined as [4]:

 

	(9)

 

 

 

 

Where l(rk) is the length of the codeword used in pixel r_k , and Pr(r_k) is the occurrence probability of each r_k . 

 Lavg = (0.5)*(1) +(0.26)*(2) + (0.15)*(3) + (0.05)*(4) + (0.04)*(4) = 1.83 bits/pixel.But the entropy of the source pixels is 1 . 817684 bits/pixel. So the resulting huffman code efficiency is 1 . 817684 / 1.83 = 0.9933.

The major advantages of huffman coding are easy to implement, produce a lossless compression of images [20].So it is widely used in many applications suc as JPEG , DEFLATE [20],and compression softwars like pkZIP, lha, gz, zoo and arj [22].On contrast , the main drawbacks of huffman coding are a relatively slow process , its efficacy depends on the accuracy of the statistical model used and type of image ,decoding is difficult due to different code lengths , causing overhead due to code table must be transmitted at the beginning of the compressed file [20],using an integral number of bits in each code, and not producing very good compression ratios [23].

As illustared previously, AMBTC produce a fixed length binary representation of each block  according to the block size of the image , for example (compression ratio = 4;block size = 4*4). On the contrary huffman coding produce a variable length binary representation , based on the statistical nature of the image.So our scheme proposed to improve the performance of AMBTC to produce a variable length compression by applying the combination of delta encoding , and huffman coding. In the following section we will introduce the propsed scheme.

3.   Proposed scheme

In a color image high correlation exists among R, G, and B planes, so a high compression ratio can be achieved by exploiting the psychovisual , spatial correlations , and the coding redundancies. In the proposed method the psychovisual redundancy is reduced by converting RGB to a less correlated color space such as YCbCr. On the other hand the spatial redundancy is reduced by block quantization using AMBTC method , and delta encoding. Finally the coding redundancy is reduced by huffman coding.

The procedures of proposed scheme  for color image compression are shown in figure 3. In the pre-processing step, color image is transformed from RGB color space into another less correlation color space such as  luminance/chrominance to generate Y,Cb,Cr components.  Since the human eye is more sensitive to  luminance changes than chrominance variance, the chrominance components  are downsampled  to reduce the size of the original image. In step 1, the three components (Y,Cb,Cr ) go through AMBTC encoder independently to reduce the spacial redundancy. In the second step, delta encoding is used to incraese the apperance of each piexl in the image.Finally in step 3, huffman coding is applied to achieve compression.

 

 

 

 

 

 

Figure 3. Compression steps using the proposed method.

The details of the encoding and decoding phases of the proposed algorithm will be illstated in the next section.

3.1.  Encoding Phase

1.Convert the color image from RGB color space to YC_bC_r color space for better coding efficiency. Since the human eye is more sensitive to small changes in luminance but not in chrominance, so the chrominance part can lose much data, without affect on image quality.

2. Divide the converted image is into three matrices: Y,C_b,C_r.

3. Downsample the chroma matrices (C_b, C_r) at a ratio of 2:1 both horizontally and vertically (is called 2h2v).

4. Compress each matrix of the three matrices(Y,C_b,C_r),  independently , as follows :

4.1. Divide the given image into a set of non overlapping blocks, the size of a block could be 4*4 or 8*8.

4.2 .Apply AMBTC principles for each block , as follows:

4.2.1. Calculate the average gray level of the block () .

4.2.2. Compute the lower mean X_L , and higher mean X_H of the block.

4.2.3. Use the binary block to represent  the pixels, where “1” is used to represent a pixel whose gray level is grater than or equal to () , and ”0” is used to represent a pixel whose gray level is less than () .

4.3.Construct the three vectors (BitmapVec, HighVec, LowVec).Where BitmapVec involves all binary values in all blocks of the image.High includes all X_H values in the image, and LowVec contais  all X_L values in the image

4.4. Convert the BitmapVec to DecVec (from binary representation to deciaml system ).

4.5. Calculate the difference between HighVec, LowVec , to get the difference vector (DiffVec). Which contains small close values.

4.6.Apply delta encoding for each vector of the three vectors (DecVec , LowVec, DiffVec), to get the delta vectores(DeltaDec , DeltaLow , DeltaVec).

4.7. Apply huffman coding  for each vector of  the three vectors(DeltaDec , DeltaLow , DeltaDiff) , independently to generate compressed files (Comp1, Comp2, Comp3).

4.8. Combine the encoded data and side information for compreseed files into a single component.

5. Finally,combine the compressed components (C_Y,C_Cb,C_Cr) , which are generated from previous steps to generate the compressed file , before storage or transmission process.

3.2.  Decoding Phase

The procedures of the decoding phase is reverse to that of the encoding phase as shown in fig4. The details of the decoder algorithm are described as follows:

1. Apply the following steps for each component of the compressed file (C_Y,C_Cb,C_Cr) independently:

1.1. Apply  huffman decoding into compressed files(Comp1-Comp2-Comp3) , to restore original vectors (DeltaDec, DeltaLow, DeltaDiff).

1.2. Apply delta decoding into vectors(DeltaDec, DeltaLow, DeltaDiff), to get the three vectors (DecVec, LowVec, DiffVec).

1.3. Constrct  the high values vector (HighVec) .

1.4. Convert the decimal numbers vector (DecVec) to  binary matrix (BitmapMat).

1.5. Apply AMBTC decoding principles into BitmapMat .Where it is divided into M*N non-overlapping blocks . For each each block , 0’s  is replaced by the corresponding number in a LowVec , and 1’s  is replaced by the corresponding number in a HighVec .

2. Upsabmle the chroma matrices (       ,        ) , both horizontally and vertically , to get the reconstructed chroma matrices with original image dimensions(R_Cb , R_Cr).

3. Combine the three reconstructed matrices (R_Y , R_Cb , R_Cr)  into a single matrix.

4. Convert the reconstructed  matrix from YC_bC_r  color space to RGB color space.

 

 

 

 

 

Figure 4. Decompression steps using the proposed method

4.   Experimental Results

To evaluate the performance of the proposed image compression scheme, we took seven standard color images of size 512*512 (24 bit per pixel) namely “Lena” , ”Peppers” , “Mandrill” , “Girl” , ”Airplane” , ”House“ , and “Sailboat “ which are shown in Fig 5. The Measurement criteria required to assess the performance of proposed method are: the compression ratio (CR) given by Eq. (10) , the root mean square error (RMSE) given by Eq. (11) , and the peak signal to noise ratio (PSNR) given by Eq. (12) [13 , 24 , 25].

 

 

 

 

 

 

Where  is the original image , is the reconstructed image, and X*Y is the dimensions of the images.                                                          

 

(12)

 

           

                                                                                   

Where typical PSNR values range between 20 and 40 dB.

A smaller value of the RMSE means that the reconstructed image has less distortion. In contrast a higher value of PSNR means lesser error in the reconstructed image. So a compression method having lower RMSE and correspondingly high PSNR values could be recognized as a better scheme.

 

 

 

 

 

 

 

 

 

Figure 5: Standard color images used for experiment (*).

Experimental  results using the proposed scheme on the taken standard color images are compared with the original BTC and AMBTC .The obtained results for block size of 4*4 are given in table 4 , and presented in fig 6 and fig 7.While the results for block size of 8*8 are given in table 5 , and presented in fig 8 and fig 9.

Table 4:Exprimental results of BTC,AMBTC , proposed method at block size 4*4.

 

 

 

 

 

 

 

Figure 6.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 7.

 

Table 5:Exprimental results of BTC,AMBTC , proposed method at block size 8*8.

 

 

 

 

 

 

 

 

 

 

Figure 8.

 

 

 

 

 

 

 

 

 

 

 

Figure 9.

 

 

The above tables assure that the image compression using AMBTC provides better image quality than image compression using BTC at the same compression ratio. While , the proposed scheme achieves higher compression ratio than that of AMBTC scheme with small degradation of image quality. Since the human eye is more sensitive to small changes in luminance but not in chrominance, so the chrominance part can lose much data, without introducing noticeable degradation in the reconstructed images.

5.   Conclusion

A modified method for improving the conventional AMBTC has been proposed. The proposed method uses delta encoding to increase the appearance degree of image pixels , and huffman coding to generate variable length compression. The performance of the proposed method has been compared with conventional  BTC and  AMBTC  and it is found that it achieves a higher compression ratio than that of both BTC , and AMBTC  with low distortion of decoded images. Experimental results, by applying our scheme on standard seven color images, show that an average compression ratio for block size 4*4 is 12.641 while for block size 8*8 is 19.031.On the other hand an average PSNR value for block size 4*4 is 26.845 , and for block size 8*8 is 25.361. Our compression scheme may be useful for low cost handheld devices with low computational power to handle images.

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