On the Relationship Between Feature‐Recognizing MRI and MRI Encoded by Singular Value Decomposition

Abstract

This paper describes the similarity between two methods of non-Fourier MRI: feature-recognizing MRI (FR MRI) and MRI with encoding by singular value decomposition (SVD MRI). Both methods represented images as truncated expansions of non-Fourier basis functions; these basis images were derived from prior image data by using closely-related mathematical techniques: the Karhunen-Loeve decomposition (or principal components analysis) and singular value decomposition, respectively. We demonstrate that FR and SVD MRI are equivalent in the following sense: given the same prior image data, they lead to exactly the same basis functions. FR MRI utilized prior images of the same body part in many "training" subjects, thought to be similar to the "unknown" subject to be imaged. SVD MRI utilized a single prior image of one subject in order to perform dynamic imaging of that subject. We demonstrate that the basis function expansion derived from a single prior image may not be capable of representing new features (features not found in the prior image). Therefore, the SVD basis functions may be inappropriate for dynamic imaging.

Applicability and efficiency of near‐optimal spatial encoding for dynamically adaptive MRI

“…Based upon this, the mathematics of our approach is clearly differentiated from the statistical approach of others (11). We do note (5), however, that the specific use of the SVD to compute encodings bears a formal relationship to the single training image limiting case of the Karhunen-Loeve statistical approach employed in (11).…”

“…We therefore describe and apply a rigorous mathematical analysis of the principal angles between vector subspaces to assess the suitability and efficiency of basis sets for encoding a given FOV in circumstances of ideal and nonideal encoding. The theory and results presented in this work specifically address several adaptive encoding issues raised in a recent critique (11), which claimed our approach to be "inappropriate" for dynamic imaging. …”

Proper Orthogonal Decomposition and Its Applications—part I: Theory

“…If the methods are actually applied in the same way they may lead to exactly the same basis functions. If the methods are applied slightly di!erently, yet in equivalent ways, then the equivalence is more hidden [33].…”

O-space with high resolution readouts outperforms radial imaging

“…But if we simulate radial imaging with 16, 32 and 64 points per echo, the rank of each encoding matrix E is 954, 1609 and 2461, respectively. More efficient encoding in MRI is possible, since MR images typically represent rank deficient matrices [29-30], and a higher rank encoding matrix E generally denotes a more efficient encoding [19, 31-32]. Therefore, the rank of the encoding matrices suggests that the proposed strategy, parallel imaging with O-Space acquisitions, is superior to parallel imaging with radial encoding, and the difference is increasingly pronounced at higher resolution readouts.…”