Our objective is to improve the time-resolution of functional magnetic resonance imaging by sampling only a small fraction of the Fourier transform of the spin density and using a prolate wavelet (P-wavelet) filter to approximately obtain not the usual susceptibility map, but instead the integral, I(t), of the difference in susceptibility between task and pretask, over a prespecified region of interest in the brain at successive time-points, t. This space-time trade-off thus allows us to obtain, at high time-resolution, the total activity, I(t), in a specified region, B, of the brain which processes the specific stimulus or task to learn or verify where the brain function takes place. We find that for a typical region, B, of the brain, say describing the hippocampus, believed to be involved in memory, consisting of say 100 points in a 64×64 brain image space, that our optimal choice of Fourier sampling region, A, has a=400 points, which then gives a 10-fold speed-up compared to the usual method of sampling, since the usual sampling needs 642 points which exceeds 10 times 400. Of course we get this speed-up at the price of spatial resolution. Even faster sampling of the integral of the susceptibility difference ought to be possible for a set B of 100 points in this level of pixelization. Once A is fixed, the mathematical problem of choosing the optimal P-wavelet filter can be viewed as a natural generalization to 2 and 3 dimensions of the theory of prolate spheroidal wave functions, due to Landau, Pollak, and Slepian. The first prolate spheroidal wave function for fixed A, B is that function φ which is maximally concentrated on B and whose Fourier transform vanishes except on A. The problem of choosing the sampling region A, for fixed size a=A, to maximize the concentration on the given B remains open and is probably NP-complete, i.e., probably involves exponential search. However, we give a very simple heuristic for choosing a set A, which seems to give excellent results in the examples we have studied, and we give the optimal P-wavelet filter, φ=φB, for the choice of A, based on our heuristic. The heuristic is to take as it A the scaled polar set of B, i.e., A=k: max x,y∈B(k,x-y)<c, (1) where c is chosen so that the size of A is a, as desired to achieve a given time-resolution for I(t). We give evidence to support our claim that A gives good results.
All Science Journal Classification (ASJC) codes
- Applied Mathematics