Jiayang Wang^{1} and Justin P. Haldar^{1}

^{1}Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA, United States

The g-factor is commonly used for quantifying the noise amplification associated with accelerated data acquisition and linear image reconstruction, and is frequently used to compare different k-space sampling strategies. While previous work computes g-factors in the image domain, we observe in this work that g-factors can also be used to quantify uncertainty in various transform domains (e.g., the wavelet domain and the multi-channel Fourier domain). These transform-domain g-factor maps provide complementary information to conventional image-domain g-factor maps, and are potentially useful for k-space sampling pattern design.

The conventional g-factor

In this work, inspired by recent developments in uncertainty quantification and experiment design

The above approach enables calculating the image-domain g-factor, but in this work, we are interested in estimating transform-domain g-factors. Consider a transformation $$$\mathbf{T}$$$ that we can apply to the image $$$\mathbf{f}$$$ to get transform coefficients that we are interested in (e.g., Fourier-domain coefficients or Wavelet coefficients). A generalized statement of the Gauss-Markov theorem

Fig. 2 shows a comparison between image-domain g-factors and Wavelet-domain g-factors for this same SENSE problem. Notably, the Wavelet-domain g-factor captures much of the same information as the image-domain g-factor, although provides additional information about uncertainty at different resolution scales. Notably, the top-left region in the wavelet domain corresponds to low-resolution image features, while the remaining regions correspond to progressively higher-resolution features, and e.g. we can clearly see that variable-density random sampling has less uncertainty for low-resolution image features and more uncertainty for high-resolution image features than the other two sampling patterns.

These ideas can also be applied naturally to other linear reconstruction formulations, as illustrated for phase-constrained SENSE

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Fig. 1. Illustration of image-domain and Fourier-domain g-factors for SENSE reconstruction with an acceleration factor of 8. The proposed new Fourier-domain g-factor clearly shows k-space locations that are difficult to estimate and which may benefit from additional data acquisition.

Fig. 2. Illustration of
image-domain and Wavelet-domain g-factors for SENSE reconstruction with an
acceleration factor of 8. The proposed
new Wavelet-domain g-factor clearly demonstrates that different sampling
patterns can have different uncertainty characteristics at different spatial
resolution scales. For example, with
variable-density sampling, the low-resolution wavelet coefficients are easier
to estimate than the high-resolution wavelet coefficients.

Fig. 3. Illustration of various g-factors for
phase-constrained SENSE reconstruction with an acceleration factor of 4. As can be seen, the transform domain
representations provide some potentially surprising insights. For example, it may not have been obvious
that partial Fourier sampling is associated with similar amounts of uncertainty
on both sides of k-space, and has better g-factors for low-resolution image
features than it does for high resolution image features.