Energy Efficient DWT Algorithm for Image ...

International Conference on Electrical, Electronics, Computers, Communication, Mechanical and Computing (EECCMC)

Energy Efficient DWT Algorithm for Image Transmission in Real Time Applications Khamees Khalaf Hasan1

Lujain Sabah Abdullah2

Electrical Engineering Department Engineering College, University of Tikrit Tikrit, Iraq [email protected]

Electrical Engineering Department Engineering College, University of Tikrit Tikrit, Iraq [email protected]

Abdumuttalib A. Hussen3 Electrical Engineering Department Engineering College, University of Tikrit, Iraq Tikrit, Iraq [email protected] Abstract— The main objective of this article is to deal with the energy and bandwidth communication problems of image data transmission; by creating 2D-DWT image codec architecture which can save both the computational and communication energy. The second objective is to recommend a comprehensive mathematical modeling of DWT energy consumption by computing the computational and data access loads to approximate the energy efficiency of the proposed techniques. Simulations were performed using grayscale images that authenticate the proposed design and attain a computational performance appropriate for numerous real-time applications. Keywords— FDWT; IDWT; 5/3 filter; LS; computational loads

I. INTRODUCTION Future wireless applications point towards a growing demand for multimedia content such as video and audio streams, still images, and scalar sensor data. However, mobile multimedia communication has several bottlenecks including low power constraints and bandwidth requirements [1], [2]. As wireless radios built for mobile multimedia communication will be powered primarily by battery, the energy consumed must be minimal. In addition, a large amount of information is needed to represent multimedia data that needs to be transmitted wirelessly. Therefore, both the communication energy (energy consumed in wirelessly transmitting information) and computation energy (energy consumed in processing information to be transmitted) can be very high [3]. The need and ability for fast, low-energy, wireless multimedia communication, is fueling the need for novel techniques for data processing and communication. However, multimedia data can be compressed in a manner, leading to compact representations of the multimedia data than is available with traditional data compression. The compression technique allows effective use of bandwidth so

the data can be transmitted at lower bit rate. Therefore, source coding (compression) plays an important role in communicating multimedia information [4]. Recently, a substantial portion of wireless data is comprised of images. For the real-time image transmitting via low bandwidth wireless channel, a more effective method of communicating can be acquire based on image compression [5]. The function of the image compression is to decrease the amount of bits required to depict an image by eliminating the unnecessary information in the raw image data using transform coding. Therefore, image compression will lead to decrease in memory required for computational analysis as well as communication costs [6]. Presently, transform-based image compression methods garner lots of attention. The discrete wavelet transform DWT has become a very versatile signal processing tool in signal and image processing. The DWT performs a multiresolution signal analysis into multiple resolution sub bands decomposition which has adjustable locality in both the space (time) and frequency domains [6]. The full-frame nature of DWT de-correlates the image over a larger scale and eliminates blocking artifacts at high compression ratios which is a major shortcoming of blockbased transformation standards [7]. 2D-DWT is usually applied by convolution method, which is the initial method of finite impulse response (FIR) filter bank structures [7]. A new method in use is the lifting scheme (LS), which is based on a spatial construction of the wavelet. The fundamental principle governing the lifting based scheme is to decompose the finite impulse response (FIR) filters in wavelet transform into a finite sequence of simple filtering steps. It is very flexible scheme with simple adders and shifters substituting

978-1-5386-4304-4/18/$31.00 ©2018 IEEE.

Page 1 of 433

International Conference on Electrical, Electronics, Computers, Communication, Mechanical and Computing (EECCMC) multipliers [8]. Therefore, it mostly decreases the amount of multiplication and accumulation entailed in computing a DWT with a convolution technique. Basically, the beginning of the prepared LS is involves breaking down the input signal into odd and even indexed components. This is followed by the modification of the indexed components using alternating prediction and updating steps. The wavelet based image compression performance is dependent to a great degree on the choice of the wavelet filter type in a platform with restricted resources and power inhibited applications [9] [10]. The 9/7 and the 5/3 lifting scheme DWT filters are utilized as the JPEG2000 standard filters for lossy and lossless image compression respectively. The 5/3 filter comprise one prediction and one updating processes while the 9/7 filter has two predictions This paper is outlined as follows. Section two explains the wavelet based image compression system. The design methodologies of the 2D-DWT implemented algorithms are elucidated in section three. Section four discusses the performance results while conclusions are made in section five. II. 2D- DWT POWER CONSUMPTION MODELING ANALYSIS The estimating power dissipation model for filters and transform type DSP algorithms of VLSI Chips in earlier study [16] allocates the evaluation of power consumption for different architectures in the realization of the wavelet transform algorithms. The major sources of the total power dissipated by a DSP system are approximately: (1) The DWT algorithm processing power requires data storage memory and transfer operations

and

arithmetic operations (additions and shifting)

, and

this allows us to follow the mentioned analysis model [16]. A complete power consumption mathematical model was designed and utilized for analytical equations with factors derived from both the 2D-DWT algorithm and the implemented hardware architecture. The 2D-DWT Lifting scheme implementation of 5/3 filter on an image is shown in Figure 1.

filter as a direct function of the consumed coefficients units. The power consumption of the low and high-pass decomposition was modeled by tallying the amount of arithmetic operations and designating this as the computational load [15]. The whole computational load can be described bases on the assumption that the input image size is of M×N pixels and the 2-D DWT is iteratively applied L times. Based on the notion that the image size declines by a factor of 4 in each transform level, the dimension of each subimage signal at level J of decomposition is 1/4J times that of the original image. The decomposition computational load is calculated as a weighted sum of the number of L times that specific basic low and high-pass operations are executed on the input image pixel samples. The (5, 3) filter needs two adders and a shifter in every basic PE [17]. Therefore, for 5/3 filter implementation; units of computational load are requisite in a unit pixel of the low-pass decomposition and units for the high-passes. Hence, the whole computational energy consumption load for the forward 2-DDWT process, which is iteratively, applied L times on the M×N pixels of input image are approximately specified by the equation below:

(2) Given that in the multilevel decomposition, the amount of processing data in each level is just a one-fourth of the preceding level. For mathematics, a geometric series exhibits an invariable ratio between consecutive terms. To obtain this equation, a general geometric series is first derived as:

(3) Where a is a scale factor, equal to the start term value of the series, and each term after the first is deduced by multiplying the preceding term by the constant mutual ratio r between consecutive terms. Therefore, Equation 2 can be written as: (4)

HH

A more straightforward formula for this sum can be obtained. If r ≠ 1, the sum can be restructured to acquire a more convenient formula for a geometric series that calculates the sum of n terms by multiplying both sides of the above equation by 1 – r:

HL X(n) input LH

LL

PREDICT

UPDATE

(5)

Fig.1. 2D-DWT Lifting scheme block diagram of 5/3 filter

Nevertheless, we compute the wavelet algorithm power adapting the 2D-DWT using an integer 5-tap/3-tap Page 2 of 433

(6) (7)

International Conference on Electrical, Electronics, Computers, Communication, Mechanical and Computing (EECCMC) (8) When the first term value of the series a = 1, this simplifies to: (9) To conclude the derivation in the series as a particular case of the sum formula for a geometric series, each consecutive term is a quarter of the preceding term with common ratio r =

.

(10) (11) Thus, the entire amount of arithmetic operations for processing data computational load is calculated in simpler formula as follows: (12) Apart from other different arithmetic operations, the transform step entails a huge volume of memory accesses. Given that the energy dissipated in external and internal data transfers can be considerable, the data-access load was approximated by calculating the sum total of memory accesses for the period of the wavelet transform. In the case regarding memory accesses, each pixel is read and written two times at a transform level, with the whole even and odd pixels. Therefore, under a similar condition as the above estimation method, the total data-access load is provided by the number of read and writes operations:

(13)

(14) The total computation energy utilized by hardware architecture for the LS 5/3 2D-DWT is computed as a weighted sum of the computational load and data-access load:

and to present the compression results. Several common classes of images usually have different statistical properties compared to photographic images. The wavelet-based image compression performance is largely based on the selection of the wavelet type [8]. However, this option depends on the image contents in order to assess the 2DDWT performance [19] [20]. Majority of wavelet-based image coders utilizes standard wavelets that are recognized for being effective on photographic images. Nonetheless, these wavelets do not achieve the same effectiveness for other common image classes, for example medical images or scanned documents [18]. III.ENERGY EFFICIENT ALGORITHM FOR IMAGE TRANSMISSION To achieve wavelet computation with high efficiency, it is imperative to remove redundant computations. A critical evaluation of the forward and reverse transforms demonstrates that approximately half the operations either results in lost data or null operations (as in multiplication by zero) [21]. To aid the recognition of these redundant operations, a modified 5/3 2-D DWT computation was developed in this study. The proposed technique utilized the numerical distribution of the high-pass coefficients to perform intricate removal of redundant samples during the image compression process. The transform itself does not compress the image, but provides a structure in the form of coefficients that is much more susceptible to compression in contrast to the original data [21]. The low frequency set is an estimate of the input signal at a coarser resolution, whereas the high-frequency set comprises information that will be employed afterward during the reconstruction phase. The approximation area LL1 consists of details about the general features of the transformed image, and the other three detail bands can be distributing as of less significant information segments [22]. The filtering step can compress the important coefficients in the LL1 sub-band (i.e., the image presents a low pass spectrum). As a result, the high-pass coefficients values are commonly denoted by small integer values. Indeed, most of the high-pass coefficients are very minute values and close to zero as depicted in Figure 2.

Approximation LL1

Verrtical Detail HL1

Numerical distribution of low-pass / highpass filters after

(15)

wavelet transform

2D – DWT

Horizontal Detail LH1

Diagonal Detail HH1

Original Lena image

(16) Where, S,A,RMEM, and WMEM, represent the hardware energy consumption required for shift, add, read, and write basic operations in a unit pixel, respectively. The complexity involved in investigating an image compression system is to reach a conclusion is to choose test images to be used for assessments. Generally, image content is a significant factor that controls the efficiency of filters

Fig 2. Numerical distribution of low-pass/high-pass Lena Image filters coefficients after 2D DWT through level 1

As a result of the numerical distribution of the high-pass filter small valued coefficients, it can be assumed that the high-pass filter coefficients are zeros and acquire minimum image quality loss. Thus, the fundamental point of the proposed technique is to evade the calculation of Page 3 of 433 high-pass coefficients. This method endeavors to conserve

International Conference on Electrical, Electronics, Computers, Communication, Mechanical and Computing (EECCMC) energy by surpassing the least important sub-bands. Hence, the proposal decreases the amount of arithmetic operations and memory accesses. The proposed method is referred to as Energy Efficient Hopping High Pass Wavelet Sub-bands (EEHHPWS). This proposed approach is a straightforward and uncomplicated technique, which is implemented by creating precise modifications on the wavelet transform as shown in Figure 3. Low pass filter

Origional image

High pass filter

Horizontal Row wise Discrete Wavelet Transform Computed data

h(n)

g(n) Exceeded Data

DWT. The decomposition scheme occurs level by level and explained as follows: For 1D-DWT, the processing computation unit chooses data (pixels) form input image in first-level decomposition. The horizontal row transform module processors decomposes to the two sub-bands Li Band and Hi Band which is passed over, and saves Li band to the memory module. Following the completion of the1DDWT, the control unit collects data from memory module. Thereafter, the Li band is transmitted to the vertical column module transform to execute the 2D-DWT. The vertical column transform module divides the Li band into the two sub-bands: LLi and LHi which is not calculated, and stores the LLi band entirely to the memory module for next decomposition phase. This process was replicated till the required final level decomposition was complete as shown in Figure 5. Image width (W)

L1

L1

H1

H1

1-D FDWT vertical filtering stage

LL1

H1

LH1

(H/2)

Vertical Column wise Discrete Wavelet Transform

1-D FDWT horizontal filtering stage

(W/2) (H/2)

Hi

(W/2)

(W/2) Image height (H)

Image height (H)

Li

(W/2)

LEVEL 1 Decomposition

h(n)

g(n)

residual details

LH1

Fig. 3. The DWT data flow steps with (Hi) sub-bands removal procedures of the proposed EEHHPWS

This approach has two major benefits. Firstly, as the high pass filter coefficients requires no calculation, EEHHPWS aids the decrease of computation energy dissipated throughout the wavelet image compression procedure by reducing the amount of implemented computation operations necessary to compress an image. Secondly, given the knowledge of the estimation technique by both the encoder and decoder, no detail information is needed to be transmitted across the wireless channel using the technique, hence reducing the necessary communication energy. The modified architecture in Figure 4 shows a one level decomposition but is reconstructed for the multi-levels decomposition to suit the objectives of this study.

H1

L2

H2

LH1

H1

w/4 w/4 1-D FDWT vertical filtering stage

w/2

LL2 LH2

H2

H1

LH1

LEVEL 2 Decomposition

Fig.5. Descriptive for two decomposition levels representation of the EEHHPWSA technique

IV.

ENERGY EFFICIENCYN OF EEHHPWS TECHNIQUE Adapting the modified 5/3 2-D DWT technique, the computation load during the horizontal 1-D wavelet transforms is similar to the traditional 5/3 transform. The total computational energy involved in horizontal decomposition can be expressed as follows:

(17) The volume of operations in the orthogonal (vertical) direction decomposition is similar to that in the horizontal operations, as indicated below:

Hi

Pixels LHi

Hardware

Li LLi

Fig. 4. EEHHPWS specific wavelet modifications

(18) First of all, the decomposition was applied in the row (horizontal) direction. Given that all even placed image pixels are decomposed into the low-pass coefficients and odd-positioned image pixels are decomposed into the highpass coefficients. Hence, only evenly placed low-passed pixels are involved in the row transform in the horizontal direction, and then used as input details in subsequent column transform in the vertical direction. Nonetheless, also during the column transform the odd-positioned highpassed pixels are surpassed, given that these pixels denote

Using the EEEHPWS 5/3 2-D DWT technique, the approach was developed through efforts to conserve energy by evading the computation of high-pass coefficients. The design of this approach offers different modifications such as the 1D-DWT, 2D-DWT and multi-level decompositionPage of 4 of 433

(H/2)

details

LL1

w/2

(H/2)

HHi Diagonal

1-D FDWT horizontal filtering stage

(H/2)

Horizontal residual details

HLi Vertical residual

w/4 w/4

(W/2)

(H/2)

Approximation of the input

(W/2)

(H/2)

LHi

g(n)

(H/2)

LLi

h(n)

International Conference on Electrical, Electronics, Computers, Communication, Mechanical and Computing (EECCMC) the high-pass sub-band (HH) and (HL) that are not calculated, and only even-columned low-passed pixels are transformed leading to less computed operation and thus saving computational load. This is attributable to the fact that the benefit of skipping high-pass coefficients Hi is more considerable at lower transform levels. This prevents the creation of a diagonal HH sub band and vertical HL sub band. The computation load saves 25% of operation duration in contrast to the traditional 5/3DWT, and is expressed as follows:

(19) This study assumes that the EEHHPWSA technique is utilized for the first F transform level from the L total transform levels. To evaluate the energy efficiency of the proposed technique, the computational and data access loads were measured by combining the computation of the second and subsequent computation levels of the first level F. Consequently, for the proposed approach, the overall computational load and data access load is represented by a general equation of computation energy shown below:

(20)

(21)

(22)

(23) The total Computation Energy Saving (CES) was derived from Equations (16) for the traditional LS 5/3 2D-DWT computation energy and (23), using the modified EEEHPWS 5/3 2-D DWT, as shown below:

(24)

(25) The impact of different compression levels on computation energy savings CES using the adapted EEEHPWS as normalized to the traditional LS 5/3 2DDWT are presented in Table 1. Table 1 Compression levels effects on computational energy saving Compression Computational Energy Saving CES CES levels (%) L=1 25.00 1 4

MN ( 8 A  4 S  2WMEM  2 R MEM )

L=2

5 MN (8 A  4 S  2WMEM  2 R MEM 16

)

31.25

L=3

21 MN (8 A  4 S  2WMEM  2 R MEM ) 64

32.81

L=4

85 MN ( 8 A  4 S  2WMEM  2 R MEM ) 256

33.20

L=5

341 MN ( 8 A  4 S  2WMEM  2 R MEM ) 1024

33.30

L=6

1356 MN ( 8 A  4 S  2WMEM  2 R MEM ) 4096

33.32

L=7

5461 MN ( 8 A  4 S  2WMEM  2 R MEM ) 16384

33.33

It is deduced that most of the image energy is found in LL sub-band. From Table 1, it is obvious that most details (75%) will not be transmitted across the network. The energy concentration in the image by consecutive decomposition levels will reduce the quantity of information being transferred to the target. For a 1-level 2D DWT, the proposed approach ensures transmission for just 25% of the image information (LL1), while for a 2level 2-D DWT; only 6.25% of the original visual information (LL2) is assured of complete transmission. The computed amount is divided by 4 at every decomposition level. This is a major goal to be realized, given that the energy consumption is relative to the information quantity being transmitted. Thus, decreasing the amount of transmitted data will reduce communication energy and hence, expand the general wireless network lifetime. The computation energy savings differs with decomposition levels and become almost stable from the third decomposition level. This is because the benefit of the image transmission system is more important at lower transform levels and then image sizes decrease by 1/4 at every level. Consequently, added decomposition is ineffective and will dissipate energy without extracting extra energy saving. In order to understand the compression levels effects on image quality, a visual comparative analysis of the five levels was performed. In this study, the Lena image of 512×512 sample size version was utilized while the computation energy was calculated under different stages of reduction as shown in Figure 6.

Page 5 of 433

International Conference on Electrical, Electronics, Computers, Communication, Mechanical and Computing (EECCMC) Image width (W) Image height (H)

L=1, PSNR=34.1025dB

L=2, PSNR=25.1478dB L1

H1

L=3, PSNR=19.3839 dB

L=4, PSNR=15.8845 dB

L=5, PSNR=12.4523dB

Fig.6. Comparison of image quality after EEEHPWS technique using the Lena 512×512 version

Critical observations show that the EEEHPWS technique provides considerable energy savings at negligible loss in image quality. The PSNRs of the images are 34.1025 dB and 25.1478 dB at level 1 and level 2, in that order. Although, the energy saving between the two approaches is significant (25%), it is observed there is almost no recognizable disparity in the quality of the two images, given that the image quality degradation is within only 9 dB. The savings in communication energy increases simultaneously with the rise in volume of removed levels. Nonetheless, the quality of the image degrades concurrently with the transform process, thereby demonstrating an exchange between communication energy and quality of image obtained. This study evaluated the computational energy saving produced by the proposed technique with the results analyzed with the 5/3 DWT algorithm. The computational energy of the EEEHPWSA 5/3 2-D DWT technique is linked to the reduction in image quality. The relation between computation energy and image quality when processing and transmitting an image at different compression levels is resolved using the wireless application prerequisites and the predictable usage of the extracted images [22]. Actually, a number of applications have a preference for saving computation energy over image quality for remote wireless sensors network application, while other applications focus on high quality feature of images rather than the amount of energy consumed energy [22]. Combined potential wireless access applications and enhancements in battery technology will delay the rapidly increasing energy requirements of future wireless data services [15]. Hence, the technique proposed in this study will be appropriate for portable wireless applications which need high energy that cannot be satisfied by restricted expansion in battery technologies such as Wireless Sensor Network WSN applications that reduce the network energy expenditure function and support reduction in the image quality.

hardware utilization for the implementation of a time consuming software algorithm, the DWT, which enhances the general performance and energy requirements of the entire compression system. Lifting theorem was employed via LS 5/3 wavelet transform to develop a design where multipliers have been substituted with shifters, thereby decreasing the volume of operations involved in computing a DWT to approximately one-half of those required by a convolution approach. Therefore, less number of computations is needed and control complexity becomes simple. Moreover, the lifting scheme is adaptable to inplace computation, in order for DWT to be executed in low memory systems. The effectiveness of the proposed DWT energy consumption was modeled and applied in mathematical illustrations to evaluate the computational and data access loads. By implementing detailed modifications in the proposed architecture to evade computing high frequency subbands, the proposed EEHHPWS technique aids the transmission of image data, with considerable reduction in energy consumption, at the same time satisfying the accessible bandwidth and data quality limitations. The future extension recommended by this work includes further work on the transformation phase as well the coding phase, to be able to develop a comprehensive scheme for image compression by utilizing a LS architectural structures that are suitable for various portable and embedded wireless devices.

Acknowledgment The authors wish to express their sincerest gratitude to the Electrical Engineering Department, Engineering College, and University of Tikrit for giving an opportunity to work on this paper and for Engineering College contribution.

References [1]

RASHID, Bushra; REHMANI, Mubashir Husain. Applications of wireless sensor networks for urban areas: A survey. Journal of Network and Computer Applications, 2016, 60: 192-219.‫‏‬ [2] GHAZALIAN, Reza; AGHAGOLZADEH, Ali; ANDARGOLI, Seyed Mehdi Hosseini. Wireless Visual Sensor Networks Energy Optimization with Maintaining Image Quality. IEEE Sensors Journal, 2017, 17.13.‫‏‬ [3] HASAN, Mohammed Zaki; AL-RIZZO, Hussain; AL-TURJMAN, Fadi. A Survey on Multipath Routing Protocols for QoS Assurances in Real-Time Wireless Multimedia Sensor Networks. IEEE Communications Surveys & Tutorials, 2017.‫‏‬ [4] HASAN, Khamees Khalaf; NGAH, Umi Kalthum; SALLEH, Mohd Fadzli Mohd. Efficient hardware-based image compression schemes for wireless sensor networks: A survey. Wireless personal communications, 2014, 77.2: 1415-1436.‫‏‬ [5] REDDY, B. Eswara; NARAYANA, K. Venkata. A lossless image compression using traditional and lifting based wavelets. Signal & Image Processing, 2012, 3.2: 213.‫‏‬ [6] SRISOOKSAI, Tossaporn, et al. Practical data compression in wireless sensor networks: A survey. Journal of Network and Computer Applications, 2012, 35.1: 37-59.‫‏‬ V. CONCLUSIONS [7] MALLAT, Stephane G. A theory for multiresolution signal While the transform operation itself is incapable of decomposition: the wavelet representation. IEEE transactions on performing the compression procedure, it is the computationpattern analysis and machine intelligence, 1989, 11.7: 674-693.‫‏‬ intensive dominant phase relating to energy consumption of [8] HASAN, Khamees Khalaf; NGAH, Umi Kalthum; SALLEH, Mohd Fadzli Mohd. The most proper wavelet filters in low-complexity and the wavelet based image compression system. The major an embedded hierarchical image compression structures for wireless significance of this paper is to propose a design for effective Page 6 of 433 sensor network implementation requirements. In: Control System,

International Conference on Electrical, Electronics, Computers, Communication, Mechanical and Computing (EECCMC)

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21] [22]

Computing and Engineering (ICCSCE), 2012 IEEE International Conference on. IEEE, 2012. p. 137-142.‫‏‬ RAMAIAH, K. Dasaradha; VENUGOPAL, T. A novel approach to detect most effective compression technique based on compression ratio and time complexity with high image data load for cloud migration. In: Colossal Data Analysis and Networking (CDAN), Symposium on. IEEE, 2016. p. 1-5.‫‏‬ AL MUHIT, Abdullah; ISLAM, Md Shabiul; OTHMAN, Masuri. VLSI implementation of discrete wavelet transform (DWT) for image compression. In: Proc. International Conference on Autonomous Robots and Agents, ICARA. 2004.‫‏‬ LEE, Dong-Gi; DEY, Sujit. Adaptive and energy efficient wavelet image compression for mobile multimedia data services. In: Communications, 2002. ICC 2002. IEEE International Conference on. IEEE, 2002. p. 2484-2490.‫‏‬ BUI, Vy, et al. Comparison of lossless video and image compression codecs for medical computed tomography datasets. In: Big Data (Big Data), 2016 IEEE International Conference on. IEEE, 2016. p. 39603962.‫‏‬ YANG, Ming; BOURBAKIS, Nikolaos. An overview of lossless digital image compression techniques. In: Circuits and systems, 2005. 48th Midwest symposium on. IEEE, 2005. p. 1099-1102.‫‏‬ TAUBMAN, David; MARCELLIN, Michael. JPEG2000 image compression fundamentals, standards and practice: image compression fundamentals, standards and practice. Springer Science & Business Media, 2012.‫‏‬ LEE, Dong-Gi; DEY, Sujit. Adaptive and energy efficient wavelet image compression for mobile multimedia data services. In: Communications, 2002. ICC 2002. IEEE International Conference on. IEEE, 2002. p. 2484-2490.‫‏‬ POWELL, Scott R.; CHAU, Paul M. A model for estimating power dissipation in a class of DSP VLSI chips. IEEE Transactions on Circuits and Systems, 1991, 38.6: 646-650.‫‏‬ ANDRA, Kishore; CHAKRABARTI, Chaitali; ACHARYA, Tinku. A VLSI architecture for lifting-based forward and inverse wavelet transform. IEEE Transactions on Signal Processing, 2002, 50.4: 966977.‫‏‬ MARTINA, Maurizio; MASERA, Guido. Multiplierless, folded 9/7– 5/3 wavelet VLSI architecture. IEEE Transactions on Circuits and Systems II: Express Briefs, 2007, 54.9: 770-774.‫‏‬ REIN, Stephan; REISSLEIN, Martin. Low-memory wavelet transforms for wireless sensor networks: A tutorial. IEEE Communications Surveys & Tutorials, 2011, 13.2: 291-307.‫‏‬ AL MUHIT, Abdullah; ISLAM, Md Shabiul; OTHMAN, Masuri. VLSI implementation of discrete wavelet transform (DWT) for image compression. In: Proc. International Conference on Autonomous Robots and Agents, ICARA. 2004.‫‏‬ LEPLEY, Margaret A.; FORKERT, Richard D. Awic: Adaptive wavelet image compression. Technical report, 1997.‫‏‬ ANGELOPOULOU, Maria E., et al. Implementation and comparison of the 5/3 lifting 2D discrete wavelet transform computation schedules on FPGAs. Journal of signal processing systems, 2008, 51.1: 3-21.‫‏‬

Page 7 of 433