On instabilities of deep learning in image reconstruction and the potential costs of AI

Abstract

Deep learning, due to its unprecedented success in tasks such as image classification, has emerged as a new tool in image reconstruction with potential to change the field. In this paper, we demonstrate a crucial phenomenon: Deep learning typically yields unstable methods for image reconstruction. The instabilities usually occur in several forms: 1) Certain tiny, almost undetectable perturbations, both in the image and sampling domain, may result in severe artefacts in the reconstruction; 2) a small structural change, for example, a tumor, may not be captured in the reconstructed image; and 3) (a counterintuitive type of instability) more samples may yield poorer performance. Our stability test with algorithms and easy-to-use software detects the instability phenomena. The test is aimed at researchers, to test their networks for instabilities, and for government agencies, such as the Food and Drug Administration (FDA), to secure safe use of deep learning methods.

Keywords: AI; deep learning; image reconstruction; instability; inverse problems.

Fig. 1.

Perturbations $r_j$ (created to simulate worst-case effect) with $\left|r_1\right|<\left|r_2\right|<\left|r_3\right|$ are added to the image $x$. (Top) Images 1 to 4 are original image $x$ and perturbations $x+r_j$. (Bottom) Images 1 to 4 are reconstructions from $A\left(x+r_j\right)$ using the Deep MRI (DM) network $f$, where $A$ is a subsampled Fourier transform (33% subsampling); see Methods and SI Appendix for details. (Top and Bottom) Image 5 is a reconstruction from $Ax$ and $A\left(x+r_3\right)$ using an SoA method; see Methods for details. Note how the artifacts (red arrows) are hard to dismiss as nonphysical.

Fig. 2.

A random Gaussian vector $e\in {\mathbb{C}}^m$ is computed by drawing (the real and imaginary part of) each component independently from the normal distribution $\mathcal{N}\left(\mathrm{0,10}\right)$. We let $v=A*e$, and rescale $v$ so that $\Vert {v\Vert }_2=\frac{1}{4}\Vert {r1\Vert }_2$, where $r_1$ is the perturbation from Fig. 1. The Deep MRI network $f$ reconstructs from the measurements $A\left(x+r_1+v\right)$ and shows the same artifact as was seen for $r_1$ in Fig. 1. Note that, in this experiment, $A\in {\mathbb{C}}^{m\times N}$ is a subsampled normalized discrete Fourier transform (33% subsampling), so that $AA*=I$ that is, $e=Av$.

Fig. 3.

Perturbations ${\stackrel{\sim }{r}}_j$ (created to simulate worst-case effect) are added to the measurements $y=Ax$, where $\left|{\stackrel{\sim }{r}}_1\right|<\left|{\stackrel{\sim }{r}}_2\right|<\left|{\stackrel{\sim }{r}}_3\right|<\left|{\stackrel{\sim }{r}}_4\right|$, and $A$ is a subsampled Fourier transform (60% subsampling). To visualize, we show $|x+r_j|$, where $y+{\stackrel{\sim }{r}}_j=A\left(x+r_j\right)$. (Top) Original image $x$ with perturbations $r_j$. (Middle) Reconstructions from $A\left(x+r_j\right)$ by the AUTOMAP network $f$. (Bottom) Reconstructions from $A\left(x+r_j\right)$ by an SoA method (see Methods for details). A detail in the form of a heart, with varying intensity, is added to visualize the loss in quality.

Fig. 4.

(Top) Perturbations $r_1,r_2$ (created to simulate worst-case effect) are added to the images $x$ and $\stackrel{\sim }{x}$. (Middle) The reconstructions by the network $f$ (MRI-VN), from $Ax$ and $A\left(x+r_1\right)$, and the network $\stackrel{\sim }{f}$ (MED 50), from $Ã\stackrel{\sim }{x}$ and $Ã\left(\stackrel{\sim }{x}+r_2\right)$. $A$ is a subsampled discrete Fourier transform, and $Ã$ is a subsampled Radon transform. (Bottom) SoA comparisons. Red arrows are added to highlight the instabilities.

Fig. 5.

(Top, Upper Middle, Middle, and Lower Middle) Images $x_j$ plus structured perturbations $r_j$ (in the form of text and symbols) are reconstructed from measurements $y_j=A_j\left(x_j+r_j\right)$ with neural networks $f_j$ and SoA methods. The networks are $f_1=\text{Ell}50$, $f_2=\text{DAGAN}$, $f_3=\text{MRI}-\text{VN}$, and $f_4=\text{Deep MRI}$. The sampling modalities $A_j$ are as follows: $A_1$ is a subsampled discrete Radon transform, $A_2$ is a subsampled discrete Fourier transform (single coil simulation), $A_3$ is a superposition of subsampled discrete Fourier transforms (parallel MRI simulation with 15 coils elements), and $A_4$ is a subsampled discrete Fourier transform (single coil). Note that Deep MRI has not been trained with images containing any of the letters or symbols used in the perturbation, yet it is completely stable with respect to the structural changes. The same is true for the AUTOMAP network (see first column of Fig. 3). (Bottom) PSNR as a function of the subsampling rate for different networks. The dashed red line indicates the subsampling ratio that the networks were trained for.