# Analysis of the Quantization Noise in Discrete Wavelet Transform Filters for 3D Medical Imaging
## 1 Introduction  
Medical imaging uses many different methods such as magnetic resonance (MR) imaging [1; 2; 3; 4; 5; 6; 7; 8], radiography [4; 9; 10; 11], radionuclide [8; 12], optical [11; 13; 14], ultrasound [1; 15] and medical robotics [16; 17]. The typical medical imaging system consists of three components (Figure 1): data acquisition, data consolidation and data processing. The data acquisition card, which filters incoming data, is the most cost-sensitive system card. Usually, a diagnostic imaging system will consist of multiple data acquisition cards. Once the data is compensated and filtered in scanners, it is sent to the data consolidation card for buffering and data alignment. Once the data has been collected, it is sent to the image processing cards [18]. These cards perform heavy-duty filtering and the most algorithm-intensive image reconstruction. Modern field-programmable gate array (FPGA) devices are widely used in data consolidation, and image processing for sophisticated application algorithms implementation including pattern recognition, image enhancement and data compression [19; 20]. Denoising of 2D and 3D medical images is an important problem in modern medical imaging systems. The noisy pattern is not always bad in medical images, but in most cases is a problem. MR images are inherently noisy and thus filtering methods are required to improve the data quality [5]. Rheological methods of increasing MR elastography resolution determine viscoelastic properties through wave inversion, which is highly ill posed and sensitive to noise [1]. In radiology using computed tomography (CT) or related morphological imaging modalities, noise affects the analysis ofanatomical structures and thus impedes diagnostic applications [11]. Low dose radiation exposure for patient safety leads to noisy and low-contrast fluoroscopic sequences [11]. The reconstruction process of the positron emission tomography images includes inherent multiplicative noise, which prevents the analysis of visual data [12]. In optical CT for retinal imaging as another example use case, noise limits the measurement of structural features in the human eye, e.g., retinal layer properties [11]. Denoising facilitates visual data interpretation from echocardiography [15]. Medical imaging systems produce increasingly accurate images with improved quality using higher spatial resolutions and bit-depths with advances in scanning technology and digital devices. Such improvements increase the amount of information that needs to be processed, transmitted and stored. This is especially true when using 3D scanning technology [4]. For example, four sets of positron emission tomography medical images of one patient may require more than 4 GB of storage space [21]. Video recording of a relatively short retinal peeling procedure may require over 40 GB of memory storage [14]. The capacity of hard drives is on average 1-2 TB with the current level of storage technology development. Thus, the compression of 3D medical images is also an important problem in modern medical imaging systems. Various transforms are used to solve problems of 2D and 3D medical images denoising and compression in practice. The most common of them are discrete Fourier transform (DFT) [3; 7; 14; 22] and discrete wavelet transform (DWT) [1; 9; 11; 14]. DFT is widely used in the frequency domain but the domain characteristics disappeared after it. We cannot determine the time position and the degree of intensity after signal DFT. It is not possible to describe the local properties of the time domain of the image. DWT solves these problems because it allows obtaining both frequency and time information about a signal [23; 24]. 2D and 3D images DWT is performed by convolution with a pair of lowpass and highpass wavelet filters of filter bank that highlight main and detailed information respectively.