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Proc. Intl. Soc. Mag. Reson. Med 9 (2001) 1581

The Equivalence of SVD and Fourier Deconvolution for Dynamic Susceptibility Contrast Analysis

David ALSOP1,Gottfried SCHLAUG1

1Beth Israel Deaconess Medical Center, Harvard Medical School, One Deaconess Road /W/CC090, Boston, MA USA;

IntroductionThe quantification of cerebral blood flow by observing the passage of a

bolus of intra-vascular contrast agent requires removing the

broadening effect of the bolus width on the otherwise brief transit timethrough the tissue. Deconvolution techniques are advantageous for

this purpose because they make few assumptions about the shape of

the tissue clearance curve.

Fourier and matrix methods can be used for this purpose, but since the

smoothing of the tissue clearance curve by the bolus causes a loss of

information, simple deconvolution will produce extremely high noise

and sometimes divide by zero errors. Ostergaard et al (1) have

evaluated a number of different deconvolution techniques and haveargued that singular value decomposition (SVD) is the most accurate.

Later studies have compared the SVD and Fourier approaches (2) and

have demonstrated the importance of choosing the SVD cutoff tobalance minimizing noise and introducing errors in the flow

measurement(3).

Below we demonstrate that SVD deconvolution is very closely related

to Fourier deconvolution and can be mathematically equivalent for a

certain Fourier filtering and SVD thresholding approach. Thisobservation yields insights into the effects of singular value

thresholding with SVD and the optimal Fourier filtering for blood flow

measurement.

TheoryIn the absence of noise, the measured concentration-time curve is aconvolution of the tissue response function, R, with the arterial input

function. This convolution can be written as a matrix multiplication,

s=MR, where s is the tissue concentration and M is the convolutionmatrix.

In an experiment, we know M and s and want to measure R. Direct

inversion of M will generally cause excessively noisy results. Instead,

one can choose to find an approximation to R, r, which minimizes the

squared difference between the measured and theoretical tissueconcentration curves subject to a constraint on the square magnitude of

r. This can be achieved by minimizing an error function, eq [1]

including lambda, a Lagrange multiplier. This error function will be aminimum when the derivative is zero, eq [2].

Now suppose M can be diagonalized with a unitary transformation, eq[3], where W is a diagonal matrix and U and V are unitary matrices.

Then we can substitute for M in eq. [2] solve for r, eq. [4]

A real square matrix M can always be decomposed into the form of

equation [3] where W is real and positive and U and V are real. This is

referred to as the singular value decomposition. In the solution to r,

singular values smaller than the square root of lambda will have little

affect on the solution. Hence, choosing lambda is nearly equivalent to

the editing of singular values typically performed with SVD matrixinversion.

M can also be diagonalized by the Fourier transformation as a result of

the convolution theorem. In this case, U and V are the forward andreverse Fourier transforms and the diagonal values of W are simply the

Fourier transformation of the input function.

DiscussionThe equivalence of Fourier and SVD deconvolution suggest that earlier

findings of differences in the two techniques were the result ofparticular implementations. In particular, the filter implied by eq. [5]

is not an optimal filter for Fourier deconvolution for an exponentialclearance. Instead it represents an optimal filter for a spike, i.e. a

waveform with a flat power spectrum. The advantage of this filter for

flow quantification is intuitive since the flow measurement is based on

the first or alternatively the peak value of the deconvolved tissue

concentration curve.

The relationship between SVD and Fourier deconvolution and the

form of eq. [5] also provide a direct relationship between SVD

thresholding and Wiener filtering for a spike waveform. This mayprovide a simpler method than simulations for determining the optimal

thresholds.

The derivation above assumed that the form of the matrix M was

identical for Fourier and SVD deconvolution. For the Fourier

convolution theorem to be valid, M must represent a cyclicconvolution. In contrast, SVD is not constrained in this way. This

may make SVD more advantageous in some situations, though zero

padding and windowing can often be used to avoid edge effects in

Fourier deconvolution.

References1.Ostergaard L et al, Magn Reson Med 36:715 (1996)

2.Wirestam R et al, Magn Reson Med 43:691 (2000)

3.Liu HL et al, Magn Reson Med 42:167 (1999)