Introduction
Fourier analysis is a field of mathematics that studies how general functions can be approximated or represented by the addition of simple trigonometric functions. Fourier analysis is named after a man who proved that representing a mathematical function in the form of a sum of a trigonometric function simplifies the inquiry into heat propagation. Joseph Fourier envisioned the Fourier series, a concept from which Fourier analysis grew. Presently, vast spectrums of mathematics involve Fourier analysis as a subject. The subject spans through the field of engineering and sciences. In these two fields, Fourier analysis involves the process through which a function is decomposed into simpler functions. The inverse of this, which involves the operations involved in rebuilding the small pieces into a whole function is called Fourier synthesis (Stein & Rami, 2003).
The field of mathematics conceptualizes Fourier analysis as the two operations, that is Fourier synthesis and Fourier analysis all studied as one subject. The decomposition of the function into smaller pieces is called Fourier transform. The transform is generally referred to using a more precise name that is dependent on the domain in addition to other properties that the function under transformation has. In addition to this, the initial model of Fourier analysis has been used over time in general and abstract situations. The general field of the concept is called harmonic analysis. As espoused earlier, each transform has its inverse. The corresponding inverse transform can be utilized in synthesizing the whole function from the small pieces. New directions inspired by Fourier analysis are continuously being discovered and exploited in various systems in physical, engineering, mathematical, biological, chemical and financial realms.
Relevance of Fourier analysis in medical imaging
Fourier analysis is of great importance in the medical fraternity. Its applications in the field have enabled the delivery of better services. Firstly, the Fourier transform together with the complex-valued neural network are used in the detection of any alterations in medial images that are watermarked. Medical images enclose diagnostic information that is important for timely detection of diseases. Such medical images are watermarked so as to ascertain their integrity. The watermarks are also used to ensure that no modifications are made by unauthorized persons, to ensure authenticity, correct pairing of medical images to the right patients, and to ensure that such images are sourced correctly (Dhawan & Dae-Shik, 2007).
Currently, the distortion that is introduced when embedding patient information or data in the images is the main problem facing watermarking systems that are employed for use in medical images. This has hindered the proper treatment and subsequent treatment of diseases. An algorithm that is based on a Fast Fourier Transform (FFT) that is free from distortion and Complex-Valued Neural Networks (FFT-CVNN) can be employed for the function of watermarking medical images. Using this method produces perceptually indistinguishable watermarked images when compared to host images. In addition to this, the tamper detector that is input when Fast Fourier Transform and Complex-Valued Neural Networks is used in watermarking can detect forgeries of any form or tempering on the medical image that is watermarked.
Medical imaging is faced with challenging problems especially the forming of reconstruction algorithms for geometries that are nonstandard. Fourier analysis offers a solution to the recurrent problems of rebinning and resampling. Conventional resampling methods use variants of interpolation thereby losing resolution and clarity in the tomographic images. Through the use of Fourier Transform resampling, potential improvements are experienced since the modulation transfer function in Fourier Transform resampling process acts like a low pass filter. In instances where the coordinate transformation in the process is nonlinear, the modulation transfer function is nonstationary. The Fourier Transfer resampling generalizes the linear coordinate transformations in the standard Fourier analysis (International Workshop on Medical Imaging and Augmented Reality & Hongen, 2010)
In a study carried out to show the potential solutions that Fourier Transform resampling can offer in medical imaging, simulated modulation transfer functions were acquired through the projection of point sources from different transverse positions on a detector of flat fan beam geometry. The study compared the modulation transfer functions to closed form expressions of Fourier Transform resampling. The obtained frequencies were in excellent agreement below or at the approximate cutoff frequency. The resultant Fourier Transform resampling algorithm when applied to simulations that had symmetric fan beam geometry showed an elliptical orbit with uniform attenuation and a normalized root mean square error equal to 0.036. At addition point source studies show that hybrid reconstructions compare favorably with direct reconstructions. The action of low pass filtration by the Fourier Transform resampling helps improve the reconstruction of algorithms in medical images.
The processing of medical images is done using both analogue and digital methods. The limitations and effectiveness of both methods have been the subject of debate and research for a while now. The analysis of the different and numerous type of errors that are introduced during the processing of medical images shows that a hybrid of both digital and analogue methods produces superior results in processing medical images. The optical Fourier Transform, an analogue method introduces errors in medical image processing through lens aberration and vignetting effects. The digital Fourier Transform introduces errors through the band limit and the aliasing effect. Using the two methods to process X-ray images both digitally and optically offers a chance to compare the limitations and effectiveness of the two methods.
The study used an optical system with a big Fourier telephoto lens to process the x-rays optically. A personal computer that uses a Fourier Transform algorithm was used to process the x-ray images digitally. An analysis of the veracity of the digital Fourier and optical spectra shows that optical methods have high speeds because of parallel processing. The optical system can also obtain high veracity in regions of high frequency. Comparatively, the digital methods offer high precision in processing medical images. The digital methods are also programmable but have lower processing speeds. The two methods using Fourier analysis can be used in different clinical settings to process medical images. The optical Fourier Transform is more suitable when compared to the digital Fourier Transform. However, a combination of both the analog optical Fourier Transform and the digital Fourier Transform offers the best results in processing medical images (Bui & Ricky, 2010).
Fourier image analysis has many useful properties. For example, in the spatial domain, convolution corresponds to multiplication that is used in the frequency domain. This is of significance because convolution is a more complicated mathematical operation when compared to multiplication (Epstein, 2008). Similar to one-dimensional signals, the property espoused above enables the different deconvolution techniques and the Fast Fourier Transform convolution.the Fourier Slice Theorem is another relevant property that the frequency domain offers the medical field among other fields. The Fast Fourier Transform is important in the processing of medical imaging. The Fourier Slice Theorem exemplifies the relationship between images and their projections. It enables images to be viewed from their sides (João et al., 2011).
This forms a basis for computed tomography. Computed Tomography is an imaging technique that is used in developing x-rays. This technique is widely used in the field of medicine. The Fourier Transform, a variant of Fourier analysis is a fundamental mathematical tool that is widely used in the analysis of signals. The Fourier Transform is ever-present in radiology and an integral part in the formation of modern magnetic resonance images. A knowledge of the accomplishments of the Fourier Transform is important in understanding the techniques of Magnetic Resonance Imaging (Poldrack et al., 2011). The Fourier Transform is the basis of encoding Magnetic Resonance Imaging. This is testament to the relevance of Fourier Analysis in processing medial imaging.
Conclusion
The discussion shows the varied uses of Fourier analysis in its different variants in different fields and especially in the field of medicine to process medical images. Fourier analysis is not only important in processing the images but also ensuring their authenticity and correct pairing of images with the patients. The discussion also shows the potential solutions to a wide array of problems that a hybrid between digital and analogue methods techniques using the Fourier Transform offer the medical fraternity (Bourne, 2010). The ability to combine the two methods in order to exploit their advantages and different levels of effectiveness offers medical image processing a new lease of life. This is because the speed offered by analogue optical techniques can be coupled with the processing precision from digital techniques. Al in all, Fourier analysis as revolutionized the process of medical imaging.
Bibliography
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