While the PFT reconstruction algorithm has been formulated and applied offline in recent years, its direct implementation on MRI scanners had not previously been reported, limiting prior validation to simulations or offline tests. In this work, we implemented it directly on the scanner, enabling experimental validation of several previously hypothesized properties of PFT—including central focusing behavior, spatially dependent artifact patterns, and distinct SNR characteristics [16].

We first demonstrated the feasibility of implementing PFT on MRI scanners by achieving practical reconstruction times and adequate computational efficiency. Building on both phantom and human studies, we then systematically compared PFT with gridding reconstruction in terms of artifact behavior and spatial distribution of SNR.

PSF and spatial dependency in PFT

To understand the mechanisms underlying the spatially varying behavior observed in PFT reconstructions—including artifact distribution and SNR patterns—we analyzed the system’s point spread function (PSF) using targeted simulations. Specifically, we reconstructed point sources (delta functions) placed at various radial distances from the image center (Fig. 7) to visualize how the reconstruction algorithm responds locally in undersampling conditions. For this analysis, 63 diagonal spokes were used to highlight undersampling effects, with all other acquisition parameters matching those of our experimental studies.

Fig. 7

Comparison of the point spread function between PFT and gridding techniques for points located at x0 = 10 mm, x0 = 30 mm and x0 = 55 mm (from left to right respectively). (a) PFT reconstruction in different stages of the algorithm for two different source points located at locations (r,θ) = (10,−π/2) and (r,θ) = (30,−π/2). (b) PFT (top) and gridding (bottom) reconstructions for three different source point located at locations (r,θ) = (10,−π/2), (r,θ) = (30,−π/2), and (r,θ) = (55,−π/2)

Both PFT and gridding reconstructions were simulated for delta functions positioned at two representative locations—10 mm and 30 mm from the center—corresponding to “central” and “peripheral” regions, respectively. Panel (a) of Fig. 7 shows the results of the PFT reconstruction at each step, illustrating how aliasing artifacts manifest in this method. At this level of undersampling, the peripheral delta function (30 mm) exhibits substantial broadening and aliasing artifacts that are absent near the center (10 mm). These artifacts appear as secondary intensity peaks located symmetrically opposite the original delta location (indicated by yellow arrows).

To facilitate side-by-side comparison, PSFs from three radial distances were also mapped onto a Cartesian grid and shown in panel (b), alongside those from gridding-based reconstructions. The gridding method exhibits a space-invariant PSF characterized by a central circular artifact-free region, known as rFOV, whose radius is calculated as 47.6 mm based on our simulation parameters [7]. In contrast, the PFT reconstruction begins to show aliasing effects beyond a radius of approximately 23.8 mm—corresponding to half the rFOV calculated in gridding.

This comparison highlights a key distinction: gridding reconstructions exhibit shift-invariance across the field of view, enabling image formation to be modeled using convolution with a fixed PSF kernel. Our PFT implementation, however, exhibits strong spatial variation in its PSF—particularly in the presence of undersampling. This space dependency violates the assumptions required for convolution-based modeling with a spatially invariant kernel and calls for spatially adaptive interpretation of the reconstruction behavior.

As the radial distance increases in PFT-reconstructed images, two principal effects emerge:

1.

Azimuthal resolution degrades, as reflected in broader PSF profiles.

2.

Signal leakage becomes more pronounced, with energy spreading across a wider region, particularly in directions opposite the original point source.

These effects directly influence image resolution as well as the nature and visibility of aliasing artifacts. In gridding, the space-invariant PSF produces structured artifacts—sharp, high-frequency streaks that maintain fixed positions and orientations, regardless of the signal’s origin. This invariance may cause such artifacts to consistently overlap with anatomical features (e.g., vessels or tissue boundaries), amplifying their visibility and making them more visually disruptive. Because of this spatial invariance, prior studies have shown that an object smaller than half the rFOV can be imaged without aliasing [7].

In contrast, PFT exhibits a spatially varying PSF that not only spreads aliasing energy more diffusely but also reduces azimuthal resolution in the periphery. This results in broader, low-frequency artifacts whose shape and orientation vary with location. Owing to this spatial variability, residual aliasing in PFT often appears as subtle intensity variations or noise-like texture, and less like coherent streaks—a behavior reminiscent of the diffuse artifact suppression observed with variable-density sampling [12]. While some structured components may still emerge, the combination of spatial variance and peripheral blurring prevents consistent alignment with anatomical features, thereby reducing the visual salience of the artifacts. As a result, PFT artifacts blend more subtly into the background and, regardless of object size, fine central features—such as the basilar artery in Fig. 6—can remain visible when they fall within the high-resolution central region.

This PSF behavior also helps explain the SNR patterns observed in Fig. 5. Within the central region, azimuthal resolution remains relatively high, preserving fine structural detail. In contrast, moving outward, progressive blurring along the angular direction smooths local intensity fluctuations, resulting in artificially elevated SNR in the periphery. The boundary of this high-resolution region corresponds to rFOV/2 and gradually shrinks with increasing undersampling. The distinct undersampling-induced blurring of PFT—illustrated in Fig. 3—also stems from its spatially varying PSF.

From a complementary perspective, the radial variation in resolution observed in PFT can also be explained as a direct consequence of the polar sampling geometry and the transform structure. In particular, the 1D FFT applied along the azimuthal direction preserves the number of angular samples during the transition from k-space to image space. As a result, pixel size in the angular direction increases proportionally with radius, leading to reduced azimuthal resolution toward the periphery. However, this does not imply enhanced resolution near the center. Despite smaller pixel sizes in central regions, the true spatial resolution is fundamentally constrained by the highest acquired spatial frequency \(_\). These smaller pixels reflect interpolation rather than actual resolvable detail.

Maintaining consistent resolution across the entire field of view, as defined by \(_\), requires a fully sampled radial acquisition. Based on the Nyquist criterion, this corresponds to \(_=_*\frac\) (≅ 400 in our experiments), which is approximately 56% more than the number of phase-encoding steps used in a Cartesian acquisition with the same base resolution. (Notably, the Nyquist condition for azimuthal sampling corresponds to a pixel size at full FOV equal to 1/\(_\).) In practice, however, radial imaging is typically performed with a number of spokes no greater than that of a fully sampled Cartesian acquisition, leveraging its robustness to undersampling. To accelerate acquisition further, some degree of artifact is typically accepted. While gridding distributes these artifacts as sharp streaks across the entire image, PFT manifests them primarily as a combination of low-frequency intensity variations, peripheral blurring, and occasional structured (streak-like) artifacts, depending on the interaction between image structure and PSF.

Dynamic imaging and ROI-focused applications of PFT

The spatially variant behavior of PFT, as established above, offers a distinct advantage for dynamic MRI applications where temporal resolution is critical and the diagnostic region of interest (ROI) is spatially localized. Our results demonstrate that reducing the number of radial spokes in PFT has less adverse impact on image quality within the central region—typically encompassing the ROI—while the periphery undergoes progressive blurring. This leads to a natural “ROI-focusing” effect: image fidelity is preferentially preserved in the center of the field of view, where the region of interest is most often located.

This characteristic makes PFT particularly appealing for dynamic studies, such as perfusion imaging, functional MRI, or contrast-agent tracking, where the ROI is confined to a central organ section or small anatomical region. Peripheral degradation, when it occurs, can often be tolerated or selectively mitigated using techniques such as view-sharing [27] or deep learning-based resolution enhancement. In such cases, any compromise in temporal resolution is localized to the periphery, while the central region retains high temporal fidelity. Moreover, dynamic imaging typically involves reconstructing a series of frames, allowing the one-time computation of Bessel coefficients to be amortized across the entire dataset—further improving computational efficiency.

Importantly, the high-resolution center in PFT does not need to coincide with the scanner isocenter, as the imaging frame can be positioned freely during scan planning to ensure that the ROI is centered within the acquisition field.

Signal and SNR behavior

The signal intensity profiles in Fig. 2, along with the uniform brightness of the water region in Fig. 3 and the higher measured values in Fig. 5, consistently show that PFT reconstruction yields stronger signal intensity in the central image region compared to gridding. This enhancement aligns with the concentration of low-frequency content near the image center—where true signal often dominates—and appears consistently across different undersampling levels and experimental conditions.

One plausible contributor to this trend is the treatment of low-frequency information. Gridding reconstructions typically apply density compensation to correct for the non-uniform k-space sampling inherent in radial acquisitions. This step is often implemented as a pre-weighting of the k-space data that functions similarly to high-pass filtering and may unintentionally attenuate true low-frequency signal. In contrast, PFT performs spatial encoding natively in the polar domain, where sampling density is inherently addressed through the Hankel transform itself. This structure avoids explicit density compensation and may therefore preserve low-frequency content more effectively, potentially leading to enhanced signal retention in regions dominated by low-frequency content.

At the same time, other mechanisms may also contribute to the observed signal behavior, though they reflect side effects rather than intentional design features. One such factor is gradient delay, which, if uncorrected, causes slight shifts in the k-space trajectory and, based on our simulations, may lead to biased signal intensity near the image center. While our gridding pipeline incorporates gradient delay compensation, our current PFT implementation does not. Another contributor may be PSF-related signal leakage: the space-variant point spread function in PFT tends to spread peripheral energy inward in conditions of undersampling. Both mechanisms could artificially boost central signal levels, not by improving reconstruction fidelity but by redistributing peripheral signal content. These effects, therefore, represent potential confounding influences that complicate the interpretation of signal enhancement.

Given these multiple overlapping factors—some inherent to the reconstruction strategy, others introduced by system imperfections or PSF behavior—the current observations should be interpreted as empirical findings rather than definitive conclusions. While the evidence points toward meaningful SNR advantages for PFT in central regions, further analytical work is needed to fully characterize these effects and disentangle their origins. Such studies could guide future refinements to both PFT methodology and its implementation in dynamic or high-precision applications.

Practical considerations and reconstruction efficiency

The primary computational cost in PFT reconstruction stems from the Hankel transform. In our implementation, runtime was significantly improved by precomputing Bessel function values and reusing them across reconstructions. Nevertheless, high-resolution PFT reconstructions currently require approximately 6–9 × the acquisition time.

This overhead is partially offset by system dead time, such as during positioning or updating acquisition parameters between successive sequences— a window that can be effectively utilized for computation. Further acceleration is feasible through parallel processing or deployment on dedicated hardware platforms, such as system-on-chip (SoC) architectures, which have previously demonstrated 10–100 × speed improvements for image reconstruction tasks [28].

Notably, dynamic protocols benefit even more from these optimizations, as Bessel values only need to be computed once and can be reused across multiple time frames—further enhancing overall reconstruction efficiency.

Limitations and future work

The Hankel transform remains the primary bottleneck in PFT reconstruction. While our kernel-based acceleration strategy improved performance, alternative methods—such as numerical approximations of the discrete Hankel Transform—may further reduce complexity. Reference [29] introduces a discrete polar Fourier transform using special radial grids based on Bessel function zeros. While mathematically robust, such sampling schemes are incompatible with the uniform radial readouts typically used in MRI.

A more viable solution may lie in custom SoC platforms [28], which could substantially accelerate Bessel and Hankel computations.

A critical challenge in radial imaging is gradient delay, which causes trajectory deviations and artifacts [30,31,32]. In scanner-based reconstruction (gridding), this issue appears to be addressed through built-in correction algorithms, as evidenced by the absence of characteristic gradient-induced artifact patterns often seen in uncorrected radial data. In contrast, our PFT implementation did not include any gradient delay correction to evaluate its intrinsic sensitivity to such imperfections. In principle, delay correction strategies similar to those used in gridding—such as trajectory calibration or precompensation—could be adapted for PFT, although they were not applied in this study. Figure 8 shows that the associated artifacts, which were clearly visible under monopolar acquisition, were significantly reduced by simply alternating the gradient direction. This straightforward adjustment proved remarkably effective—reducing gradient delay artifacts to a level lower than that observed in gridding-based reconstructions.

Fig. 8

Comparison of acquisitions with odd and even number of spokes in different reconstruction techniques. PFT reconstruction with an even number of spokes shows the gradient delay artifact as two pronounced bright parallel lines on the top and bottom of the phantom edges, while for odd number, the artifact is not as evident. (The contrast and brightness of the whole image are increased by 40% for a better visualization of artifacts.)

Although not a primary focus of this study, the extension of PFT to accommodate non-uniform angular sampling, such as Golden Angle acquisitions, is conceptually straightforward. In prior work [33], where Golden Angle sampling was employed, PFT was tested in parallel and found to be compatible with the proposed reconstruction framework, though the final results were not explicitly reported. Those preliminary tests indicated that even with the standard PFT—without correction for angular aperiodicity—the reconstruction quality remained acceptable. More accurate outcomes could be achieved by adapting the angular transform to handle non-uniform sampling directly, albeit with additional computational complexity.

As a methodological study, this work focuses on artifact behavior, resolution, and SNR/CNR characteristics under undersampling rather than clinical evaluation. Broader studies involving diverse subjects or clinically relevant endpoints remain an important direction for future work.

Finally, estimation errors from Bessel function computation and interpolation should be noted. Our recursive Bessel function evaluation was benchmarked against MATLAB’s built-in routines and showed a maximum absolute error of 1.41 × 10⁻⁶, which is negligible for imaging purposes. Likewise, the polar-to-Cartesian interpolation step introduced minimal artifacts due to the smoothness of the underlying image and the use of zero padding before the final Fourier transform. Nevertheless, more advanced interpolation schemes could improve numerical precision, though possibly at the cost of introducing smoothing effects that could complicate quantitative interpretation.