From Fourier (k space) to wavelet domain in MRI sensing - Signal Processing Stack Exchange most recent 30 from dsp.stackexchange.com 2019-11-19T11:57:44Z https://dsp.stackexchange.com/feeds/question/50500 https://creativecommons.org/licenses/by-sa/4.0/rdf https://dsp.stackexchange.com/q/50500 0 From Fourier (k space) to wavelet domain in MRI sensing Antoni Parellada https://dsp.stackexchange.com/users/27561 2018-07-13T04:05:25Z 2018-07-13T14:21:23Z <p>In <a href="https://onlinelibrary.wiley.com/doi/pdf/10.1002/mrm.21391" rel="nofollow noreferrer">compressed sensing MRI (cSENSE MRI)</a> technology the idea seems to entail sampling from the Fourier domain (k space) in a way that, when transformed to the wavelet domain ("sparsification"), the sparsity is maximized.</p> <p>The inverse recovery problem becomes exact provided that $\mu$ is small:</p> <p>$$\mu \left( \mathcal F W^\top \right)=\max_{i,j}\vert \langle W_i, \mathcal F_j\rangle \vert$$</p> <p>i.e. the dot products of the column of the Fourier transform and wavelet transform are minimal ("mutual incoherence").</p> <p>The idea seems to stem from <a href="http://statweb.stanford.edu/~candes/papers/StableRecovery.pdf" rel="nofollow noreferrer">this paper by Candes, Romberg and Tao</a>.</p> <p>In the wavelet domain the coefficients include scale and translation, while in Fourier space the coefficients belong to different frequencies without temporal support.</p> <p>I would like to confirm that there is indeed a triple step: First (partially) filling in k-space with Fourier coefficients; second, randomly sampling these coefficients; and third, transforming them to wavelet space <strong>in each MRI image acquisition</strong> (as opposed to being an assumption based on the nature of MRI images). </p> <p>And if this is the case, how does the step from Fourier to wavelet takes place (a reference would be OK).</p> <p>Or, contrarily, whether the signal is primarily analyzed as wavelets?</p>