# Analysis of the Quantization Noise in Discrete Wavelet Transform Filters for 3D Medical Imaging
## 1 Introduction
Denoising and compression of images are performed by detailed information manipulating in modern algorithms such as set partitioning in hierarchical trees (SPHITs) [25] and embedded zerotrees of wavelet transforms (EZWs) [26]. The convolution operation has high computational complexity. Hardware implementation on modern microelectronic devices such as field-programmable gate arrays (FPGAs) and application-specific integrated circuits (ASICs) working with fixed-point numbers is one of the ways to improve its characteristics [27; 28; 29]. Quantization noise occurs when converting  
Figure 1: The typical medical imaging system. wavelet filters coefficients into this format, due to which convolution is performed with an error. The question arises about the accuracy of wavelet filters coefficients representation in the device's memory, which is efficient in terms of resources and enough to achieve the required quality of image processing. A novel area-efficient high-throughput 3D DWT architecture for real-time medical imaging based on distributed arithmetic is proposed by the authors [30]. The design and implementation of 3D Haar wavelet transform with transpose based computation and dynamic partial reconfiguration for 3D medical image compression are presented in [31]. The implementation of positron emission tomography using DWT on FPGA is proposed by authors [32]. In paper [33] described the architecture based on the use of DWT for biomedical signals compression. The design and implementation of context-based adaptive variable length coding and comparative analysis of trade-off offered by DWT for 3D medical image compression systems are described by authors [34]. In [35] presented the design and implementation of 3D DWT with a transpose-based method for medical image compression on FPGA. Experimental results from [36] showed that the system constructed a 1D DWT system based on FPGA can filter the noise and extract the electroencephalogram (EEG) signal well. The design and implementation on FPGA of 3D DWT using Daubechies wavelets with a transpose-based method for medical image compression are presented in [37]. The design and implementation of distributed arithmetic architectures of 3D DWT with a hybrid method for medical image compression are presented in [38]. Authors [39] presented the FPGA-based embedded system design using DWT and its evaluation for a pre-processing stage of EEG signal analysis. A detailed review of FPGA and ASIC architectures for DWT implementation in biomedical and intelligent applications, which can be designed either for higher-accuracy or for low-power consumption is provided by the authors [29]. In [40], the authors showed that DWT along with Gaussian filtering shows better results in removing the noise and smoothes the electrocardiogram signals. Authors [41] described the design and implement a complete hardware model based on DWT for EEG data compression and reconstruction on FPGA. A framework is offered in [42] based on DWT using linear and non-linear classifiers for detecting an epileptic seizure from EEG data recorded from normal subjects and an epileptic patient. There are no references to selected bit-width of wavelet filters coefficients in the materials studied about the hardware implementation of medical images DWT on FPGA and ASIC [29; 30; 31; 32; 34; 35; 36; 37; 38; 39; 40; 41; 42]. Authors [33] quantized wavelet filters coefficients by 16 bits, but there is no rationale for this choice. The problem of analyzing the quantization noise effect in wavelet filters coefficients for 2D grayscale and color images DWT with 8 bits per color (BPC) was solved in [43]. Analysis of 3D medical images DWT result quality dependence on noise arising from filters coefficients quantizing of wavelet with compact support is the purpose of this work.