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Discrete Transforms
Published in Scott E. Umbaugh, Digital Image Processing and Analysis, 2017
This transform is typically implemented in the spatial domain by using 1D convolution filters. In the section on edge detection, we looked at 2D convolution masks that mark places in the image where the gray levels are changing rapidly. These rapid changes correspond to high-frequency information, so edge detectors are basically high-pass filters. To do this, we apply the convolution theorem, which is an important Fourier transform property. As we have seen, the convolution theorem states that convolution in the spatial domain is the equivalent of multiplication in the frequency domain. We have seen that multiplication in the frequency domain is used to perform filtering; the convolution theorem tells us that we can also perform filtering in the spatial domain via convolution, such as is done with spatial convolution masks. Therefore, if we can define convolution masks that satisfy the wavelet transform conditions, the wavelet transform can be implemented in the spatial domain. We have also seen that if the transform basis functions are separable, the 2D transform can be performed with two 1D transforms applied successively. An additional benefit of convolution versus frequency domain filtering is that if the convolution mask is short, it is much faster.
Image Visualization
Published in Alexandru Telea, Data Visualization, 2014
If we denote the Fourier transforms of f and g by F and G, the convolution theorem says that the Fourier transform of the convolution f ∗ g equals the product F · G of the corresponding Fourier transforms F and G. Having this result, we immediately see that the filtering operation can now be implemented much more easily by simply computing the convolution f ∗ ϕ of the desired input signal f(x) with the filter function ϕ(x), which is the inverse Fourier transform of the frequency transfer function Φ(ω). Using convolutions to filter signals has the main advantage of computing the filter function ϕ from the frequency transfer function Φ only once. After we have ϕ, we simply convolve the input signal f with ϕ to filter it.
Frequency Domain Filtering
Published in Elizabeth Berry, A Practical Approach to Medical Image Processing, 2007
It was hinted earlier in this chapter that filtering in the frequency domain would give the same results as convolution filtering in the spatial domain. The results of the two methods are indeed the same, and this can be explained using the convolution theorem. The convolution theorem states that the result of the convolution operation between two functions in the spatial domain is equal to the inverse Fourier transform of the result of multiplying together the Fourier transforms of the two functions. For those happy with mathematics, this is expressed using the mathematical notation below.
Development of a digital astronomical intensity interferometer: laboratory results with thermal light
Published in Journal of Modern Optics, 2018
Nolan Matthews, David Kieda, Stephan LeBohec
We have successfully employed two different types of digital correlators, off-line and real-time. In the off-line correlator, the digitized data from each channel are scaled, truncated to 8-bits and merged into a single continuous data stream by a Virtex-5 FPGA (PXIe7965R). The data stream is then recorded to a high speed (700 MB/s) 12TB RAID disc. A software routine using LabVIEW can later be used to retrieve intensity correlations between channels as a function of the digital time lag, typically up to 1s in steps of 4 ns. Due to the large number of samples, the data are read in blocks of 512 samples. The convolution theorem gives the correlation between two signals as the inverse Fourier transform of the product of the Fourier transforms of the signals. This is implemented by use of the NI Multi-core Analysis and Sparse Matrix toolkit cross-correlation virtual instrument (VI), which optimizes the computation by utilizing separate computing cores.