Complex-valued neural network with learnable non-linearities in medical imaging
For machine training and application of a trained complex-valued machine learning model, an activation function of the machine learning model, such as a neural network, includes a learnable parameter that is complex or defined in a complex domain with two dimensions, such as real and imaginary or magnitude and phase dimensions. The complex learnable parameter is trained for any of various applications, such as MR fingerprinting, other medical imaging, or non-medical uses.
1. A method for application of a complex-valued neural network for a medical imaging system, the method comprising:
scanning an internal region of a patient by the medical imaging system;
applying the complex-valued neural network to scan data from the scanning and representing the internal region of the patient, the complex-valued neural network including one or more learned, complex-valued activation functions; and
displaying an image having an indication output by the complex-valued neural network from the applying.
2. The method of claim 1 wherein scanning comprises scanning with the medical imaging system comprising a magnetic resonance scanner, wherein applying comprises inputting the scan data as complex values to the complex-valued neural network, and wherein displaying the image comprises displaying a magnetic resonance image.
3. The method of claim 2 wherein applying comprises applying with the complex-valued neural network having been trained for outputting values of multiple parameters for magnetic resonance fingerprinting, and wherein displaying the image comprises displaying the image for one of the parameters and displaying another image for another of the parameters.
4. The method of claim 1 wherein applying comprises applying with the complex-valued neural network having been trained for flow, Dixon, fingerprinting, or processing in k-space.
5. The method of claim 1 wherein applying comprises applying with the one or more learned, complex-valued activation functions each comprising a learned parameter for a relationship between real and imaginary components or between magnitude and phase components.
6. The method of claim 1 wherein applying comprises applying with the one or more learned, complex-valued activation functions comprising learned non-linearities.
7. The method of claim 1 wherein applying comprises applying with the one or more leaned, complex-valued activation functions comprising a machine-learned parameter in a two-dimensional complex grid.
8. The method of claim 7 wherein applying comprises applying with the machine-learned parameter comprising a machine-learned angle as a bias term for rotation in phase of a Cardioid function.
9. The method of claim 7 wherein applying comprises applying with the machine-learned parameter comprising a machine-learned mixing coefficient with shifts in the two-dimensional complex grid, a machine-learned kernel with two-dimensional vectors in the two-dimensional complex grid, or both in a kernel activation function.
10. The method of claim 9 wherein applying comprises applying the machine-learned mixing coefficient with shifts in the two-dimensional complex grid and the machine-learned kernel with two-dimensional vectors in the two-dimensional complex grid in the kernel activation function as a bivariate kernel activation function.
11. The method of claim 7 wherein applying comprises applying with the machine-learned parameter comprising a machine-learned parameter comprising a separable kernel of a kernel activation function, the separable kernel having a polar representation for complex numbers.
12. The method of claim 1 wherein applying the complex-valued neural network comprises applying a fully connected, dropout, batch-normalization, multi-dimensional convolution, average pooling, magnitude-max pooling, or magnitude transformation neural network.
13. The method of claim 1 wherein applying comprises applying with the one or more leaned, complex-valued activation functions comprising a ReLU with a learned rotation.
14. A medical imaging system for operating on complex-valued data, the medical imaging system comprising:
a medical scanner configured to scan a patient and generate the complex-valued data from the scan;
an image processor configured to apply the complex-valued data to a machine-learned model, the machine-learned model including a two-dimensional activation function with two dimensions being real and imaginary or magnitude and phase, at least one learned parameter of the two-dimensional activation function learned to relate between the two dimensions; and
a display configured to display a medical image from an output of the application.
15. The medical imaging system of claim 14 wherein the two-dimensional activation function comprises a Cardioid activation function, and the at least one learned parameter comprises a rotation in the two dimensions.
16. The medical imaging system of claim 14 wherein the two-dimensional activation function comprises a kernel activation function, and the at least one learned parameter comprises a mixing coefficient in both of the two dimensions, a kernel in the two dimensions, or both.
17. The medical imaging system of claim 14 wherein the two-dimensional activation function comprises a kernel activation function, and the at least one learned parameter comprises a separable kernel in a polar representation.
18. A method for machine training in a complex-valued neural network, the method comprising:
defining the complex-valued neural network with a Cardioid or kernel activation function, the Cardioid or kernel activation function having a learnable parameter in a real and imaginary or magnitude and phase grid;
machine training, by a machine, the complex-valued neural network, the machine training including training the learnable parameter; and
storing the neural network as trained.
19. The method of claim 18 wherein defining comprises defining the Cardioid or kernel activation function as the Cardioid activation function with the learnable parameter comprising a rotation in the grid.
20. The method of claim 18 wherein defining comprises defining the Cardioid or kernel activation function as the kernel activation function with the learnable parameter comprising grid shifts in both directions of the grid, variance in both the directions, or both.