# Block based image compression technique using rank reduction and wavelet difference reduction

Anastasia Bolotnikova, Pejman Rasti, Andres Traumann, Iiris Lüsi, Morteza Daneshmand, Fatemeh Noroozi, Kadri Samuel, Suman Sarkar, Gholamreza Anbarjafari

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Anastasia Bolotnikova, Pejman Rasti, Andres Traumann, Iiris Lüsi, Morteza Daneshmand, et al. Block based image compression technique using rank reduction and wavelet difference reduction. Seventh International Conference on Graphic and Image Processing Seventh International Conference on Graphic and Image Processing (ICGIP 2015), 2015, Singapour, Singapore. 10.1117/12.2227994. hal-02527962

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# Block Based Image Compression Technique Using Rank Reduction and Wavelet Difference Reduction

Anastasia Bolotnikova, Pejman Rasti, Andres Traumann, Iiris Lusi, Morteza Daneshmand, Fatemeh Noroozi, Kadri Samuel, Suman Sarkar, and Gholamreza Anbarjafari,
ICV Research Group, Institute of Technology, University of Tartu, Nooruse 1, Tartu 50411, Estonia

## ABSTRACT

In this paper a new block based lossy image compression technique which is using rank reduction of the image and wavelet difference reduction (WDR) technique, is proposed. Rank reduction is obtained by applying singular value decomposition (SVD). The input image is divided into blocks of equal sizes after which quantization by SVD is carried out on each block followed by WDR technique. Reconstruction is carried out by decompressing each blocks bit streams and then merging all of them to obtain the decompressed image. The visual and quantitative experimental results of the proposed image compression technique are shown and also compared with those of the WDR technique and JPEG2000. From the results of the comparison, the proposed image compression technique outperforms the WDR and JPEG2000 techniques.

**Keywords:** Lossy Image Compression, Wavelet Difference Reduction, Block Processing, Singular Value Decomposition.

## 1. INTRODUCTION

With the growing demand for multimedia applications especially high definition images, efficient storage and transmission of images have been issues of great concern [1]–[4]. Image processing deals with the reduction of the amount of bits used to represent an image.

There are two categories of image compression techniques namely lossless and lossy image compression techniques [5], [6]. In lossless image compression, the original image can be perfectly recovered from the compressed image while in lossy compression the original image cannot be perfectly recovered from the compressed image because some information is lost as a result of compression. Lossless compression is used in applications with high requirements such as medical imaging. Lossy compression techniques are very popular because they offer higher compression ratio. The objective of image compression is to achieve as much compression as possible with little loss of information [7], [8].

In this research work, a new lossy compression technique which employs Singular value decomposition (SVD) and wavelet difference reduction (WDR) is presented. SVD is a lossy image compression technique which can be regarded as a quantization process where it reduces the physical visual redundancies of the image [9], [10]. Wavelet transforms are used frequently in many image processing applications [11]–[13]. One of these applications is in image compression. WDR is one of the state-of-the-art techniques in image compression which uses wavelet transform. It is a lossy image compression technique which achieves compression by first taking the wavelet transform of the input image and then applying the difference reduction method on the transform values [14]–[17].

In the proposed compression technique, the input image is firstly divided into the equal sized blocks and each block is quantized using the SVD technique and then WDR compression is carried out on the output image from the SVD compression for further compression. The proposed technique has been tested on several well-known images such as, Lena, Peppers, Boat, and Airfield. The results of this technique have been compared with those of JPEG2000 and WDR techniques. The quantitative experimental results based on PSNR show that the proposed technique overcomes the aforementioned techniques. The SVD and WDR image compression techniques are discussed in the next section.
## 2. RELATED WORK

### 2.1 Singular Value Decomposition

SVD has been used in many image processing application [18-20]. Due to its nature, SVD has been always an important mathematical tool for image compression [21-23]. An image is represented by a matrix in digital form. SVD is a technique used to decompose a given matrix A, into three matrices U, Σ, and V. U and V are orthogonal while Σ is a diagonal matrix containing the singular values A [24-26]. Σ can be approximated by ignoring small singular values. This approximation will result into a lower rank singular matrix. In order to reconstruct the input matrix, U and V matrices also need to be updated. Eqn. (1) shows the reconstruction of the input matrix with approximated Σ.

$$A_{m \times n} = U_{m \times p} \bar{\Sigma}_{p \times q} (V_{n \times q})^T \quad (1)$$

$$\text{where } U_{m \times m} = [U_{m \times p} \quad U_{m \times (m-p)}] , \quad V_{n \times n} = [V_{n \times q} \quad V_{n \times (n-q)}]$$

The SVD compression is a lossy compression technique because singular values of the image are ignored in the procedure cannot be recovered, which results in losses [21], [26].

### 2.2 Wavelet Difference Reduction

Wavelet transformations have been employed in many of applications of computer vision in last couple of decays [27-29]. The WDR is a compression technique based on the difference reduction method. The wavelet transform of the input image is first taken; bit plane encoding is then applied to the transform values. The bit plane encoding procedure starts with the initialization stage, where a threshold $T_o$ is chosen such that $T_o$ is greater than all the transform values and at least one of the transform values has a magnitude of $T_o/2$. The next stage is the initialization stage where the threshold $T = T_{k-1}$ is updated to $T = T_k$, where $T_k = T_{k-1/2}$. New significant transform values (w(i)) which are satisfying $T \le |w(i)| \le 2T$ are then identified at the significant pass stage. The transform values of these significant transform values are then encoded using the difference reduction method. At the significant pass stage, already quantized values ($w_Q$) which satisfy $|w_Q| \ge 2T$ are then refined in order to reduce error [30-32].

## 3. THE PROPOSED TECHNIQUE

The proposed image compression technique is a lossy compression technique. It begins by dividing the input image into equal sized blocks. In this work block sizes 16x16, 32x32, 64x64, and 128x128 are investigated for input images of size 256x256. SVD compression is then carried out on each block. As it was mentioned in the previous section, the SVD based compression is lossy due to the nature of the process. However the qualitative loss is not psychovisually noticeable up to some point. After SVD compression, WDR compression is carried out to further compress the image. The compression ratio is obtained by multiplication of the compression ratio obtained by using SVD with that of WDR. The compression ratio of WDR in the proposed block based compression technique is calculated by dividing the total number of bits required for representing the original image by the total number of bits stream which obtained by WDR. Decompression is carried out by taking the inverse WDR of the bit streams in order to reconstruct each block and then all blocks are combined together to form the decompressed image.

The block diagram of the proposed image compression technique is shown in Figure 1. The experimental qualitative and quantitative results are represented and discussed in the next section.
<figure><img src="image_2.png" /><figcaption>Figure 1: The block diagram of the proposed blocked based lossy image compression technique.</figcaption></figure>

## 4. EXPERIMENTAL RESULTS AND DISCUSSION

The proposed lossy image compression, as it was mentioned in the Introduction, was tested on Airfield, Boats, Lena and Peppers images. All the input images are 256x256, 8-bit grey scale. Table 1 and 2 are showing the quantitative comparison between the proposed techniques and JPEG2000 and WDR by use of PSNR for compression ratio of 60:1 and 80:1 respectively.

<table><caption>Table 1: PSNR values in dB for 60:1 compression</caption><thead><tr><td>Image/Technique</td><td>Proposed Technique</td><td>JPEG2000</td><td>WDR</td></tr></thead><tbody><tr><td>Airfield</td><td>42.23</td><td>23.76</td><td>23.71</td></tr><tr><td>Boats</td><td>43.45</td><td>28.26</td><td>28.14</td></tr><tr><td>Lena</td><td>44.75</td><td>31.18</td><td>31.71</td></tr><tr><td>Peppers</td><td>44.50</td><td>30.96</td><td>30.29</td></tr></tbody></table>

<table><caption>Table 2: PSNR values in dB for 80:1 compression</caption><thead><tr><td>Image/ Technique</td><td>Proposed Technique</td><td>JPEG2000</td><td>WDR</td></tr></thead><tbody><tr><td>Airfield</td><td>41.45</td><td>22.64</td><td>22.71</td></tr><tr><td>Boats</td><td>43.00</td><td>26.76</td><td>26.82</td></tr><tr><td>Lena</td><td>44.24</td><td>29.62</td><td>29.71</td></tr><tr><td>Peppers</td><td>44.22</td><td>29.54</td><td>28.93</td></tr></tbody></table>

As the PSNR values shows in Table 1 and 2 the performance of the proposed technique overcomes the JPEG2000 and WDR based image compression techniques. Figure 2 is showing a portion of the magnified Lena image being compressed with compression ratio of 60:1 by using JPEG2000, WDR, and the proposed technique with block size of 64x64.
<figure><img src="image_3.png" /><figcaption>Figure 2: Zoomed (a) original Lena image, and compressed images by using (b) JPEG2000, (c) WDR, and (d) the proposed image compression technique using 64x64 block sizes at 60:1 compression ratio.</figcaption></figure>

Figure 3 is illustrating PSNR versus the compression ratio for the aforementioned images comparing the JPEG2000 and WDR techniques and the proposed block based lossy image compression technique with 128x128 block sizes. The graphs are showing the superiority of the proposed image compression over the other state-of-the-art techniques.

<figure><img src="image_4.png" /><figcaption>Figure 3: The PSNR (in dB) versus compression ratio for (a) Lena, (b) Boat, (c) Peppers, and (d) airfield images in which the block size of the proposed technique is 128x128.</figcaption></figure>
## 5. CONCLUSION

This paper was proposing a new block based lossy image compression technique which uses rank reduction and WDR technique. The input image is divided into blocks of equal sizes. Then each block was quantized by reducing its rank using SVD and then compressed by applying WDR technique. Reconstruction was carried out by decompressing each blocks bit streams and then merging all of them to obtain the decompressed image. The proposed image compression technique was compared with WDR technique and JPEG2000. The excremental results were showing that the proposed image compression technique outperforms the WDR and JPEG2000 techniques.

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