Introduction

Magnetic resonance imaging (MRI) allows versatility to acquire volumetric images under various tissue contrasts, but prolonged data acquisitions limit the quality and diversity of the images collected. The native approach mitigating this limitation involves recovering diverse arrays of high-quality images from acquisitions undersampled across tissue contrasts^1-4 or k-space^5-8. The recovery methods process data either volumetrically^2,3,5 or cross-sectionally^1,4,6-8. Volumetric models offer advanced global consistency, yet they may manifest increased model complexity, yielding suboptimal learning. Cross-sectional models suggest reduced complexity with improved training, but they may suffer from loss of contextual consistency across the longitudinal dimension. Here we proposed a novel progressively volumetrized model⁹, ProvoGAN, for data-efficient image recovery in accelerated multi-contrast MRI. ProvoGAN decomposes volumetric image recovery tasks into a sequence of cross-sectional mappings optimally ordered across rectilinear orientations (Fig. 1) to effectively capture global-local details while demonstrating reduced model complexity⁹.

Methods

Progressively Volumetrized Generative Model for MR Image Recovery

First Progression: Given a progression sequence of the rectilinear orientations $(o_1\rightarrow~o_2\rightarrow~o_3)$, ProvoGAN first learns a cross-sectional mapping in $o_1$ via a generator $(G_{o_1}:x_{o_1}^i\rightarrow\hat{y}_{p_1,\,o_1}^i)$ and a discriminator $(D:\{x_{o_1}^i,\,y_{o_1}^i\}\rightarrow[0,1]\,$and$\,\{x_{o_1}^i,\,\hat{y}_{p_1,\,o_1}^i\}\rightarrow[0,1])$, where $x_{o_1}^i-y_{o_1}^i$ denote the $i$th cross-sections in $o_1$ of the source$-$target volumes, and $\hat{y}_{p_1,\,o_1}^i$ denotes the $i$th cross-section of the target volume in $o_1$ recovered in the first progression. $G_{o_1}$ and $D_{o_1}$ are trained with adversarial and pixel-wise losses: $${L_{p_1}=\underbrace{\mathrm{E}_{x_{o_1}^i,y_{o_1}^i}[|y_{o_1}^i-\hat{y}_{p_1,o_1}^i|]}_{\mathrm{pixel-wise\,loss}}-\underbrace{\mathrm{E}_{x_{o_1}^i,y_{o_1}^i}[(1-D_{o_1}(x_{o_1}^i,y_{o_1}^i))^2]-\mathrm{E}_{x_{o_1}^i}[D_{o_1}(x_{o_1}^i,\hat{y}_{p_1,o_1}^i)^2]}_{\mathrm{adversarial\,loss}}}$$ Once $L_{p_1}$ is optimized, $I$ cross-sections in $o_1$ are separately recovered, and then reformatted with a concatenation block $f$ to generate the target volume: $${\hat{Y}_{p_1}={f(\hat{y}_{p_1,o_1}^1,\hat{y}_{p_1,o_1}^2,\dots,\hat{y}_{p_1,o_1}^I)}}$$ Second Progression: ProvoGAN then learns a separate recovery model in $o_2$ to progressively enhance global and local context: $\hat{y}_{p_2,o_2}^j=G_{o_2}(x_{o_2}^j,\hat{y}_{p_1,o_2}^j)$, where $x_{o_2}^j-\hat{y}_{o_2}^j$ denote the $j$th cross-sections of the source and recovered target volumes in $o_2$, and $\hat{y}_{p_1,o_2}^j$ denotes the $j$th cross-section in $o_2$ of the previously recovered volume incorporated to further leverage priors. $G_{o_2}$ and $D_{o_2}$ are again trained in an adversarial setup.

Third Progression: ProvoGAN lastly performs a cross-sectional mapping in $o_3$ with a third generator $G_{o_3}$ receiving as input cross-sections of the source and previously recovered target volumes: $\hat{y}_{p_3,o_3}^k=G_{o_3}(x_{o_3}^k,\hat{y}_{p_2,o_3}^k)$, where $x_{o_3}^k-\hat{y}_{o_3}^k$ denote the $k$th cross-sections of the source and recovered target volumes in $o_3$. An adversarial setup is used in this progression. The final output of ProvoGAN is then generated by concatenating the recovered $K$ separate cross-sections in $o_3$: $${\hat{Y}_{p_3}={f(\hat{y}_{p_3,o_3}^1,\hat{y}_{p_3,o_3}^2,\dots,\hat{y}_{p_3,o_3}^K)}}$$ where $\hat{Y}_{p_3}$ denotes the final output volume recovered by ProvoGAN. In MRI reconstruction, consistency of the recovered and acquired k-space coefficients is ensured as follows: $$\mathcal{F}_u(\hat{Y}_{p_n}):=\mathcal{F}_u(X)$$ where $\mathcal{F}_u$ denotes the Fourier operator, and $n$ denotes the ongoing progression index.


Implementation Details

Each cross-sectional model in ProvoGAN contained a generator with $3$ conv-layers, $9$ ResNet blocks, and $3$ conv-layers, and a discriminator with $5$ conv-layers in series. Trainings were continued for $100$ epochs with a batch size of $1$. Learning rate was $0.0002$ in the first $50$ epochs, and linearly decreased to $0$ in the remaining epochs with the ADAM optimizer $(\beta_1=0.5,\,\beta_2=0.999)$. Comprehensive experiments were performed on the IXI dataset¹⁰ $(\mathrm{T_1}-,\,\mathrm{T_2}-$weighted images of $52$ subjects, training: $37$, validation: $5$, test: $10$) to demonstrate recovery quality of ProvoGAN on two synthesis tasks: $\mathrm{T_1}\rightarrow\mathrm{T_2}$ and $\mathrm{T_2}\rightarrow\mathrm{T_1}$ and two reconstruction tasks: $\mathrm{T_1\,(R=4,8)}\rightarrow\mathrm{T_1\,(R=1)}$ and $\mathrm{T_2\,(R=4,8)}\rightarrow\mathrm{T_2\,(R=1)}$, $R$ denoting acceleration rate. Optimal progression sequence of ProvoGAN was determined to be $(C\rightarrow~A\rightarrow~S)$ for $\mathrm{T_1}\rightarrow\mathrm{T_2}$, $(S\rightarrow~A\rightarrow~C)$ for $\mathrm{T_2}\rightarrow\mathrm{T_1}$, $(C\rightarrow~A\rightarrow~S)$ for $\mathrm{T_1\,(R=4)}\rightarrow\mathrm{T_1\,(R=1)}$, $(A\rightarrow~C\rightarrow~S)$ for $\mathrm{T_1\,(R=8)}\rightarrow\mathrm{T_1\,(R=1)}$, $(C\rightarrow~S\rightarrow~A)$ for $\mathrm{T_2\,(R=4)}\rightarrow\mathrm{T_2\,(R=1)}$, and $(A\rightarrow~C\rightarrow~S)$ for $\mathrm{T_2\,(R=8)}\rightarrow\mathrm{T_2\,(R=1)}$, where $A$ denotes axial, $C$ coronal, and $S$ sagittal orientations. ProvoGAN was evaluated against cross-sectional sGAN and volumetric vGAN models. The network architectures and hyperparameters of these models were adopted from a previous study successfully demonstrated for MRI recovery¹, where the 2D conv-layers used for sGAN were replaced with 3D ones for vGAN. Separate sGAN models were trained for each individual orientation in synthesis (sGAN-A: axial, sGAN-C: coronal, sGAN-S: sagittal), whereas a single sGAN model was trained for the transverse orientation (axial) in reconstruction.

Results

We first performed demonstrations on the synthesis tasks $(\mathrm{T_1}\rightarrow\mathrm{T_2}\,$and$\mathrm{T_2}\rightarrow\mathrm{T_1})\,$in IXI. Table 1 lists the volumetric PSNR$-$ measurements of the models under comparison, where ProvoGAN achieves $0.58\,dB$ higher PSNR and $\%2.32$ higher SSIM compared to the second-best method, on average. Representative results are displayed in Fig. 2 for $\mathrm{T_1}$-weighted image synthesis.

Next, we demonstrated the reconstruction quality of ProvoGAN in IXI $(\mathrm{T_1\,(R=4,8)}\rightarrow\mathrm{T_1\,(R=1)}\,$and$\,\mathrm{T_2\,(R=4,8)}\rightarrow\mathrm{T_2\,(R=1)})$. PSNR$-$SSIM measurements reported in Table 2 for the reconstruction tasks indicate that ProvoGAN yields, on average, $1.78\,dB$ higher PSNR, and $\%4.61$ higher SSIM compared to the second-best method. Representative results are displayed in Fig. 3 for $\mathrm{T_1}$-weighted image reconstruction.

The overall results suggest that sGAN suffers from suboptimal recovery in the longitudinal dimensions due to loss of contextual consistency, whereas vGAN suffers from poor recovery of fine-structural details and deteriorated resolution due to expanded model complexity with suboptimal learning behaviour. Meanwhile, ProvoGAN advances capture of global consistency in all rectilinear orientations via progressively volumetrized cross-sectional mappings and offers enhanced recovery of fine-structural details by manifesting reduced complexity.

Discussion

Here, we introduced a progressively volumetrized model (ProvoGAN) for image recovery in MRI. ProvoGAN decomposes volumetric image recovery tasks into a task-optimally ordered series of simpler cross-sectional mappings which empowers ProvoGAN to leverage global contextual information and enhance fine-structural details in each orientation while maintaining reduced complexity.

Conclusion

The proposed method holds a great promise to advance practicality and utility of MR image recovery.