Design and Simulation of a Low Power and High Speed Fast Fourier Transform for Medical Image Compression

For front-end wireless application in small battery-powered devices, the discrete Fourier (DFT) transform is a critical processing method for discrete time signals. Advanced radix structures are created to reduce the impact of transistor malfunction. To develop DFT, with radix sizes 4, 8, etc. is complex and tricky issue for algorithm designers. The main reason for this is that the butterfly algorithm's lower radix level equations were manually estimated. This requires the selection of new design process. As a result of fewer calculations and smaller memory requirements for computationally intensive scientific applications, this research focuses on Radix-4 Fast Fourier Transform (FFT) technique. A new 64-point DFT method based on Radix-4 FFT and multi-stage strategy to solving DFT-related issues is presented in this paper. Based on the results of simulations with Xilinx ISE, it can be concluded that the algorithm developed is faster than conventional approaches, with18.963 ns delay and 12.68 mW of power consumption. It was found that the computed picture compression drops ratios of 0.10, 0.31, 0.61 and 0.83 had direct relationship to the varied tolerances tested 0.0007625, 0.003246, 0.013075 and 0.03924. Fast reconstruction techniques, wireless medical devices, and other applications benefit from this FFT's low power consumption, little storage requirements, and high processing speed.


Introduction
One of the most widely utilized mathematical operations is Fast Fourier Transform. Several medical applications use Fast Fourier Transform for image reconstruction and frequency domain analysis.
Image processing applications such as filtering, compression, and de-noising all rely on FFT to certain extent. Figure 1 shows the usage of FFT, an improved version of the traditional discrete signal processing tool (Discrete Fourier Transform), for medical image compression with various drop ratios. FFT is widely used in medical imaging, engineering, communication, and other fields because it transitions quickly from the T-domain to the F-domain and vice versa [1][2][3][4][5][6].
The medical imaging method provides images of the human body and its components for clinical application. Computer tomography (CT), Magnetic Resonance Imaging (MRI), Ultrasound and Optical Imaging Technology are the most prevalent modes of medical imaging that produce a prohibitive amount of data. The images produced by these instruments are pixels representing the operations of human organs in terms of their visual depiction. They are also the patient's most vital information and demand high storage and transmission width [7,8].
FFT-based compression is a compression algorithm which can process the image quickly coupled with the transformed domain compression. The modified domain includes coefficients of both low and high frequencies that are measured. Various quantified coefficient values of high frequency are unimportant and almost equivalent to zero and remove them from the modified image. This preprocessing step leads to the compression platform. By supplying different symbols, the FFT method accomplishes compression. The majority of appearing symbols is assigned to be shorter while the less are assigned longer size symbols. The variable-length compressed data subsequently is stored on transmission media.
As hospitals are progressing into digitization, filmless imaging and telemedicine, the medical imagery becomes significantly important in the health sector. This has led to the major difficulty of developing compression algorithms that prevent diagnostic errors and have a high compression ratio for lower bandwidth and storages. In the medical area particularly, quick diagnosis is only achievable when the required diagnostic information is maintained by the compression approach. These images help physicians to easily diagnose the inner parts of the body. It also helps to perform keyhole procedures without too many incisions to reach the inner sections of the body. They can be processed fast, analysed objectively, and made available in numerous places simultaneously by means of communication protocols and networks likes Digital Imaging and Communications in Medicine (DICOM) protocol and Picture Archiving and Communication Systems (PACS) networks respectively. The X-ray, CT, MRI or Ultrasound images contain huge amounts of data that demand vast channels or storage capacities. The implementation costs limit storage capacity even with the progress in storage capacity and connectivity [8][9][10]. There are certain approaches that create imperceptible variations and acceptable fidelity that can lead to medical picture low loss compression. In this article an FFT based compression is proposed. Many different mathematical FFT algorithms vary from easy theory of complex numbers arithmetically to group and numerical theory; this paper provides an available technical outline and few characteristics while explaining the algorithms in the subsidiary sections. The DFT is obtained via the decomposition of a series of values into various frequency components as given in Eq. (1) & (2). This operation is useful in several fields, but it is always too slow to be practical for computing it directly from given description.    (4). Thus, Final representation of X(k) is, According to (4)-(6) the process is called decimation in time because of the samples of time are arranged into groups. The basic operation of R4 butterfly is shown in fig. 3 [17]. The decimation-intime process consolidates the inputs at each stage of decomposition, resulting in "input order that is bit-reversed" at the end. This set-up allows for the intermediate outputs to be stored in the same memory regions as the inputs (in-place algorithm). Radix-4 FFT's slight reorganization allows the inputs to be redirected from digit to bit [18] as shown in table 2.   (2) at 45°, 135°, 225°, and 315° in a unit circle, despite the fact that the number of radix minimizes the number of computation steps [19,20].

2.Development of a lossless medical image compression using FFT algorithm
Medical image compression using FFT is developed as shown in the flow chart given in the figure 4.
Load/Read any medical input image and convert it into 2D array of doubles image. Compress the loaded image with different tolerance values. While compressing the image it takes as inputs the original image X and the drop tolerance parameter and outputs a compressed image Y. It also returns the drop ratio given in (7) which is defined to be as the ratio of "Total number of nonzero Fourier coefficients dropped to the Total number of initially nonzero Fourier coefficients".
For every drop count for a compressed image apply FFT to each sub block.

Drop ratio = (Total number of nonzero Fourier coefficients dropped/Total number of initially nonzero Fourier coefficients) (7)
Figure.4: Medical image compression using FFT algorithm flowchart.

Results and Discussion
In

DFT Four
DFT four is the basic unit in radix-4 structure, because it transfers the input value to output. Each stage having this unit, four inputs and four outputs are present in this unit. The RTL view and simulation results of DFT four are shown in Fig. 18 & 19 respectively.

Even and Odd Parts
Even and odd parts are the two different functioning in the entire butterfly unit. The combination of even and odd part unit presents in all modules (M2, M3, M4 and M5). In four modules, four even and odd part combinations are present. In the different parts, the divided sequence can be ordered into even and odd parts/places to save the memory requirements. The even and odd part has two instances internally to make easy execution. The RTL view and their simulation results are shown in

Design summary
The design summary of 64-point Radix-4 DIT-FFT algorithm is given in

Timing report
The timing report is generated under speed grade -5. This report includes all the input and output cells and their fan-out. Each gate delay and net delay is also considered, and the summation of gate delay gives the timing delay of the project as 19ns. The comparison between the performance of radix-2 and radix-4 algorithm based on different aspects like number of slices used, LUTS, bonded IOBs, flip flops, global clocks for their operations is given in Table 4.  Table 4, it is observed that the minimum delay for functioning of inputs and outputs for radix-4 is very less when compared with radix-2 and memory usage is almost same with radix-2 even radix-4 using a greater number of inputs. And observed that 75% computations were saved in Radix-4 even though device utilization is more.

Medical image Compression
The proposed algorithm for image compression is simulated using the same targeted device given in

4.Conclusion
In this article the new high-speed DIT-FFT algorithm based on radix-4 algorithm for medical image compression was proposed and simulated on a target device xc3s500e-5fg320. The simulation results show that radix-4 processes the input with less delay. From the time delay table, it is very clear that approximately 75% of processing time is saved with less memory usage. Proposed radix algorithm also shown low power consumption than the existing radix2 this makes the use of the present algorithm in medical field where low power devices are preferable. This is another milestone for this article. Due to these advantages the proposed algorithm used in the medical image compression.