Patent Publication Number: US-2022215600-A1

Title: Data-consistency for Image Reconstruction

Description:
RELATED APPLICATION 
     This application claims the benefit of EP 21150346, filed Jan. 5, 2021, which is hereby incorporated by reference in its entirety. 
     TECHNICAL FIELD 
     Various examples generally relate to image reconstruction. Various examples specifically relate to enforcing data consistency in connection with the image reconstruction. 
     BACKGROUND 
     Image reconstruction is used for various use cases. Examples include increasing an image resolution and reducing artifacts or noise included in an image. For example, image reconstruction can be used for medical image datasets. Image reconstruction can also be used for post-processing microscopy images or movie clips. 
     Various reconstruction algorithms are known. One example type of reconstruction algorithm employs machine learning. Such machine-learned reconstruction algorithms often show improved image quality compared to traditional reconstruction techniques, however, one of the limitations is the risk of hallucinating structures, or other stability issues such as e.g., an unpredicted behavior when an input is too different from what the network has been trained on, which is of special concern for medical image application. 
     For example, image reconstruction using generative adversarial networks (GANs) has been described in: Goodfellow, Ian; Pouget-Abadie, Jean; Mirza, Mehdi; Xu, Bing; Warde-Farley, David; Ozair, Sherjil; Courville, Aaron; Bengio, Yoshua (2014). Generative Adversarial Networks. Proceedings of the International Conference on Neural Information Processing Systems (NIPS 2014). pp. 2672-2680. 
     Other examples include black-box networks such as AUTOMAP, see B. Zhu, J. Z. Liu, S. F. Cauley, B. R. Rosen and M. S. Rosen, ‘Image reconstruction by domain-transform manifold learning’, Nature, vol. 555, no. 7697, p. 487, March 2018. Unrolled networks have been used to reconstruct magnetic-resonance imaging datasets, see Antun, V., Renna, F., Poon, C., Adcock, B., &amp; Hansen, A. C. (2019). On instabilities of deep learning in image reconstruction-Does AI come at a cost?. arXiv preprint arXiv:1902.05300. 
     Generally, for various types of image reconstruction algorithms, it is desirable to enforce consistency between the input image and the reconstructed image. In other words, artifacts stemming from the reconstruction algorithm itself should be avoided or reduced. Unforeseen deviations between the input image and the reconstructed image should be avoided. This is generally achieved by a data-consistency operation (DCO). 
     SUMMARY 
     Accordingly, there is a need for advanced techniques of determining a reconstructed image based on an input dataset defining an input image. In particular, there is a need for advanced techniques that help to enforce consistency between an input image defined by an input dataset and the reconstructed image. 
     Various examples of the disclosure generally relate to image reconstruction. According to various examples, a reconstruction algorithm is used to determine a reconstructed image. A DCO—that may be embedded into the reconstruction algorithm or may be separately defined—is applied to the reconstructed image, to thereby enforce a consistency between an input image that is defined by an input dataset, as well as the reconstructed image. 
     The DCO may also be termed data-fidelity operation or forward-sampling operation. 
     The DCOs described herein facilitate accurate reconstruction with little or no artifacts stemming from the reconstruction algorithm itself. The DCOs described herein are computationally inexpensive. 
     According to various examples, a specific type of DCO is used. According to various examples, the DCO determines, for multiple K-space positions at which the input dataset includes respective source data, a contribution of respective K-space values associated with the input dataset to a K-space representation of the reconstructed image. 
     In other words, the DCO can, at least to a certain degree, override or undo contributions of the reconstruction algorithm, so that at K-space positions where source data is available, the respective values of the K-space representation of the reconstructed image obtained from the reconstruction algorithm are replaced fully or partly by respective values of the source data. 
     Thus, the reconstructed image obtained from the reconstruction algorithm prior to the DCO can be labelled preliminary reconstructed image or pre-DCO reconstructed image; and the reconstructed image obtained after applying the DCO is the final reconstructed image or post-DCO reconstructed image. I.e., where source data is available in the input dataset, this source data can prevail in the final reconstructed image or, at least, significantly contribute to the K-space values of the K-space representation of the final reconstructed image. On the other hand, where the source data is not available in the input dataset, no contribution of respective K-space values associated with the input dataset may be made to the K-space representation of the final reconstructed image. 
     Various examples of the disclosure also relate to a method of training of a machine-learned reconstruction algorithm. According to various examples, a machine-learned reconstruction algorithm is trained using a loss function. The loss function is based on a difference between a ground truth image and the reference image. The reference image is determined based on a training dataset that is associated with the ground truth image and using the machine-learned reconstruction image. The reference image is further determined based on executing a data-consistency operation. The data-consistency operation determines, for multiple K-space positions at which the training dataset includes respective source data, a contribution of respective K-space-values associated with the training dataset to a K-space representation of the reference image. 
     By considering the data-consistency operation in the training process, i.e., by using a loss function that is based on the difference between the ground truth image and the reference image that has been determined based on executing the data-consistency operation, the machine-learned reconstruction image can be accurately trained, in particular, taking into consideration alterations to its output made by the data-consistency operation. For example, spatial frequencies in the reference image that are modified by the contribution determined by the data-consistency operation may not be overemphasized in the training process. The training can focus on other spatial frequencies not modified by the contribution determined by the data-consistency operation. 
     It is to be understood that the features mentioned above and those yet to be explained below may be used not only in the respective combinations indicated, but also in other combinations or in isolation without departing from the scope of the invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  schematically illustrates a reconstruction algorithm according to various examples. 
         FIG. 2  schematically illustrates a device according to various examples. 
         FIG. 3  is a flowchart of a method according to various examples. 
         FIG. 4  schematically illustrates a K-space representation of a reconstructed image prior to executing a data-consistency operation according to various examples. 
         FIG. 5  schematically illustrates an example K-space representation of a K-space representation of an example input image and an example mask segmenting into K-space positions for which the K-space representation includes K-space values and further K-space positions for which the K-space representation does not include K-space values. 
         FIG. 6  schematically illustrates a K-space representation of a reconstructed image after executing a data-consistency operation according to various examples. 
         FIG. 7  schematically illustrates a dependency of values of a weighting factor on K-space positions according to various examples. 
         FIG. 8  is a flowchart of a method implementing a reconstruction algorithm including multiple iterations or cascades according to various examples. 
         FIG. 9  is a flowchart of a method according to various examples. 
         FIG. 10  schematically illustrates a training process for training a machine-learned reconstruction algorithm according to various examples. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
     Some examples of the present disclosure generally provide for a plurality of circuits or other electrical devices. All references to the circuits and other electrical devices and the functionality provided by each are not intended to be limited to encompassing only what is illustrated and described herein. While particular labels may be assigned to the various circuits or other electrical devices disclosed, such labels are not intended to limit the scope of operation for the circuits and the other electrical devices. Such circuits and other electrical devices may be combined with each other and/or separated in any manner based on the particular type of electrical implementation that is desired. It is recognized that any circuit or other electrical device disclosed herein may include any number of microcontrollers, a graphics processor unit (GPU), integrated circuits, memory devices (e.g., FLASH, random access memory (RAM), read only memory (ROM), electrically programmable read only memory (EPROM), electrically erasable programmable read only memory (EEPROM), or other suitable variants thereof), and software which co-act with one another to perform operation(s) disclosed herein. In addition, any one or more of the electrical devices may be configured to execute a program code that is embodied in a non-transitory computer readable medium programmed to perform any number of the functions as disclosed. 
     In the following, embodiments of the invention will be described in detail with reference to the accompanying drawings. It is to be understood that the following description of embodiments is not to be taken in a limiting sense. The scope of the invention is not intended to be limited by the embodiments described hereinafter or by the drawings, which are taken to be illustrative only. 
     Various examples of the disclosure generally relate to image reconstruction. An input dataset is obtained. As a general rule, the techniques described herein can be applicable to various image reconstruction tasks. For instance, image reconstruction of medical images, e.g., MRI image or computer tomography (CT) images would be possible. Microscopy images could be reconstructed, e.g., to achieve super resolution and/or remove artifacts. Photography is could be reconstructed, e.g., to remove flares. X-ray images could be reconstructed, e.g., to make certain structures otherwise hidden visible. 
     The input dataset can be obtained from a data storage. The input dataset can be obtained from a measurement device, e.g., a multi-pixel detector, radio-frequency receivers, etc. 
     The input dataset can include source data. The source data could be implemented by K-space data samples or by image-domain pixels or voxels. 
     The input dataset defines an input image. According to some examples, the input dataset may include the input image. I.e., the input dataset may be defined in spatial domain. For instance, an optical microscope may be used to obtain the input image. According to further examples, the input dataset may be defined in K-space. Examples would include acquisition of a magnetic resonance imaging (MRI) measurement dataset. Here, a Fourier transformation is required to obtain the input image from the input dataset. 
     Example input datasets include source data such as raw measurement in an MRI K-space-to-image reconstruction, or, e.g., a low-resolution image in the case of a super-resolution image-to-image reconstruction. 
     As a general rule, the particular type of the reconstruction algorithm is not germane for the functioning of the techniques described herein; in other words, the techniques described herein can be flexibly combined with various kinds and types of reconstruction algorithms. For instance, depending on the particular source of the input dataset, different types of reconstruction algorithms may be appropriate to use. For instance, to suppress artifacts in a microscopy image, a respective image reconstruction algorithm may be used, e.g., correcting aberrations introduced by the objective lens, creating super-resolution, etc. Differently, to remove undersampling aliasing artefacts from an MRI measurement data acquired with an undersampling K-space trajectory, a respective image reconstruction algorithm tailored to this task may be used. 
     According to various examples, a machine-learned reconstruction algorithm may be used. Examples would include variational or unrolled networks for reconstruction of undersampled MRI measurement data, see, e.g., Hammernik, Kerstin, et al. “Learning a variational network for reconstruction of accelerated MRI data.”  Magnetic resonance in medicine  79.6 (2018): 3055-3071. Another example would be GANs or AUTOMAP algorithms, as described above. 
     Irrespective of the particular implementation of the image reconstruction algorithm, according to various examples, a DCO is used, to thereby enforce a consistency between the reconstructed image and the input image defined by the input dataset. 
     For instance, the preliminary reconstructed image obtained from the image reconstruction algorithm may be altered or amended, at least at certain positions in image domain or in K-space (i.e., for certain spatial frequencies), when applying the DCO. Such altering or amending of the data underlying the preliminary reconstructed image can be done to increase a similarity value between the input image and the final reconstructed image. 
     Thus, by the DCO, generally the quality of the image reconstruction can be increased. 
     The DCOs described herein can operate in K-space. I.e., different K-space positions may be treated differently by the DCO. Different spatial frequencies may be affected differently by the DCO. 
     According to various examples, the DCO determines, for multiple K-space positions at which the input dataset includes respective source data (e.g., measurement data or generally data that is initially available before applying any reconstruction), a contribution of respective K-space values associated with the input dataset to a K-space representation of the reconstructed image. 
     Thus, the DCO can select, for each one of the multiple K-space positions where source data is natively available, to include the respective source data in the reconstructed image, at least to a certain degree. 
     As a general rule, the DCO may be defined separately from the reconstruction algorithm or may be integrated into the reconstruction algorithm. 
     For example, in case of an iterative optimization including multiple iterations, the DCO can be applied in each iteration. 
     For example, in case of an unrolled neural network, each layer of the unrolled neural network may be followed by applying the DCO. 
     For a K-space-to-image reconstruction, the input dataset is natively defined in the K-space and the available source data includes the acquired K-space values. In the case of an image-to-image reconstruction, the source data or more specifically the input image included in the input dataset is transformed to K-space, to obtain the K-space representation of the input image; here, it is possible to provide a mask indicating the K-space positions natively included as or defined by the source data to the DCO. Natively included could mean, e.g., for an input image such K-space positions captured by the input image up to a cut-off spatial frequency defined by Nyquist&#39;s theorem. K-space values can be selected from the K-space representation of the input image by the DCO within the mask. For example, for a super-resolution reconstruction without partial Fourier, the mask includes all the K-space positions of the input low-resolution image. If there is partial Fourier, only the acquired measurements defining the source data will be included in the mask. 
     Using DCOs as outlined above, the reconstruction can have an increased stability and robustness. For example, considering the case of the super-resolution reconstruction, the hard DCO according to Eq. 2 would ensure that the reconstruction algorithm only changes the high spatial frequencies in the reconstructed image vis-á-vis the input image, which high spatial frequency are not available in the source data of the input dataset defining the input image. The reconstruction would not change the low spatial frequencies where most of the energy of the input image and the reconstructed image is located. The low spatial frequencies are fixed, so that it is not possible to add something that was not present in the original input image or remove something that was present in the original input image in this frequency band. 
     More formally, in the case of an image-to-image reconstruction, the DCO can be summarized with the following pseudo code where y is the input of the DCO (i.e., the pre-DCO reconstructed image; which may be labeled a preliminary reconstructed image) and ŷ its output (i.e., the post-DCO final reconstructed image), X is the K-space representation of the input image using zero-padding so that the size of X matches the size of the K-space representation of y (denoted Y), mask being a binary matrix the size of Y with ones where the source data is available from the input dataset and zero when not: 
         Y←FFT ( y )  (1)
 
         Y ←where(mask, X,Y )  (2)
 
         ŷ←IFFT ( Y )  (3)
 
     FFT means fast Fourier transformation; IFFT means inverse fast Fourier transformation and the operation. Where (condition, m,n) is defined as: 
     
       
         
           
             
               out 
               i 
             
             = 
             
               { 
               
                 
                   
                     
                       
                         m 
                         i 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       if 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         condition 
                         i 
                       
                     
                   
                 
                 
                   
                     
                       
                         n 
                         i 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       otherwise 
                     
                   
                 
               
             
           
         
       
     
     I.e., where (mask,X,Y) returns a tensor of elements selected from either X or Y, depending on the mask value. 
     In such a scenario, according to Eq. 2, the contribution determined by the DCO replaces, for each one of the multiple K-space positions for which the input dataset includes respective source data (i.e., inside the mask), a further respective K-space value provided by the reconstruction algorithm (i.e., the respective element of the preliminary reconstructed image Y) by the respective K-space value associated with the input dataset (i.e., with the respective element of X). Such operation according to Eq. 2 can be labeled as “hard” DCO, because it is a binary selection between either the K-space value of the source data or the K-space value of the K-space representation of the preliminary reconstructed image. 
     Such a strict replacement of the respective K-space values in the pre-DCO reconstructed image to obtain the post-DCO reconstructed image according to Eq. 2 is only one option. It would also be possible to determine a “soft” contribution of the respective K-space values associated with the input dataset to the K-space representation of the reconstructed image. Such a “soft” DCO is illustrated below: 
         Y←FFT ( y ) 
         Y←Y −λ·where(mask, Y−X, 0)
 
         ŷ←IFFT ( Y )  (4)
 
     The weighting parameter λ defines a weighted combination of the K-space value associated with the input dataset (i.e., X) and a respective further K-space value provided by the reconstruction algorithm (i.e., the preliminary reconstructed image Y), according to Eq. 4. 
     The weighting parameter is configured to suppress the respective further K-space values if compared to the respective K-space value, as evidenced by the subtraction of Eq. 4. 
     Note that the hard DCO of Eq. 2 is equivalent to the soft DCO of Eq. 4 when λ=1. 
     Next, details with respect to the weighting parameter will be explained. 
     The weighting parameter λ may be predefined. It would also be possible that the weighting parameter is trained using machine-learning techniques. For instance, the weighting parameter may be trained end-to-end together with the reconstruction algorithm. I.e., while varying weights of the reconstruction algorithm, it is possible to also vary the value of the weighting parameter. 
     It would be possible that the value of the weighting parameter varies as a function of the multiple K-space positions. I.e., it would be possible that depending on the particular K-space position, a different value of the weighting parameter is used. In other examples, the value of the weighting parameter may be fixed, i.e., not vary as a function of the K-space position. 
     By such techniques, it would be possible to define a transition regime of spatial frequencies where the DCO operates less strict or particularly strict. For instance, the transition regime may be located at the edge of the mask, i.e., at the edge of the regime of multiple K-space positions for which the input dataset includes source data. Thereby, a gradual change between preserving source data and relying on reconstructed data can be achieved, overall leading to an increase in quality of the reconstructed image. 
     As a general rule, various options are available for implementing the image reconstruction algorithm. Some of the options are summarized in TAB. 1 below. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Examples of image reconstruction algorithms. In examples 
               
               
                 III and IV, the DCO can thus be implemented as a sequence 
               
               
                 of gradient steps with a step-size of one. 
               
            
           
           
               
               
               
            
               
                   
                 Brief 
                   
               
               
                   
                 description 
                 Example details 
               
               
                   
                   
               
            
           
           
               
               
               
            
               
                 I 
                 Classic 
                 A manually parameterized reconstruction 
               
               
                   
                 reconstruction 
                 algorithm may be used. For instance, 
               
               
                   
                 algorithm 
                 such reconstruction algorithm may rely 
               
               
                   
                   
                 on operations such as interpolation, 
               
               
                   
                   
                 edge sharpening, contrast enhancement, 
               
               
                   
                   
                 low-/high-pass filtering, blurring 
               
               
                   
                   
                 operations, etc. to name just a few 
               
               
                   
                   
                 options. 
               
               
                   
                   
                 The DCO can be sequentially applied, 
               
               
                   
                   
                 after the reconstruction algorithm 
               
               
                 II 
                 Machine- 
                 For instance, it would be possible to 
               
               
                   
                 learned 
                 use a neural network to implement the 
               
               
                   
                 reconstruction 
                 reconstruction algorithm. 
               
               
                   
                 algorithm 
                 For example, if the reconstruction 
               
               
                   
                   
                 algorithm is implemented as a multi- 
               
               
                   
                   
                 layer deep neural network, it would be 
               
               
                   
                   
                 possible that one or more layers 
               
               
                   
                   
                 towards the output layer or even the 
               
               
                   
                   
                 output layer itself implement the DCO. 
               
               
                   
                   
                 For example, the following 
               
               
                   
                   
                 implementation would be possible for a 
               
               
                   
                   
                 multi-layer deep neural network: a 
               
               
                   
                   
                 layer at the end of the multi-layer 
               
               
                   
                   
                 deep neural network may be added to 
               
               
                   
                   
                 implement the DCO; this DCO-layer may 
               
               
                   
                   
                 transform the input of that layer into 
               
               
                   
                   
                 K-space and then replace all the values 
               
               
                   
                   
                 at those K-space positions for which 
               
               
                   
                   
                 there is a corresponding source data 
               
               
                   
                   
                 available in the input dataset being 
               
               
                   
                   
                 input into the input layer of the 
               
               
                   
                   
                 multi-layer deep neural network. Such 
               
               
                   
                   
                 source data may be natively defined in 
               
               
                   
                   
                 K-space. It would also be possible to 
               
               
                   
                   
                 determine the K-space representation of 
               
               
                   
                   
                 the input image to determine these 
               
               
                   
                   
                 values. In either case, a skip 
               
               
                   
                   
                 connection from the input of the multi- 
               
               
                   
                   
                 layer deep neural network to the DCI- 
               
               
                   
                   
                 layer may be implemented. After 
               
               
                   
                   
                 replacing all the values at those K- 
               
               
                   
                   
                 space positions at which corresponding 
               
               
                   
                   
                 source data is available, the DCO layer 
               
               
                   
                   
                 can transform the result of this 
               
               
                   
                   
                 operation back to the image domain. 
               
               
                   
                   
                 During a training phase, weights of 
               
               
                   
                   
                 each layer of the neural network may be 
               
               
                   
                   
                 set. During the training phase, it 
               
               
                   
                   
                 would be possible to rely on ground 
               
               
                   
                   
                 truth data defining a reference 
               
               
                   
                   
                 reconstructed image of high quality. 
               
               
                   
                   
                 For instance, it would be possible to 
               
               
                   
                   
                 obtain the input image by artificially 
               
               
                   
                   
                 downsampling or otherwise preprocessing 
               
               
                   
                   
                 the reference reconstructed image. 
               
               
                   
                   
                 Noise or artifacts could be 
               
               
                   
                   
                 synthesized. 
               
               
                   
                   
                 The machine-learned reconstruction 
               
               
                   
                   
                 algorithm may be trained end-to-end 
               
               
                   
                   
                 including the DCO, so during the 
               
               
                   
                   
                 training process the machine-learned 
               
               
                   
                   
                 reconstruction algorithm is not 
               
               
                   
                   
                 required to learn to reconstruct the 
               
               
                   
                   
                 low spatial frequencies and could then 
               
               
                   
                   
                 use all of its capacity to only focus 
               
               
                   
                   
                 in predicting the high spatial 
               
               
                   
                   
                 frequencies, i.e. a much simpler task 
               
               
                   
                   
                 for which less training data are needed 
               
               
                   
                   
                 to obtain a satisfying generalization. 
               
               
                   
                   
                 This can increase the accuracy. 
               
               
                 III 
                 Iterative 
                 The iterative optimization can include 
               
               
                   
                 optimization 
                 multiple iterations, e.g., so-called 
               
               
                   
                   
                 Landweber iterations. Other examples 
               
               
                   
                   
                 include prima-dual or alternating 
               
               
                   
                   
                 direction method of multipliers. 
               
               
                   
                   
                 Thereby, a gradient descent 
               
               
                   
                   
                 optimization can be implemented. By 
               
               
                   
                   
                 the iterative optimization, it is 
               
               
                   
                   
                 possible to maximize or minimize a 
               
               
                   
                   
                 respective goal function that defines 
               
               
                   
                   
                 the reconstructed image. 
               
               
                   
                   
                 The multiple iterations can yield a 
               
               
                   
                   
                 sequence of (temporary) images. 
               
               
                   
                   
                 Each iteration can include a 
               
               
                   
                   
                 regularization operation and the DCO. 
               
               
                   
                   
                 The DCO can be implemented by an 
               
               
                   
                   
                 appropriately configured forward- 
               
               
                   
                   
                 measurement model. 
               
               
                   
                   
                 For example, the DCO executed in a 
               
               
                   
                   
                 given iteration is based on difference 
               
               
                   
                   
                 between the K-space source data and 
               
               
                   
                   
                 synthesized K-space source data of that 
               
               
                   
                   
                 respective iteration. 
               
               
                   
                   
                 The synthesized K-space source data is 
               
               
                   
                   
                 based on a K-space representation of a 
               
               
                   
                   
                 prior image of the sequence of images 
               
               
                   
                   
                 and the forward-measurement model. The 
               
               
                   
                   
                 forward-measurement model can then 
               
               
                   
                   
                 suppress contributions to the 
               
               
                   
                   
                 synthesized K-space source data at K- 
               
               
                   
                   
                 space positions at which the K-space 
               
               
                   
                   
                 source data has been sampled using the 
               
               
                   
                   
                 undersampling trajectory, to thereby 
               
               
                   
                   
                 implement the DCO. 
               
               
                   
                   
                 For example, such techniques can be 
               
               
                   
                   
                 used to implement a reconstruction 
               
               
                   
                   
                 algorithm to reconstruct an MRI image 
               
               
                   
                   
                 based on an input dataset including K- 
               
               
                   
                   
                 space source data that is undersampled 
               
               
                   
                   
                 in accordance with a respective 
               
               
                   
                   
                 undersampling trajectory. Acquisition 
               
               
                   
                   
                 of MRI data can require significant 
               
               
                   
                   
                 time. To accelerate the data 
               
               
                   
                   
                 acquisition, it is known to undersample 
               
               
                   
                   
                 K-space. Missing data can be 
               
               
                   
                   
                 reconstructed (MRI reconstruction). 
               
               
                   
                   
                 Various techniques for implementing MRI 
               
               
                   
                   
                 reconstruction are known. One prior art 
               
               
                   
                   
                 technique is referred to as compressed 
               
               
                   
                   
                 sensing. See, e.g., Lustig, Michael, 
               
               
                   
                   
                 David Donoho, and John M. Pauly. 
               
               
                   
                   
                 “Sparse MRI: The application of 
               
               
                   
                   
                 compressed sensing for rapid MR 
               
               
                   
                   
                 imaging.”  Magnetic Resonance in   
               
               
                   
                   
                 
                   Medicine: An Official Journal of the 
                 
               
               
                   
                   
                 
                   International Society for Magnetic 
                 
               
               
                   
                   
                   Resonance in Medicine  58.6 (2007): 
               
               
                   
                   
                 1182-1195; also see Lustig, Michael, et 
               
               
                   
                   
                 al. “Compressed sensing MRI.” IEEE 
               
               
                   
                   
                 signal processing magazine 25.2 (2008): 
               
               
                   
                   
                 72-82. 
               
               
                   
                   
                 In further detail, such reconstruction 
               
               
                   
                   
                 relies on representation of MRI images 
               
               
                   
                   
                 in a wavelet basis in connection with 
               
               
                   
                   
                 the regularization operation. As 
               
               
                   
                   
                 described in id., page 13, section 
               
               
                   
                   
                 “Image Reconstruction”, an optimization 
               
               
                   
                   
                 problem - typically defined in an     1 - 
               
               
                   
                   
                 norm - can be defined. The 
               
               
                   
                   
                 regularization operation is 
               
               
                   
                   
                 conventionally based on a non-linear 
               
               
                   
                   
                      1 -norm. 
               
               
                   
                   
                 A classic formulation of the 
               
               
                   
                   
                 regularization operation is based on 
               
               
                   
                   
                 sparsity of the MRI image in a 
               
               
                   
                   
                 transform domain such as a wavelet 
               
               
                   
                   
                 domain in combination with pseudo- 
               
               
                   
                   
                 random sampling that can introduce 
               
               
                   
                   
                 aliasing artifacts that are incoherent 
               
               
                   
                   
                 in the respective transform domain. 
               
               
                   
                   
                 Another example would be a Fourier 
               
               
                   
                   
                 domain, in particular for acquisitions 
               
               
                   
                   
                 of a dynamically moving target. Another 
               
               
                   
                   
                 example would be total variation (TV) 
               
               
                   
                   
                 used in connection with non-Cartesian 
               
               
                   
                   
                 K-space trajectories such as radial and 
               
               
                   
                   
                 spiral trajectories. 
               
               
                 IV 
                 Unrolled deep 
                 The unrolled deep neural network can 
               
               
                   
                 neural network 
                 include multiple cascades, 
               
               
                   
                 (sometimes 
                 corresponding to the iterations of 
               
               
                   
                 also referred 
                 scenario III. Here, each cascade can 
               
               
                   
                 to as 
                 include a respective regularization 
               
               
                   
                 variational 
                 operation implemented, e.g., by 
               
               
                   
                 neural 
                 convolutions and/or activations of a 
               
               
                   
                 network) 
                 respective layer of the unrolled deep 
               
               
                   
                   
                 neural network. 
               
               
                   
                   
                 See Hammernik, Kerstin, et al. 
               
               
                   
                   
                 “Learning a variational network for 
               
               
                   
                   
                 reconstruction of accelerated MRI 
               
               
                   
                   
                 data.” Magnetic resonance in medicine 
               
               
                   
                   
                 79.6 (2018): 3055-3071, as well as 
               
               
                   
                   
                 Knoll, Florian, et al. “Deep learning 
               
               
                   
                   
                 methods for parallel magnetic resonance 
               
               
                   
                   
                 image reconstruction.” arXiv preprint 
               
               
                   
                   
                 arXiv: 1904.01112 (2019). 
               
               
                   
                   
                 Such techniques are based on the 
               
               
                   
                   
                 finding that wavelet compositions - as 
               
               
                   
                   
                 described in example III- can be 
               
               
                   
                   
                 expressed as a subset of trainable 
               
               
                   
                   
                 convolutions of a deep neural network 
               
               
                   
                   
                 such as a convolutional neural network 
               
               
                   
                   
                 and that soft-thresholding can be used 
               
               
                   
                   
                 as an activation function in the deep 
               
               
                   
                   
                 neural network. 
               
               
                   
                   
                 for implementing the regularization 
               
               
                   
                   
                 operation is the U-net, see 
               
               
                   
                   
                 Ronneberger, Olaf, Philipp Fischer, and 
               
               
                   
                   
                 Thomas Brox. “U-net: Convolutional 
               
               
                   
                   
                 networks for biomedical image 
               
               
                   
                   
                 segmentation.” International Conference 
               
               
                   
                   
                 on Medical image computing and 
               
               
                   
                   
                 computer-assisted intervention. 
               
               
                   
                   
                 Springer, Cham, 2015. The U-net employs 
               
               
                   
                   
                 skip-connections between hidden layers 
               
               
                   
                   
                 and down-sampling and up-sampling of 
               
               
                   
                   
                 feature maps. 
               
               
                   
               
            
           
         
       
     
     The implementation of the DCO using the forward-measurement model (FMM) according to examples III and IV of TAB. 1 is explained below for an undersampled MRI input dataset acquired using multiple RF receiver coils having different sensitivity maps. 
     As the FMM (excluding the undersampling step M) A=FS, with F being the Fourier transform, and S the coil sensitivity mapping is a Parseval tight frame, i.e. such that A*A=S*F*FS=S*S=I, where A*=S*F* is the adjoint operator of A, and I is the identity matrix. The FMM thus yields synthesized source data that can be compared with the source data y. 
         x←x−A *( MAx−y )= A *(( I−M ) Ax+y ),  (5)
 
     where y is the source data of the input dataset (i.e., the MRI raw data) which contains zeros where there is no measurement, x is the input into the DCO (in image space; i.e., an image of the sequence of images associated with the given iteration or cascade), and M is the binary mask which contains 1 where there is a measurement (i.e. where corresponding y coefficient is non zero) and zero otherwise. 
     As can be seen from Eq. 5, the gradient step with step size 1 generalizes the hard DCO described above. 
     It includes transforming the current image of the sequence of images x to K-space after applying the sensitivity maps (operation Ax) of the FMM, and then, for the coefficients available in y (for which the mask equal 1) replacing coefficients of x with the one of y and then transforming the result back to the image domain with the operator A*. I.e., the FMM suppresses contributions included in the synthesized source data at those K-space positions at which values are available in y. 
       FIG. 1  schematically illustrates aspects with respect to a reconstruction algorithm  201 . The reconstruction algorithm  201  determines a reconstructed image  102  based on an input dataset  101 . 
     According to some examples, the input dataset  101  may directly include an input image; in such a scenario, an image-to-image reconstruction is implemented. Here, the source data corresponds to contrast values for each pixel of the input image. It would also be possible that the input dataset  101  includes raw data that implicitly defines the input image. For instance, the raw source data could be defined in K-space or could be compressed. Pixel values of the input image then have to be inferred from the input dataset, e.g., using the reconstruction algorithm or other preprocessing. 
     Also illustrated in  FIG. 1  is a DCO  205 . The illustrated scenario, the DCO  205  is embedded into the reconstruction algorithm  201 . As a general rule, it would also be possible that the DCO is executed after completing execution of the reconstruction algorithm  201 . 
     The DCO  205  can alter or change the reconstructed image  102 . This makes it possible to define a pre-DCO version of the reconstructed image  102  and a post-DCO version of the reconstructed image  102  (not illustrated in  FIG. 1  for sake of simplicity). 
     As illustrated in  FIG. 1 , the DCO  205  operates based on the input dataset  101 . In particular, the DCO can determine, for multiple K-space positions at which the input dataset includes respective source data (possibly after a Fourier transform of the input image included in the input dataset  101  and to K-space), a contribution of respective K-space values associated with the input dataset to the K-space representation of the reconstructed image  102 . 
       FIG. 2  illustrates aspects with respect to a device  501 . The device  501  includes a processor  505  and a memory  506 . The device  501  also includes an interface  507 . For instance, it would be possible that the input dataset  101  is received via the interface  507 , e.g., from a database or from a measurement equipment such as a microscope, MRI scanner, a CT scanner, to give just a few examples. The processor  505  can load program code from the memory  506  and execute the program code. Upon loading and executing the program code, the processor  505  can perform one or more of the following: executing a reconstruction algorithm such as the reconstruction algorithm  201 ; executing a DCO such as the DCO  205 ; performing a Fourier transform or an inverse Fourier Transform; executing a training process for training machine-learned algorithms; etc. 
       FIG. 3  is a flowchart of a method according to various examples. For instance, the method of  FIG. 3  could be executed by the processor  505  upon loading program code from the memory  506 . 
     The method of  FIG. 3  pertains to inference of a reconstructed image using a reconstruction algorithm such as the reconstruction algorithm  201 . 
     At box  3020 , a reconstructed image is determined based on input dataset that defines an input image. For this purpose, the reconstruction algorithm is used. Various types of reconstruction algorithms have been discussed in connection with TAB. 1 and such and other types of reconstruction algorithms can be used in connection with box  3020 . 
     At box  3025 , a DCO for enforcing consistency between the input image and the reconstructed image is executed. The DCO can operate on the reconstructed image and change one or more pixel values of the reconstructed image. The DCO can, in particular, operate in K-space. I.e., the method may include a Fourier transform of the reconstructed image into K-space, wherein then K-space values (of the thus defined preliminary version of the reconstructed image before executing the DCO) are adjusted or replaced at one or more K-space positions of the K-space representation of the reconstructed image, followed by an inverse Fourier Transform back into image domain. 
     The DCO determines, for multiple K-space positions at which the input dataset includes respective source data, a contribution of respective K-space values associated with the input dataset to a K-space representation of the reconstructed image. 
     While in  FIG. 3  the execution of the DCO at box  3025  is shown to be executed after executing the reconstruction algorithm at box  3020 , as a general rule, it would be possible that the DCO is executed while still executing the reconstruction algorithm, in particular, in a scenario in which the DCO is embedded in the reconstruction algorithm (a respective scenario will be described later in connection with  FIG. 8 ). 
     Details with respect to the operation of the DCO are explained next in connection with  FIG. 4 ,  FIG. 5 , and  FIG. 6 . 
       FIG. 4  illustrates the K-space representation of a preliminary reconstructed image  211  prior to applying the DCO (the reconstructed image  211  corresponds to the reconstructed image  102  of  FIG. 1 , but here is specifically denoting the pre-DCO version). 
     As illustrated, the K-space representation of the pre-DCO reconstructed image  211  includes K-space values at K-space positions from −Kx to Kx and from −Ky to +Ky, respectively (illustrated by the dashed filling). 
       FIG. 5  schematically illustrates a K-space representation of the input image defined by the input dataset  101 . 
     The K-space representation may be natively included in the respective source data, e.g., for MRI measurements. It would also be possible to perform a Fourier transform to obtain the K-space representation of the input image defined by the input dataset  101 , cf. Eq. 1. 
     In the illustrated scenario, the K-space representation of the input image defined by the input dataset  101  only includes K-space values at K-space positions close to the K-space center (In  FIG. 5 , the K-space positions where the respective K-space representation of the reconstructed image  102  and the input image defined by the input dataset  101  include K-space values are also illustrated with a dashed filling). 
     In  FIG. 5 , the circumference of a mask  760  that includes all K-space positions for which the K-space representation of the input image includes K-space values is labeled with dashed lines. This mask  760  can define the matrix mask of Eq. 2. 
       FIG. 6  schematically illustrates the K-space representation of the reconstructed image  212  after applying the DCO (post-DCO reconstructed image). As illustrated in  FIG. 6 , the K-space values of the K-space representation of the input image defined by the input dataset  101  have replaced the respective K-space values provided by the reconstruction algorithm, i.e., included in the pre-DCO reconstructed image  211  of  FIG. 4 . This corresponds to the DCO according to Eq. 2. Other K-space values (for which the input dataset does not include K-space values) in the pre-DCO reconstructed image  211 —i.e., away from the center, outside the mask  760 —are preserved in the post-DCO reconstructed image  212 . 
     Instead of such a replacement of the respective K-space values of the pre-DCO reconstructed image  211  by the respective K-space values associated with the input dataset, it would also be possible to implement a weighted combination, e.g., using a weighting parameter as discussed above in connection with Eq. 4. The weighting parameter can be configured to suppress the respective K-space values provided by the reconstruction algorithm  201  if compared to the respective K-space values of the input image. This means that while the pre-DCO reconstructed image  211  at and around the K-space center has K-space values determined by the reconstruction algorithm  201 , the post-DCO reconstructed image  212  includes, at and around the K-space center, K-space values determined only to a certain degree by the reconstruction algorithm  201 , but also based on the K-space representation of the input image. Thus, the influence of the reconstruction algorithm  201  is suppressed. The degree of suppression is controlled by the value or values of the weighting parameter. 
       FIG. 7  illustrates aspects with respect to the weighting parameter  750 . As illustrated in  FIG. 7 , the weighting parameter  750  (cf. Eq. 4, λ)—more accurately, its values—can vary as a function of the K-space position. 
     The specific configuration of the dependency of the values of the weighting parameter  750  on the K-space position can vary according to different examples. In the illustrated example of  FIG. 7 , the values of the weighting parameter  750  define three regimes  751 - 753 . 
     In regime  751 —at and around the K-space center—the DCO replaces the K-space values of the pre-DCO reconstructed image  211  by the K-space values of the K-space representation of the input image (λ=1). In regime  752 , the DCO implements a weighted combination of the K-space values of the pre-DCO reconstructed image  211  and the K-space values of the K-space representation of the input image. In regime  753 , the K-space representation of the input image does not include any K-space values (cf.  FIG. 5 ); and accordingly, the K-space values of the K-space representation of the pre-DCO reconstructed image  211  prevail and are immediately included in the post-DCO reconstructed image  212 . By such setting of the dependency of the values of the weighting parameter  750  on the K-space position, it is possible to smoothly transition between preserving the K-space values of the K-space representation of the input image in regime  751  and using the reconstructed K-space values of the K-space representation of the reconstructed image in regime  753 . Thereby, an overall improved reconstructed image can be obtained. 
     For instance, the shape of the spatial dependency of the values of the weighting parameter as illustrated in  FIG. 7  could be manually parametrized or predefined. It would also be possible that the value or values of the weighting parameter are determined using a machine-learned process (as will be explained later in connection with  FIG. 9  and  FIG. 10 ). In particular, and end-to-end learning of the reconstruction algorithm  2012  together with the value or values of the weighting parameter  750  could be implemented. 
       FIG. 8  is a flowchart of a method according to various examples.  FIG. 8  illustrates an example implementation of the reconstruction algorithm  201 . For illustration, the method of  FIG. 8  could be executed by the processor  505  of the device  501 , upon loading program code from the memory  506 . 
     The reconstruction algorithm  201  according to the method of  FIG. 8  implements a multi-step reconstruction, including multiple iterations or cascades  3071 . For instance, an iterative optimization or an unrolled deep neural network could be used to implement the reconstruction algorithm  201  according to the method of  FIG. 8 . 
     Each iteration or cascade  3071  includes a regularization operation, box  3050 ; and a DCO at box  3060 . 
     A concrete implementation of box  3050  and box  3060  could be based on Knoll, Florian, et al. “Deep learning methods for parallel magnetic resonance image reconstruction.”  arXiv preprint arXiv: 1904.01112 (2019): equation 12. Here, the left term included in the bracket corresponds to the regularization operation of box  3050 . The DCO applied at box  3060  can be implemented by the FMM according to Eq. 5. Eq. 5 corresponds to a modified version of the right-hand term in Knoll et al., Eq. 12 in that the FMM is modified to suppress contributions to the synthesized K-space source data at K-space positions at which the K-space source data has been sampled using the undersampling trajectory. Thereby, the FMM implements the DCO. 
     Note that in Eq. 12 of Knoll et al, the index t counts iterations/cascades and the respective sequence of MRI images is denoted with u t . 
     Then, at box  3070 , it is checked whether a further iteration or cascade  3071  is required; and, in the affirmative, box  3050  and box  3060  are re-executed. 
     It would be possible that at box  3070  it is checked whether a certain predefined count of iterations or cascades  3071  has been reached. This can be an abort criterion. Other abort criteria are conceivable, e.g., as defined by the optimization method (convergence into a minimum), e.g., gradient descent. It could be checked whether a further layer of the unrolled neural network is available. 
       FIG. 9  is a flowchart of a method according to various examples.  FIG. 9  illustrates a training phase at box  3505  and an inference phase at box  3510 . 
     In a training process of the training phase of box  3505 , weights of a machine-learned reconstruction algorithm can be set by implementing a respective training process based on a ground truth. The training process could be executed by the processor or  505  of the device  501  upon loading program code from the memory  506  and executing the program code. 
     Then, the trained reconstruction algorithm can be used for image reconstruction tasks during inference at box  3510  when no ground truth is available. 
     Details with respect to the inference have been discussed above. Next, details with respect to the training process of the training phase at box  3505  will be explained below. 
       FIG. 10  illustrates aspects with respect to training of a machine-learned reconstruction algorithm  201  configured to determine a reconstructed image based on an input dataset (cf.  FIG. 1 ). 
     As illustrated in  FIG. 10 , a ground truth image  903  is available. The ground truth image  903  can have desired properties to be achieved by the reconstruction algorithm  201 , e.g., a particularly high resolution, suppressed or no artifacts, etc. 
     The reconstruction algorithm  201  is initially in a untrained state or has only been partly trained. I.e., weights of, e.g., one or more layers have not been accurately set. The purpose of the training process is to accurately set respective parameters of the reconstruction algorithm. 
       FIG. 10  also illustrates that a training dataset  901  is available. The training dataset  901  is associated with the ground truth image  903 . Specifically, the training dataset  901  can define an input image generally corresponding to the ground truth image  903 , yet including artifacts or noise or being of lower resolution; i.e., including characteristics that are to suppressed or mitigated by the reconstruction algorithm  201 . 
     The training dataset  901  can be obtained from measurements or can be inferred from the ground truth image  903  (as illustrated in  FIG. 10 ), e.g., by artificially introducing synthesized artifacts or downsampling the resolution, to give just a few options. 
     Also illustrated is a reference image  902  determined based on the training dataset  901  by using the reconstruction algorithm  201  (in its respective training state). Further, the reference image  902  is obtained after executing the DCO  205 . 
     It is then possible to determine a loss based on the loss function  905  by comparing the reference image  902  with the ground truth image  903 . Deviations are penalized. Based on the loss obtained from the loss function  905 , the training state of the reconstruction algorithm  201  is changed, e.g., by changing weights of one or more layers, e.g., using backpropagation. 
     On the other hand, at least in some examples, the DCO can remain fixed during the training process. The DCO can be predefined, e.g., according to Eq. 2 or Eq. 4 or Eq. 5. 
     Yet, in some examples it would be possible to also change the value or values of the weighting parameter  750  being part of the DCO  205  during the training process. I.e., the weighting parameter  750  of the DCO  205  and the reconstruction algorithm can be trained end-to-end. 
     Such training process is iteratively executed in multiple iterations 999. An iterative optimization is thus implementing the training process of the training phase of box  3505 . 
     Summarizing, at least the following EXAMPLES have been disclosed: 
     EXAMPLE 1. A computer-implemented method, comprising:
         based on an input dataset ( 101 ) defining an input image, determining a reconstructed image ( 102 ,  211 ,  212 ) using a reconstruction algorithm ( 201 ), and   executing a data-consistency operation ( 205 ) for enforcing consistency between the input image and the reconstructed image ( 102 ,  211 ,  212 ), wherein the data-consistency operation ( 205 ) determines, for multiple K-space positions at which the input dataset ( 101 ) comprises respective source data, a contribution of respective K-space values associated with the input dataset ( 101 ) to a K-space representation of the reconstructed image ( 102 ,  211 ,  212 ).       

     EXAMPLE 2. The computer-implemented method of EXAMPLE 1, wherein the contribution replaces, for each one of the multiple K-space positions, a further respective K-space value provided by the reconstruction algorithm ( 201 ) by the respective K-space value associated with the input dataset ( 101 ). 
     EXAMPLE 3. The computer-implemented method of EXAMPLE 1, wherein the contribution implements, for each one of the multiple K-space positions, a weighted combination of the K-space value associated with the input dataset ( 101 ) and a respective further K-space value provided by the reconstruction algorithm ( 201 ) in accordance with a weighting parameter ( 750 ), the weighting parameter ( 750 ) being configured to suppress the respective further K-space value if compared to the respective K-space value. 
     EXAMPLE 4. The computer-implemented method of EXAMPLE 3, wherein the reconstruction algorithm ( 201 ) is machine-learned in a training phase ( 3510 ), wherein the weighting parameter ( 750 ) is trained end-to-end with the reconstruction algorithm ( 201 ). 
     EXAMPLE 5. The computer-implemented method of EXAMPLE 3 or 4, wherein the weighting parameter ( 750 ) varies as a function of the multiple K-space positions. 
     EXAMPLE 6. The computer-implemented method of any one of EXAMPLES 2 to 5, wherein the reconstruction algorithm ( 201 ) is configured to provide a preliminary reconstructed image ( 102 ,  211 ), the further K-space values being determined based on the preliminary reconstructed image ( 102 ,  211 ). 
     EXAMPLE 7. The computer-implemented method of any one of the preceding EXAMPLES, 
     wherein the input dataset ( 101 ) comprises the input image, wherein the method further comprises:
         transforming the input image into K-space to obtain a K-space representation of the input image, wherein the K-space values are selected from the K-space representation of the input image within a mask ( 760 ) limited to such K-space positions natively defined by the source data.       

     EXAMPLE 8. The computer-implemented method of any one of EXAMPLES 1 to 7, wherein the source data of the input dataset ( 101 ) defines the input image in K-space, the source data being undersampled in accordance with an undersampling trajectory, 
     wherein the reconstruction algorithm ( 201 ) comprises an iterative optimization, the iterative optimization comprising multiple iterations ( 3071 ), 
     wherein the data-consistency operation ( 205 ) is executed for each iteration of the multiple iterations, to thereby obtain a sequence of images, 
     wherein the data-consistency operation ( 205 ) is executed, in a given iteration of the multiple iterations, to enforce consistency between the source data and synthesized source data, the synthesized source data being based on a K-space representation of a prior image of the sequence of images and a forward-measurement model, the forward-measurement model suppressing contributions to the synthesized source data at K-space positions at which the source data has been sampled using the undersampling trajectory. 
     EXAMPLE 9. The computer-implemented method of any one of EXAMPLES 1 to 7, wherein the source data of the input dataset ( 101 ) defines the input image in K-space, the source data being undersampled in accordance with an undersampling trajectory, 
     wherein the reconstruction algorithm ( 201 ) comprises an unrolled neural network comprising multiple cascades ( 3071 ) associated with different layers of the unrolled neural network, 
     wherein the data-consistency operation ( 205 ) is executed for each one of the multiple cascades, to thereby obtain a sequence of images, 
     wherein the data-consistency operation ( 205 ) is executed, in a given cascade, to enforce consistency between the source data and synthesized source data, the synthesized source data being based on a K-space representation of a prior image of the sequence of images and a forward-measurement model, the forward-measurement model suppressing contributions to the synthesized source data at K-space positions at which the source data has been sampled using the undersampling trajectory. 
     EXAMPLE 10. The computer-implemented method of any one of the preceding EXAMPLES, wherein the reconstruction algorithm ( 201 ) comprises multiple iterations or cascades, wherein a regularization operation and a forward-measurement model are executed in each one of the multiple iterations or cascades, the forward-measurement model implementing the data-consistency operation ( 205 ). 
     EXAMPLE 11. The computer-implemented method of any one of the preceding EXAMPLES, wherein the reconstruction algorithm ( 201 ) comprises a machine-learned neural network, and wherein the machine-learned neural network is trained end-to-end including the data-consistency operation ( 205 ). 
     EXAMPLE 12. The computer-implemented method of any one of the preceding EXAMPLES, wherein the reconstructed image ( 102 ,  211 ,  212 ) has an increased resolution if compared to the input image. 
     EXAMPLE 13. The computer-implemented method of any one of the preceding EXAMPLES, wherein the reconstructed image ( 102 ,  211 ,  212 ) has reduced aliasing artifacts if compared to the input image. 
     EXAMPLE 14. The computer-implemented method of any one of the preceding EXAMPLES, wherein, for multiple further K-space positions at which the input image does not comprise the source data, further respective K-space values provided by the reconstruction algorithm ( 201 ) are preserved in the reconstructed image ( 102 ,  211 ,  212 ) by the data-consistency operation ( 205 ). 
     EXAMPLE 15. A computer-implemented method of training a machine-learned reconstruction algorithm ( 201 ), the method comprising:
         training the machine-learned reconstruction algorithm ( 201 ) using a loss function ( 905 ) which is based on a difference between a ground truth image ( 903 ) and a reference image ( 902 ), the reference image ( 902 ) being determined based on a training dataset ( 901 ) associated with the ground truth image ( 903 ) and using the machine-learned reconstruction algorithm ( 201 ), and further based on executing a data-consistency operation ( 205 ),       

     wherein the data-consistency operation ( 205 ) determines, for multiple K-space positions at which the training dataset ( 901 ) comprises respective source data, a contribution of respective K-space values associated with the training dataset ( 901 ) to a K-space representation of the reference image ( 902 ). 
     EXAMPLE 16. The computer-implemented method of EXAMPLE 15, wherein the contribution implements, for each one of the multiple K-space positions, a weighted combination of the K-space value associated with the training dataset and a respective further K-space value provided by the machine-learned reconstruction algorithm ( 201 ) in accordance with a weighting parameter ( 750 ), the weighting parameter ( 750 ) being configured to suppress the respective further K-space values if compared to the respective K-space value, wherein one or more values of the weighting parameter ( 750 ) are adjusted during said training. 
     EXAMPLE 17. The computer-implemented method of EXAMPLE 15, wherein the data-consistency operation ( 205 ) is predefined and not adjusted during said training. 
     EXAMPLE 18. A device ( 501 ) comprising a processor ( 502 ) configured to:
         based on an input dataset ( 101 ) defining an input image, determine a reconstructed image using a reconstruction algorithm ( 201 ),   execute a data-consistency operation ( 205 ) for enforcing consistency between the input image and the reconstructed image,       

     wherein the data-consistency operation ( 205 ) determines, for multiple K-space positions at which the input dataset ( 101 ) comprises respective source data, a contribution of respective K-space values associated with the input dataset ( 101 ) to a K-space representation of the reconstructed image. 
     EXAMPLE 19. The device of EXAMPLE 18, wherein the processor is configured to perform the method of any one of EXAMPLEs 1 to 14. 
     EXAMPLE 20. A device configured to train a machine-learned reconstruction algorithm ( 201 ), the device comprising a processor, the processor being configured to:
         train the machine-learned reconstruction algorithm ( 201 ) using a loss function which is based on a difference between a ground truth image and a reference image, the reference image being obtained by executing, to a training dataset, the machine-learned reconstruction algorithm ( 201 ) and further executing a data-consistency operation ( 205 ),       

     wherein the data-consistency operation ( 205 ) determines, for multiple K-space positions at which the training dataset comprises respective source data, a contribution of respective K-space values associated with the training dataset to a K-space representation of the reference image. 
     EXAMPLE 21. The device of EXAMPLE 20, wherein the processor is configured to perform the method of any one of EXAMPLEs 15 to 17. 
     EXAMPLE 22. A computer program or a computer-program product or a computer-readable storage medium comprising program code, the program code being loadable and executable by a processor, wherein the processor, upon loading and executing the program code performs the method of any one of EXAMPLEs 1 to 17. 
     Although the invention has been shown and described with respect to certain preferred embodiments, equivalents and modifications will occur to others skilled in the art upon the reading and understanding of the specification. The present invention includes all such equivalents and modifications and is limited only by the scope of the appended claims.