Wavelet Fourier Dsp Medical Imaging

The evolutionary step from analog to digital signal processing has revolutionized the medical imaging industry. The development of the cosine, fourier, laplace, and wavelet transforms along with digital FIR and IIR filters have made the digital realm king. The software processing behind field programmable gate arrays are fueling the industry of research in advanced mathematically intensive filtering and processing of discretely sampled analog signals.
The fourier transform was one of the most important equations of our time. It has paved the way for signal analysis in a frequency domain, the core of all digital signal processing. The wavelet transform however seems to be taking over now as it is much faster of an algorithm. The fourier transform takes O(N^2), the fast fourier transform from Cooley-Tukey takes O(nLog2(n)), and the fast wavelet transform takes O(N). The required amount of processing power is thereby lessened considerably. This movement is a part of the mathematical movement from cosine decomposition to a linear algebra and discrete graph theory quantization of the frequency components in a sampled analog line.
An analog line is represented with digital samples along discrete delays, obeying the nyquist law of the maximum frequency component in the signal being half the sample rate. For ultrasound the lines are beamformed along a delay and sum algorithm. A Computed Axial Tomography machine uses other algorithms to compute the various attenuation lines for x-ray lines much like an ultrasound transducer calculates the change in the transmit attenuation of power in a given tensor.
This is alot of crazy math and computer science vocabulary but it is rather simple when seen from the overview. The increased speeds in computational algorithms advance the industry at substantial rates which are comperable to hardware upgrades, in that inventing a mathematical algorithm which does the same operation in half the calculation density is the same as creating a processor that is twice as fast. The field of computer science still has yet to see any real boundaries as far as what can be accomplished in the software when implemented properly, and the medical imaging hardware required is getting to be smaller to the point of a new market gaining hold. The portable imaging units, medical ultrasound is a five billion dollar a year industry but all of the new growth is in the hand carry market where new machines are the size of a briefcase in a laptop fashion with the processing power of what used to take up a room less than ten years ago.
These better algorithms are leading to new technologies for testing for cancer and also observing cancer under testing conditions for angiogenesis rates, the technology is also being developed for blood testing and vascular, and cardiac imaging. These new fields are called elastography and photoacoustics, both of which have been made possible by advances in the digital signal processing capabilities of medical imaging equipment.