Patent Publication Number: US-11042803-B2

Title: Method and apparatus for using generative adversarial networks in magnetic resonance image reconstruction

Description:
BACKGROUND 
     In general, magnetic resonance imaging (MRI) examinations are based on the interactions among a primary magnetic field, a radiofrequency (RF) magnetic field, and time varying magnetic gradient fields with gyromagnetic material having nuclear spins within a subject of interest, such as a patient. Certain gyromagnetic materials, such as hydrogen nuclei in water molecules, have characteristic behaviors in response to external magnetic fields. The precession of spins of these nuclei can be influenced by manipulation of the fields to produce RF signals that can be detected, processed, and used to reconstruct a useful image. 
     In general, a neural network may be used to reconstruct the images produced by an MRI system. However, some neural networks use loss functions that incorporate pixel-wise distance to learn to reconstruct the images, which may lead to blurred images. Further, although generally producing sharper images, generative adversarial networks, may, in some scenarios take longer to train, and can sometimes produce image artifacts or unrealistic images, especially when applied to smaller-sized datasets. 
     BRIEF DESCRIPTION 
     In one embodiment, a method of reconstructing imaging data into a reconstructed image may include training a generative adversarial network (GAN) to reconstruct the imaging data. The GAN may include a generator and a discriminator. Training the GAN may include determining a combined loss by adaptively adjusting an adversarial loss based at least in part on a difference between the adversarial loss and a pixel-wise loss. Additionally, the combined loss may be a combination of the adversarial loss and the pixel-wise loss. Training the GAN may also include updating the generator based at least in part on the combined loss. The method may also include receiving, into the generator, the imaging data and reconstructing, via the generator, the imaging data into a reconstructed image. 
     In another embodiment, a tangible, non-transitory, machine-readable medium with machine-executable instructions which, when executed by at least one processor of a machine, may cause the processor to train a generative adversarial network (GAN) to reconstruct input image data. The GAN may include a generator and a discriminator, and training of the GAN may include updating the generator based at least in part on a combined loss. Additionally, the combined loss may include a combination of an adversarial loss and a pixel-wise loss. The instructions may also cause the processor to receive, via the generator, the input image data, and reconstruct, the input image data into an output image. 
     In another embodiment, a method of training a generator network may include determining, over each iteration of multiple iterations, an adaptive combined loss. The adaptive combined loss may include a pixel-wise loss and an adversarial loss. Additionally, the pixel-wise loss may correspond to a direct comparison between a generated image, generated by the generator network, and a ground truth image, and the adversarial loss may correspond to an estimated comparison between the generated image and the ground truth image, as estimated by a discriminator network. The adversarial loss may be adaptively adjusted based on a ratio between the pixel-wise loss and the adversarial loss such that the ratio is below a threshold. The method may also include updating, during each iteration, the generator network based at least in part on the adaptive combined loss such that, in subsequent iterations, the updated generator network is more likely to generate a subsequent generated image that better represents a subsequent ground truth image than the generated image represented the ground truth image. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       These and other features, aspects, and advantages of the present invention will become better understood when the following detailed description is read with reference to the accompanying drawings in which like characters represent like parts throughout the drawings, wherein: 
         FIG. 1  illustrates a magnetic resonance imaging (MRI) system having a scanner and an image reconstruction unit, in accordance with an aspect of the present disclosure; 
         FIG. 2  is a schematic diagram of a neural network architecture for use in image reconstruction, in accordance with an aspect of the present disclosure; 
         FIG. 3  is a flowchart of an example process for training a generative adversarial network (GAN) and reconstructing an image, in accordance with an aspect of the present disclosure; 
         FIG. 4  is a flowchart of an example process for determining an adaptive combined loss from an adversarial loss and a pixel-wise loss, and implementing the adaptive combined loss in GAN training, in accordance with an aspect of the present disclosure; 
         FIG. 5  is an example pseudocode depicting the adjustment to the adversarial loss, in accordance with an aspect of the present disclosure; 
         FIG. 6  is a Frèchet Inception Distance comparison during training, as shown on a validation dataset, of different implementations of GANs utilizing a WGAN, a cWGAN, and a cWGAN with an adaptive combined loss, in accordance with an aspect of the present disclosure; and 
         FIG. 7  is a Normalized Mean Square Error comparison during training, as shown on a validation dataset, of different implementations of GANs utilizing a WGAN, a cWGAN, and a cWGAN with an adaptive combined loss, in accordance with an aspect of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     One or more specific embodiments will be described below. In an effort to provide a concise description of these embodiments, all features of an actual implementation may not be described in the specification. It should be appreciated that in the development of any such actual implementation, as in any engineering or design project, numerous implementation-specific decisions must be made to achieve the developers&#39; specific goals, such as compliance with system-related and business-related constraints, which may vary from one implementation to another. Moreover, it should be appreciated that such a development effort might be complex and time consuming, but would nevertheless be a routine undertaking of design, fabrication, and manufacture for those of ordinary skill having the benefit of this disclosure. 
     When introducing elements of various embodiments of the present disclosure, the articles “a,” “an,” “the,” and “said” are intended to mean that there are one or more of the elements. The terms “comprising,” “including,” and “having” are intended to be inclusive and mean that there may be additional elements other than the listed elements. Furthermore, any numerical examples in the following discussion are intended to be non-limiting, and thus additional numerical values, ranges, and percentages are within the scope of the disclosed embodiments. 
     In general, magnetic resonance imaging (MRI) is based on the interactions of a primary magnetic field, time varying magnetic gradient fields, and a radiofrequency (RF) field with gyromagnetic material within a subject of interest (e.g., a patient). Certain gyromagnetic materials, such as hydrogen nuclei in water molecules, have characteristic behaviors in response to external electromagnetic fields (e.g., constant or time varying electric fields, magnetic fields, or a combination thereof). The precession of spins of these nuclei can be influenced by manipulation of the fields to produce RF signals that can be detected, processed, and used to reconstruct a useful image. 
     Coil data from the RF signals may be under sampled and/or include artifacts or blurring, for example, due to patient motion during imaging. Furthermore, under sampled coil data may be obtained deliberately to increase the speed at which coil data may be taken and images produced. In some scenarios, quicker image capture may prove less prone to movement between subsequent coil data capture and lead to less blurring of a rendered image. Additionally, quicker image capture may also result in faster imaging procedures, which may increase efficiency and/or patient comfort. However, under sampled or artifact ridden coil data may produce inaccurate, blurred, or artifact ridden images when reconstructed. As such, it is now recognized that a need exists for a reconstruction module capable of accurately and efficiently reconstructing coil data into a displayable image. 
     In some embodiments, image reconstruction of coil data may be accomplished, for example, via a neural network, to provide a higher quality image (e.g., higher quality than images produced from a basic reconstruction). However, in some scenarios deep neural network reconstruction models may struggle to reconstruct sharp images with fine detail while maintaining a natural appearance. As such, in some embodiments, the neural network may be trained to identify which types of characteristics of the image are expected to be present and how to more accurately reconstruct an image from under sampled, noisy, or artifact ridden coil data. 
     In some embodiments, the neural network may be trained using a generative adversarial network (GAN), which, in some embodiments, may be implemented in a reconstruction module. Furthermore, the reconstruction module may be implemented, for example, via executable code (e.g., software) stored in memory of a computer, which may include one or more processors to execute the code. In general, a GAN utilizes a generator network and a discriminator network as part of a supervised machine learning algorithm. The generator network may produce a generated image from under sampled, noisy, or artifact ridden coil data to estimate a true image and the discriminator network may receive the generated image and the true image and attempt to determine which, of the generated image and the true image, is, in fact, the true image. In learning how to generate accurate image representations and determine the difference between true images and generated images, the generator network and the discriminator network may be balanced such that each learns at a similar rate as the other. In other words, the generator network may attempt to fool the discriminator network by trying to reconstruct the under sampled coil data into an image that appears real, and the discriminator network may learn, over subsequent iterations, how to better determine which image is real and which is generated by the generator network. As discussed herein, learning may refer to updating the code (e.g., updating weighting factors within the neural networks) of a neural network, implemented, for example, on a computer. 
     The generator network may be updated after each iteration of the learning process by the discriminator with an adversarial loss. With each update, the adversarial loss may be used by the generator network to learn how to better fool the discriminator network. However, in general, GANs may incur instabilities in training, and, when applied to small datasets (e.g., less than 50, less than 100, or less than 500 samples), may produce image artifacts (e.g., jagged edges, blurring, and/or inaccuracies), which may be undesirable in radiological settings. To assist in overcoming such instabilities, preserve fine detail, and/or maintain a natural appearance the adversarial loss may be combined with a pixel-wise loss indicative of the differences between the generated image and the true image. Such combination may be accomplished by a weighted sum and/or by adaptive loss balancing. In some embodiments, the adaptive loss balancing may help the GAN converge to higher quality generated images in fewer epochs (e.g., iterations across the entire training dataset) than with a simple weighted sum. 
     As set forth above, the embodiments described herein may be implemented as a part of an MRI system, wherein specific imaging routines are initiated by a user (e.g., a radiologist). Thus, the system may perform data acquisition, data reconstruction, and in certain instances, image synthesis. Accordingly, referring to  FIG. 1 , an imaging system  10  is illustrated schematically as including a scanner  12 , scanner control circuitry  14 , and system control circuitry  16 . 
     The imaging system  10  additionally includes remote access and storage systems  18  and/or devices such as picture archiving and communication systems (PACS), or other devices such as teleradiology equipment so that data acquired by the imaging system  10  may be accessed on- or off-site. In this way, MRI data may be acquired, followed by on- or off-site processing and evaluation. While the imaging system  10  may include any suitable scanner or detector, in the illustrated embodiment, the imaging system  10  includes a full body scanner  12  having a housing  20  through which an opening (e.g., an annular opening) is formed to accommodate a bore tube  22 . The bore tube  22  may be made of any suitable material such as a non-metallic and/or non-magnetic material. A table  24  is moveable into the bore tube  22  to permit a patient  26  to be positioned therein for imaging selected anatomy within the patient. In some embodiments, the bore tube  22  may surround an entire subject or just a portion thereof (e.g., a patient&#39;s head, thorax, or extremity). In some embodiments, the bore tube  22  may support the table  24  and/or articulation components (e.g., a motor, pulley, and/or slides). 
     The scanner  12  may include a series of associated conductive coils for producing controlled electromagnetic fields for exciting the gyromagnetic material within the anatomy of the subject being imaged. Specifically, primary magnet coils  28  are provided for generating a primary magnetic field, which is generally aligned with the bore tube  22 . The primary magnetic coils  28  may be made of a superconductor, which during operation, may generate the primary magnetic field to strengths greater than 1 Tesla. A coil support structure  30  may support the primary magnetic coils  28  and maintain their position within the scanner  12  under the forces sustained during operation. 
     A series of gradient coils  32 ,  34 , and  36  (collectively  38 ) permit controlled magnetic gradient fields to be generated for positional encoding of certain of the gyromagnetic nuclei within the patient  26  during examination sequences. Additionally, an RF coil  40  may generate radio frequency pulses for exciting the certain gyromagnetic nuclei within the patient  26 . In addition to the coils that may be local to the scanner  12 , the imaging system  10  may also include a set of receiving coils  42  (e.g., an array of coils) to be placed proximal to (e.g., against) the patient  26 . As an example, the receiving coils  42  can include cervical/thoracic/lumbar (CTL) coils, head coils, single-sided spine coils, and so forth. Generally, the receiving coils  42  are placed close to or on top of the patient  26  so as to receive the weak RF signals (e.g., weak relative to the transmitted pulses generated by the scanner coils) that are generated by certain of the gyromagnetic nuclei within the patient  26  as they return to their relaxed state. In some embodiments, the RF coils  40  may both transmit and receive RF signals accomplishing the role of the receiving coils  42 . 
     The various coils of the imaging system  10  may be situated within the housing  20  of the scanner  12 , and are controlled by external circuitry to generate the desired field and pulses, and to read emissions from the gyromagnetic material in a controlled manner. In the illustrated embodiment, a main power supply  44  provides power to the primary magnetic coils  28  to generate the primary magnetic field. A driver circuit  50  may include amplification and control circuitry for supplying current to the coils as defined by digitized pulse sequences output by the scanner control circuitry  14 . 
     An RF control circuit  52  is provided for regulating operation of the RF coil  40 . The RF control circuit  52  includes a switching device for alternating between the active and inactive modes of operation, wherein the RF coil  40  transmits and does not transmit signals, respectively. The RF control circuit  52  may also include amplification circuitry to generate the RF pulses. Similarly, the receiving coils  42 , or RF coils  40  if no separate receiving coils  42  are implemented, are connected to a switch  54 , which is capable of switching the receiving coils  42  between receiving and non-receiving modes. Thus, the receiving coils  42  may resonate with the RF signals produced by relaxing gyromagnetic nuclei from within the patient  26  while in the receiving mode, and avoid resonating with RF signals while in the non-receiving mode. Additionally, a receiving circuit  56  may receive the data detected by the receiving coils  42  and may include one or more multiplexing and/or amplification circuits. 
     It should be noted that while the scanner  12  and the control/amplification circuitry described above are illustrated as being connected by single lines, one or more cables or connectors may be used depending on implementation. For example, separate lines may be used for control, data communication, power transmission, and so on. Further, suitable hardware may be disposed along each type of line for the proper handling of the data and current/voltage. Indeed, various filters, digitizers, and processors may be disposed between the scanner  12  and the scanner control circuitry  14  and/or system control circuitry  16 . 
     As illustrated, the scanner control circuitry  14  includes an interface circuit  58 , which outputs signals for driving the gradient field coils  38  and the RF coil  40  and for receiving the data representative of the magnetic resonance signals produced in examination sequences. The interface circuit  58  may be connected to a control and analysis circuit  60 . The control and analysis circuit  60  executes the commands to the driver circuit  50  and the RF control circuit  52  based on defined protocols selected via system control circuitry  16 . 
     The control and analysis circuit  60  may also serve to receive the magnetic resonance signals and perform subsequent processing before transmitting the data to system control circuitry  16 . Scanner control circuitry  14  may also include one or more memory circuits  62 , which store configuration parameters, pulse sequence descriptions, examination results, and so forth, during operation. 
     A second interface circuit  64  may connect the control and analysis circuit  60  to a system control circuit  66  for exchanging data between scanner control circuitry  14  and system control circuitry  16 . The system control circuitry  16  may include a third interface circuit  68 , which receives data from the scanner control circuitry  14  and transmits data and commands back to the scanner control circuitry  14 . As with the control and analysis circuit  60 , the system control circuit  66  may include a computer processing unit (CPU) in a multi-purpose or application specific computer or workstation. System control circuit  66  may include or be connected to a second memory circuit  70  to store programming code for operation of the imaging system  10  and to store the processed coil data for later reconstruction, display and transmission. The programming code may execute one or more algorithms that, when executed by a processor, are configured to perform reconstruction of acquired data. 
     An additional input output (I/O) interface  72  may be provided for exchanging coil data, configuration parameters, and so forth with external system components such as remote access and storage systems  18 . Finally, the system control circuit  66  may be communicatively coupled to various peripheral devices for facilitating an operator interface and for producing hard copies of the reconstructed images. In the illustrated embodiment, these peripherals include a printer  74 , a monitor  76 , and a user interface  78  including, for example, devices such as a keyboard, a mouse, a touchscreen (e.g., integrated with the monitor  76 ), and so forth. 
     In some embodiments, a reconstruction module  80  may be implemented to reconstruct coil data into a viewable image. As discussed above, in some embodiments, image reconstruction of coil data may be accomplished, for example, via a neural network to provide a sharper and more detailed image than would be produced from a basic reconstruction. As such, the reconstruction module  80  may include a generator network  82  (e.g., a convolutional neural network) to estimate a fully sampled image from under sampled coil data with minimal artifacts or blurring. Furthermore, the reconstruction module  80  may be implemented as software to be executed by one or more processors of a computer system. Additionally or alternatively, the reconstruction module may include a computer, including memory and one or more processors, on which software modules may be run. 
     In some embodiments, the reconstruction module  80  may include a GAN  84  to teach the generator network  82  characteristics of real images in the subject matter of interest. For example, the generator network  82  may be trained to expect certain image features (e.g., pictorial representations of the subject of an MRI scan) and interpolate the noisy or under sampled coil data to provide an estimation of a fully sampled image. As such, the generator network  82  may be conditioned to expect certain elements in the coil data and learn how to more accurately reconstruct an image from under sampled, noisy, or artifact ridden coil data. 
     To help teach the generator network  82  the GAN  84  may also include a discriminator network  86  (e.g., a convolutional neural network). The generator network  82  and the discriminator network  86  may teach each other as part of a supervised machine learning algorithm. As should be appreciated, the reconstruction module  80 , the generator network  82 , and/or the discriminator network  86  may be integrated, at least partially, into the imaging system  10  (e.g., in the scanner control circuitry  14 , system control circuitry, and/or remote access and storage systems  18 ) or be implemented separately from the imaging system  10 , for example, as part of stand-alone software or a stand-alone computer. For example, the reconstruction module  80  may receive coil data from an interface circuit  58 ,  64 ,  68 ,  72  of the scanner control circuitry  14  or the system control circuitry  16  or the storage system  18  such as PACS, and the generator network  82  of the reconstruction module  80  may reconstruct an image of the scanned subject (e.g., the patient  26 ). Additionally, reconstructed images may be sent, for example, to the storage system  18  and/or to the monitor  76  for viewing. Moreover, in some embodiments, the generator network  82  of the GAN  84  may be trained with the discriminator network  86 , and implemented separately from the discriminator network  86  for use with the imaging system  10 . For example, the generator network  82  may be trained with use of a GAN  84  and subsequently transferred to and implemented by memory of a reconstruction module. 
       FIG. 2  is a schematic diagram of the neural network architecture of the GAN  84  for use in training the generator network  82  to produce improved (e.g., relative to basic reconstructions) reconstructed images. During training, the GAN  84  may receive data from an under sampled dataset  88  and a fully sampled dataset  90 . The under sampled dataset  88  and fully sampled dataset  90  may stem from a common group of training images, such that each under sampled image corresponds to a fully sampled image. The training images may be derived from, for example, coil data from the storage system  18 , the scanner control circuitry  14 , and/or the system control circuitry  16 . The under sampled dataset  88  may correspond to or be indicative of data produced from the scanner  12  such as coil data. For example, the under sampled dataset  88  may be similar to under sampled coil data desired to be reconstructed that does not already have a fully sampled counterpart. Additionally, the fully sampled dataset  90  may have associated ground truth images  92  which may be of high fidelity (e.g., minimal or no artifacts or blurring) and natural in appearance. Moreover, the ground truth images  92  may be indicative of radiological images at the high fidelity that the generator network  82  is trying to replicate from the under sampled dataset  88 . 
     In one embodiment, the generator network  82  of the GAN  84  may receive an under sampled image, or coil data indicative thereof, from the under sampled dataset  88  and produce a generated image  94  therefrom. In some embodiments, the under sampled image may be zero filled (e.g., missing or unavailable sections of data may be written as logical zeros). Subsequently, the discriminator network  86  may approximate an earth mover&#39;s distance (EMD) or other suitable metric between the generated image  94  and the ground truth image  92  and estimate which of the images is indeed the ground truth image  92 . The EMD may represent a measure of the distance between two probability distributions over a region or space. From the EMD estimation, a loss function, for example an adversarial loss  96 , may be generated and fed back into the generator network  82 . Additionally, in some embodiments, a pixel-wise loss  98  may be produced and combined with the adversarial loss  96  to further the training of the generator network  82  toward production of realistic and high fidelity images. As should be appreciated, the adversarial loss  96  and/or pixel-wise loss  98  may be determined by the generator network  82 , the discriminator network, or exterior to the networks, such as by a training monitor portion of the GAN  84 . Further, the discriminator network  86  may be updated, for example based on the adversarial loss or a discriminator loss, such that, in subsequent iterations, the discriminator network  86  may better estimate the EMD between the generated image  94  and the ground truth image  92  and, thus, more accurately determine which image is, in fact, the ground truth image  92 . 
     Furthermore, the GAN  84  may be any suitable type of GAN, such as a Wasserstein GAN (WGAN), a conditional GAN (cGAN), or a combination thereof (e.g., a conditional Wasserstein GAN (cWGAN)). As illustrated, in some embodiments, the GAN  84  may be a conditional generative adversarial network (cGAN). When implemented as a cGAN, pairs of images  100 , or a set of any suitable number of images, may be supplied to the discriminator network  86  for evaluation. For example, the generator network  82  may provide the generated image  94  and a conditioning image  102  to the discriminator network  86 . Likewise, the ground truth image  92  may also be accompanied by a conditioning image  102 . In some embodiments, when the discriminator network  86  receives pairs of images  100  (e.g., with a conditioning image  102 ) the discriminator network  86  is able to enforce a higher data fidelity compared to that of a non-conditional GAN. For example, the use of a conditioning image  102  may lead to perceptually better image appearance due to the discriminator network&#39;s ability to match a specific image (e.g., the conditioning image  102 ) to the generated image  94  instead of general perceptual properties of the entire dataset (e.g., the fully sampled dataset  90  and/or the under sampled dataset  88 ). Furthermore, in some embodiments, multiple generated images  94  or pairs of images  100  may be sent in a batch to the discriminator network  86  during a single step of the training loop. For example, the generator network  82  may receive ten under sampled images from the under sampled dataset  88 , generate ten corresponding images, and the discriminator network may then estimate a score for each image, or pair of images  100 , separately, before the generator network  82  and/or the discriminator network  86  is updated. 
     Additionally, the generator network  82  may be any suitable type of image reconstruction network. For example, the generator network  82  may reconstruct images based on under sampled coil data and be or include a deep learning-based network, a sparse reconstruction network, and/or a convolutional neural network. In some embodiments, the generator network  82  may be a densely connected iterative network (DCI network). Further, in some embodiments, the DCI network may include dense connections across multiple network layers, which may strengthen feature propagation. Moreover, the DCI network may include a relatively deep architecture with multiple convolution layers (e.g., greater than 10, greater than 50, greater than 100, and so on) facilitating increased capacity. Furthermore, the discriminator network  86  may be any suitable type of discriminator network for use in GAN training. For example, the discriminator network  86  may be a convolutional network. 
     As discussed above, to update the generator network  82  and/or the discriminator network  86 , the adversarial loss  96  may be created based on the discriminator network&#39;s estimation of the EMD between the generated image  94  and the ground truth image  92 . As each network (e.g., the generator network  82  and the discriminator network  86 ) is updated, each network is trained against the other. For example, EQ. 1 illustrates an example GAN objective function,
 
   cWGAN ( G,D )=   x,y   [D ( x,y )]−   x   [D ( x,G ( x ))],  EQ. 1
 
where   is the expected value, G is a function of the generator network  82 , D is a function of the discriminator network  86 , x is the under sampled image, and y is the ground truth image. While the generator network  82  tries to minimize EQ. 1, the discriminator network  86  tries to maximize it, and, in some embodiments, the adversarial loss  96  may be given by the residual, for example, as in EQ. 2,
 
     
       
         
           
             
               
                 
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     Upon iteration of the training cycle, the generator network  82  may be trained how to trick the discriminator network  86  into determining that the generated image  94  is the ground truth image  92 . As such, the adversarial loss  96  may dynamically change during training from one iteration to the next. Simultaneously, throughout the iterations, the discriminator network  86  may learn how to better decipher which of the generated images  94  and the ground truth images  92  are, in fact, ground truth images  92 . However, in some scenarios, updating the GAN  84  based on just the adversarial loss  96  may lead to training instabilities. For example, the dynamically changing adversarial loss  96  may generate gradients with a variable norm. Therefore, the supervised machine learning process may deviate away from the ground truth images  92  spatially and, thus deviate from realistic looking images even while reducing the EMD estimated by the discriminator network  86 . As such, the training of the GAN  84  may result in perceivable artifacts appearing in the generated images  94 . To provide for more accurate generated images, the adversarial loss  96  and pixel-wise loss  98  may be combined, for example via a weighted sum, to update the generator network  82  such that subsequent iterations of image generation have a higher likelihood of reducing the EMD estimated by the discriminator network  86  and producing more realistic and higher quality images. 
     The pixel-wise loss  98  may include any suitable metric for directly comparing the generated image  94  and the ground truth image  92 . For example, the pixel-wise loss  98  may correspond to a mean-square-error (MSE) between the generated image  94  and the ground truth image  92 , for example, as given in EQ. 3, 
                         ℒ     M   ⁢   S   ⁢   E       ⁡     (   G   )       =       1   WH     ⁢         ∑   W       i   =   1       ⁢         ∑   H       j   =   1       ⁢       (       y     i   ,   j       -       G   ⁡     (   x   )         i   ,   j         )     2             ,           EQ   .           ⁢   3               
where W and H are the pixel width and pixel height of the images, respectively, and i and j are counting indices.
 
     In some embodiments, the combination of the adversarial loss  96  and the pixel-wise loss  98  may include a weighted sum, for example, as shown in EQ. 4, 
                       ℒ   G     =       arg   ⁢           ⁢       min   G     ⁢       max   D     ⁢       ℒ   cWGAN     ⁢     WGAN   ⁡     (     G   ,   D     )               +       λℒ     M   ⁢   S   ⁢   E       ⁡     (   G   )           ,           EQ   .           ⁢   4               
where λ is a weighting factor or function. The combined adversarial loss  96  and pixel-wise loss  98  may update the generator network  82  such that subsequent iterations of image generation have a higher likelihood of reducing the EMD estimated by the discriminator network  86  (e.g., from the adversarial loss  96 ) and producing a higher quality image that maintains the realism of the ground truth images  92  (e.g., via the pixel-wise loss  98 ).
 
     In some scenarios, a simple weighted sum may, eventually (e.g., over the training iterations), lead to a divergence between the adversarial loss  96  and the pixel-wise loss  98 . Further, the divergent losses may lead to instabilities in training of the GAN  84 . For example, the standard deviation of the adversarial loss  96 , or gradient thereof, may become one or more orders of magnitude greater than the pixel-wise loss  98 , or gradient thereof, which may cause the training of the GAN  84  to drift spatially away from the ground truth. As such, in one embodiment, the combined adversarial loss  96  and pixel-wise loss  98  may be adaptively weighted, for example, to maintain a moving average of the standard deviation of the gradients (MASDG) of the adversarial loss  96  less than a multiple (e.g., 0.5 times, 1 times, 5 times, 10 times, 50 times, 100 times, and so on depending on implementation) of the MASDG of the pixel-wise loss  98 . In one embodiment, the adversarial loss  96  may be reduced (e.g., divided) by a rate parameter, for example, via the reconstruction module  80 . Moreover, in some embodiments, the rate parameter may change or be adapted to each iteration of the GAN training, depending on the ratio between the adversarial loss  96  and the pixel-wise loss  98 . By maintaining the ratio of the MASDG of the adversarial loss to the MASDG of the pixel-wise loss  98  less than a threshold (e.g. 0.5, 1, 5, 10, 100), the training of the GAN  84  may be further stabilized and avoid drifting away from the ground-truth spatial information. 
     Additionally, in some scenarios, the reduction of the adversarial loss  96  (e.g., relative to a previous value) may lead to the discriminator network  86  learning faster than the generator network  82 , for example, due to the damped adversarial loss  96  supplied to the generator network  82 . As such, an adaptive loss balancing variable may be used by the GAN  84  to decay the loss updating the discriminator network  86 , for example, by the same or substantially the same (e.g., within 1 order of magnitude) rate parameter as the adversarial loss  96 . As such, the learning rate of the generator network  82  and discriminator network  86  may be approximately equal. 
       FIG. 3  is a flowchart of an example process  104 , implemented, for example, by the GAN  84 , for training the GAN  84  and reconstructing an image. In some embodiments, the generator network  82  may receive under sampled or noisy image data (e.g., from the under sampled dataset  88 ) (process block  106 ), and generate the generated image  94  from the under sampled or noisy image data (process block  108 ). Additionally, the discriminator network  86  may receive the generated image  94  and the ground truth image  92  or pairs of images  100 , each including the conditioning image  102  (e.g., a zero filled conditioning) and the generated image  94  or the ground truth image  92  (process block  110 ). The discriminator network  86  may estimate which of the images or sets of images is or contains the ground truth image  92  (process block  112 ). Additionally, the discriminator network  86  or other portion of the GAN  84  may determine the adversarial loss  96  (process block  114 ). Moreover, the pixel-wise loss  98  may also be determined (process block  116 ) by a portion of the GAN  84 , and an adaptively adjusted adversarial loss  96  may be determined, for example, from the pre-adjusted adversarial loss  96  (process block  117 ). Additionally, an adaptive combined loss may be determined from the adaptively adjusted adversarial loss  96  and the pixel-wise loss  98  (process block  118 ), for example, via the generator network  82 , discriminator network  86 , a training monitor portion of the GAN  84 , or a combination thereof. The discriminator network  86  may be updated based at least in part on the adaptively adjusted adversarial loss  96  (process block  119 ), and the generator network  82  may be updated based at least in part on the adaptive combined loss (process block  120 ). If the training is not complete, the process  104  may undergo multiple iterations. Further, if the training is complete, the generator network  82  may then be given under sampled or noisy image data and produce a reconstructed image (process block  122 ). 
       FIG. 4  is flowchart of an example process  124 , implemented, for example, by the GAN  84 , for determining the adaptive combined loss from the adversarial loss  96  and the pixel-wise loss  98 , and implementing the adaptive combined loss in training. The adversarial loss  96  may be evaluated to determine if the MASDG of the adversarial loss  96  is greater than a threshold relative to the MASDG of the pixel-wise loss  98  (process block  126 ). As discussed above, in some embodiments, the threshold may be a multiple of the MASDG of the pixel-wise loss  98 . The adversarial loss  96  may then be adaptively adjusted (e.g., reduced), for example, by a multiple of a rate parameter until the MASDG of the adversarial loss  96  is less than the threshold (process block  128 ). The adaptively adjusted adversarial loss  96  and the pixel-wise loss  98  may then be combined into an adaptive combined loss (process block  130 ), for example, via a weighed sum. The generator network may then be updated via the adaptive combined loss (process block  132 ). Additionally, a discriminator loss, generated, for example, via the discriminator network  86  or a training monitor portion of the GAN  84  to update the discriminator network  86 , may be adjusted to maintain adversarial learning between the generator network and the discriminator network at a similar rate (process block  134 ), for example, by reducing the discriminator loss by the rate parameter, and the discriminator network  86  may then be updated via the adjusted discriminator loss (process block  136 ). In some embodiments, the discriminator loss may be one-and-the-same with the adversarial loss  96 . As such, the discriminator network  86  may be updated based on the adaptively adjusted adversarial loss  96 . 
       FIG. 5  is an example pseudocode  138  used, for example, by the reconstruction module  80 , to adjust the adversarial loss  96 . As discussed above, in some embodiments, the MASDG of the adversarial loss  96  may be evaluated against a factor of the MASDG of the pixel-wise loss  98 , as shown in the pseudocode line  140 . If the MASDG of the adversarial loss  96  is greater than the threshold (e.g., the MASDG of the pixel-wise loss  98  multiplied by a threshold ratio  142  of the MASDG of the adversarial loss  96  to the MASDG of the pixel-wise loss  98 ), then the MASDG of the adversarial loss  96  may be adjusted by being reduced by a rate parameter  144 , as shown in the pseudocode line  146 . 
     Additionally, in some embodiments, the adaptive loss balancing variable  148  may be implemented to reduce the discriminator loss, and, thus, equalize the rates of learning of the discriminator network  86  and the generator network  82 , as shown in the pseudocode lines  150 ,  152 ,  154 . Moreover, the adaptive loss balancing variable  148  may be adjusted dynamically based on the adjustment to the adversarial loss  96  at the same rate or via a proportional rate parameter  144 , as shown in the pseudocode line  156 . 
       FIGS. 6 and 7  depict Frèchet Inception Distance (FID)  158  and Normalized Mean Square Error (NMSE)  160  comparisons, respectively, of different implementations of GANs  84  utilizing a WGAN  162 , a cWGAN  164 , and a cWGAN with an adaptive combined loss (ACL)  166  to compare the effectiveness and efficiency of different GANs  84 . The y-axis  168  represents the respective metric of the FID  158  and NMSE  160  comparisons, and the x-axis  170  represents the number of learning epochs completed by the respective GANs  84 . As depicted, in both the FID  158  and NMSE  160  comparisons the cWGAN with ACL  166  converges to a solution with less error and does so faster than the WGAN  162  or the cWGAN  164  without the ACL. As such, by implementing an adaptive combined loss, the GAN  84  may be able to train a generator network  82  to accurately reconstruct coil data into sharp, realistic, and high quality images. Moreover, the training of the generator network  82  may converge to produce better images faster (e.g., in less iterations) than other types of GANs  84 . 
     As should be appreciated, the adversarial loss  96 , the pixel-wise loss  98 , and/or the adjustments thereto or combinations thereof discussed herein may represent measurements, such as EMD or MSE, gradients or sums of the measurements, and/or standard deviations of the measurements or gradients, such as the MASDG. Additionally, although stated herein as generally applying to the reconstruction of under sampled magnetic resonance coil data, as should be appreciated, the techniques described herein such as the use of an adaptive combined loss may be utilized in the training of any suitable GAN, for construction or reconstruction of image data such as medical image data (e.g., computerized tomography (CT) image reconstruction) and/or non-medical image data. Furthermore, although the above referenced flowcharts of the processes  104 ,  124  are shown in a given order, in certain embodiments, the depicted steps may be reordered, altered, deleted, and/or occur simultaneously. Additionally, the referenced flowcharts of the processes  104 ,  124  are given as illustrative tools, and further decision and/or process blocks may be added depending on implementation. 
     This written description uses examples to disclose the invention, including the best mode, and also to enable any person skilled in the art to practice the invention, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the invention is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims.