Patent Publication Number: US-11035919-B2

Title: Image reconstruction using a colored noise model with magnetic resonance compressed sensing

Description:
PRIORITY CLAIM 
     This application claims priority to U.S. provisional application Ser. No. 62/812,274, filed 1 Mar. 2019, which is entirely incorporated by reference. 
    
    
     FIELD 
     The following disclosure relates to reconstructing magnetic resonance compressed sensing data using a colored noise model. 
     BACKGROUND 
     In magnetic resonance (MR) imaging, images are reconstructed from k-space measurements. Because MR images are usually highly compressible in some transform domain, one can “compress” or undersample the MR measurements before reconstruction into MR images. The process of reconstructing images from fewer measurements is referred to as compressed sensing. MR compressed sensing accelerates MR scans by reducing the number of measurements needed per image. 
     However, because fewer measurements are present in the measurements generated by compressed sensing, noise-like aliasing artifacts may be present in reconstructed images. The aliasing artifacts due to the compressed sensing process may be removed during reconstruction using a prior model of fully sampled images and a model of the aliasing artifacts. Many reconstruction algorithms are available, each affecting the performance of the reconstruction process and the resulting image quality. For example, different reconstruction algorithms may be more or less effective at removing the aliasing artifacts from the reconstructed image. 
     SUMMARY 
     By way of introduction, the preferred embodiments described below include methods, systems, instructions, and computer readable media for reconstructing magnetic resonance compressed sensing data. 
     In a first aspect, a method is provided for reconstructing medical image data. The method may include acquiring compressed sensing data, determining a sampling density of the compressed sensing data and a noise level of the compressed sensing data, transforming the compressed sensing data into an estimated image that is an unbiased estimator of a true image corrupted by noise-like aliasing, determining an intensity of the noise-like aliasing, denoising the compressed sensing data based on a noise model constructed from the sampling density of the compressed sensing data, the noise level of the compressed sensing data, and the intensity of the noise-like aliasing, checking the denoised image data for consistency with the compressed sensing data, and outputting an output image as a function of the updated image data. A result of the denoising may be denoised image data. A result of the checking may be updated image data. 
     In a second aspect, a method is provided for image reconstruction for compressed magnetic resonance data. The method may include acquiring compressed sensing data, reconstructing the compressed sensing data into a medical image, and outputting an output image based on the updated image. The reconstructing may include generating a model of aliasing artifacts in the compressed sensing data based on a noise level of the compressed sensing data and a sampling density of the compressed sensing data, denoising the compressed sensing data, the denoising based on the model, the sampling density, and the noise level of the compressed sensing data, and generating an updated image based on the compressed sensing data and the denoised image data. A result of the denoising may be denoised image data. 
     In a third aspect, a magnetic resonance compressed sensing image reconstruction system is provided. The system may include an image processor, coupled with a memory containing instructions. When executed, the instructions may cause the image processor to receive compressed sensing data generated by a magnetic resonance imager, generate a model of noise in the compressed sensing data based on a noise level of the compressed sensing data and a sampling density of the compressed sensing data, reconstruct the compressed sensing data into a medical image by denoising the compressed sensing data based on the model and by generating an updated image based on the compressed sensing data and the denoised image data, estimate an error of the denoised image data, update a denoising parameter based on the error of the denoised image, repeat the denoising based on the parameter when the error is above a threshold error, and output an output image based on the updated image. A result of the denoising may denoised image data. 
     The present invention is defined by the following claims, and nothing in this section should be taken as a limitation on those claims. Further aspects and advantages of the invention are discussed below in conjunction with the preferred embodiments and may be later claimed independently or in combination. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The components and the figures are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the invention. Moreover, in the figures, like reference numerals designate corresponding parts throughout the different views. 
         FIG. 1  illustrates one embodiment of a method of reconstructing an image using a colored noise model; 
         FIG. 2  illustrates compressed sensing data and aliasing artifacts; 
         FIG. 3  illustrates a set of different reconstructions; 
         FIG. 4  illustrates another set of different reconstructions; 
         FIG. 5  illustrates a further set of different reconstructions; and 
         FIG. 6  is a block diagram of one embodiment of a system for reconstruction using a colored noise model for compressed sensing in MR. 
     
    
    
     DETAILED DESCRIPTION OF THE DRAWINGS AND PRESENTLY PREFERRED EMBODIMENTS 
     Compressed sensing is used for acquiring MR images because it reduces the time of acquiring measurements. In compressed sensing, undersampling may be performed using an incoherent sampling pattern that generates noise-like aliasing artifacts in the acquired MR data. 
     Though deep learning may be used to reconstruct MR images, deep learning systems are trained to perform well on reconstructing a specific dataset but do not generalize well to new applications that were not part of a training dataset. The incoherent undersampling present in compressed sensing presents another challenge as deep learning reconstruction systems are better suited to reconstructions involving structured artifacts such as temporal parallel acquisition technique (TPAT) undersampling or non-cartesian imaging. 
     In another reconstruction approach, the aliasing artifacts may be removed from the image data during reconstruction, for example, using a denoiser. The denoiser may effectively remove the aliasing artifacts if the distribution and intensity of the aliasing artifacts is modeled. A model of the aliasing artifacts may be generated based on the sampling density of the compressed sensing and a noise level of the compressed sensing data. A reconstruction process modeling the aliasing artifacts due to compressed sensing and removing the artifacts may improve the quality of the image resulting from reconstruction and the speed of the reconstruction process. As an alternative to deep learning-based reconstruction techniques, a reconstruction process that models the aliasing as noise and removes the aliasing artifacts during denoising may overcome the challenges faced by a deep learning reconstruction. 
     The aliasing artifacts may be modeled for each band of the k-space data. Removing the aliasing artifacts based on the model may require, for each band, tuning multiple parameters of the denoiser. Optimal denoising parameters may be specific to the undersampling pattern, coil sensitivities of coils of the MR imager, what part of a body is being scanned, and other factors. The denoising parameters may be manually tuned for optimal image quality. However, manual tuning may be time and resource intensive, and the denoising parameters tuned for reconstruction of one set of compressed sensing data and one denoiser may not be applicable to reconstructions of other image data, for example, generated using a different undersampling pattern, with different coils, and/or on other parts of the body using different denoisers. In other words, a denoiser manually optimized for one set of imaging conditions may not provide an adequate quality reconstruction under other imaging conditions. 
     Instead, the denoising parameters may be chosen, tuned, or updated based on minimizing an error introduced by the denoiser. Though the error (e.g. the difference between the denoised image and the ground truth) may not be directly calculable when the ground truth is not known, the error may be estimated. Minimizing the estimate of the error corresponds to minimizing the actual error. In this way, the denoising parameters may be automatically tuned without manual intervention. Because the tuning is automatic, a reconstruction and denoising technique based on error-minimization may be adaptable to different imaging conditions without manual retuning and may improve the efficiency of reconstruction and the quality of the reconstructed image. 
       FIG. 1  illustrates one embodiment of a method of reconstructing an image. Reconstruction in general is discussed with respect to  FIG. 1 . The use of a colored noise model in the denoising is described for act  107  in this reconstruction context. Though  FIG. 1 , as an example, reconstructs a two-dimensional (2D) image from 2D MR data, the acts of  FIG. 1  may also apply to reconstruction of three-dimension or higher-dimension MR data. Image space, as used in the description, may also refer to object space. Pixels, as used in the description, may also refer to voxels. 
     More, fewer, or different acts may be performed. In some cases, act  125  may be omitted. The acts may be performed in a different order than shown. A processor coupled to a memory may be configured to perform one or more of the acts. For example, the image processor  603  of  FIG. 6  may be configured to perform one or more of the acts of  FIG. 1 . The processor may be part of or in communication with a MR imaging device or system. For example, the MR imaging device may be the Magnetic Resonance Imager  609  of  FIG. 6 . 
     The reconstruction process may include three phases: a phase where the transform sparsity is increased, a consistency checking phase, and a consistency increasing phase. Other representations of the reconstruction process may be used. 
     In act  101 , k-space data is acquired. A MR imaging device may scan an imaging volume and record k-space data. The k-space data may be a measurement of radiofrequency intensity in the imaging volume. However, locations in the k-space may not correspond to physical locations or pixels/voxels within the imaging volume. 
     For compressed sensing, the k-space data may be undersampled. In some cases, the undersampling may be incoherent, or random across the frequency domain. Additionally or alternatively, the sampling may be a Fourier sampling or occur over a subset of frequencies in the frequency domain. The energy of the MR measurements taken by the MR imaging device may be more concentrated in low frequencies than in high frequencies. For example, the undersampling may be performed over the lower frequencies but not the higher frequencies. 
     Noise in the k-space, for example, caused by reception coils or other measurement devices of the MR device, may be distributed over the entire frequency range. The undersampling may cause aliasing artifacts that are visible when the k-space measurements are transformed into an image or object space (e.g. in act  103 ). Where the undersampling is incoherent, the aliasing artifacts will be distributed like measurement noise and, therefore, can be removed like measurement noise (e.g. in act  107 ). As compared to white noise (or random or Gaussian noise), the aliasing artifacts due to compressed sensing look like colored noise. In some cases, the measurement noise may be determined with the acquisition of the k-space data. For example, a noise-only pre-scan may be performed. The measurement noise may be determined based on the noise-only pre-scan. The distribution and intensity of noise in the pre-scan may correspond to the distribution and intensity of measurement noise in the acquired k-space data. In some other cases, the measurement noise (and the sampling density) may be estimated based on the acquired k-space data. 
     In act  103 , the k-space data is transformed into an image or object space. A Fourier transform may be used to turn the k-space data into an image. The image may represent an amplitude or intensity of electromagnetic response measured by the MR imager as distributed over two or three dimensions. The k-space data may be the measured k-space data (e.g. from act  101 ). On a first iteration, the image transformed from the measured k-space data may contain artifacts and noise. As a result, further refinement of the k-space data may be necessary to obtain a usable image. 
     In act  105 , the image generated in act  103  is transformed into w-space. The transformation may be made using the wavelet transform (the w-transform). The output of the w-transform may be known as w-space data or the wavelet representation of the MR data. In w-space, the MR data may have increased sparsity than in image space or k-space. The increased sparsity of the w-space, e.g. where the signal is concentrated into fewer pixels while most pixels have only a lower intensity, may allow for easier separation of the signal from the noise in the MR data than in the image space. 
     In act  107 , the w-space data is denoised. Noise in the w-space data may be due to both white (e.g. Gaussian) noise and noise due to aliasing artifacts from the compressed sensing. For the aliasing noise to be removed, the distribution of the noise over the w-space data may be determined. 
     A model of the aliasing noise may reveal where the aliasing noise is located across the w-space. With compressed sensing, the location or shape of the aliasing noise may be known because the MR data is more highly concentrated in a subset of (e.g. lower) frequencies; the aliasing may also be present in those frequencies, not evenly spread across the whole frequency spectrum, as would random or white (Gaussian) noise. Because reconstruction methods like vector approximate message passing (VAMP) assume that the noise present in the compressed sensing data is random and has the same energy distribution (e.g. the noise is spread across the frequency spectrum), such assumptions do not apply to compressed sensing where the MR measurements (and the aliasing artifacts) are concentrated within a subset of the frequency spectrum. 
     With incoherent undersampling across the k-space in compressed sensing, the MR aliasing “looks” like colored noise, not white, random, or Gaussian noise. For example, the sampling may be represented by or correspond to a vector of independent Bernoulli samplings across the frequency spectrum.
 
 m   k ˜Bernoulli( p   k )  Eq. 1
 
     In Equation 1, k represents the frequency index, p represents the sampling probability, or sampling density, at frequency k, Bernoulli refers to the Bernoulli probability distribution, and m represents the binary sampling mask resulting from drawing a random realization of the Bernoulli distribution according to the sampling probability p. Additionally or alternatively, the undersampling of the frequency spectrum may be represented as Fourier sampling according to the sampling probability p. The sampling density p may represent the extent to which the compressed sensing data is undersampled. 
     Because the undersampling is performed across the image spectrum, the spectrum of the aliasing may correspond to the sampling mask m. In this way, the aliasing spectrum may be modulated by both the image spectrum and the sampling density within that spectrum.
 
 x   k   =m   k ( y   k +ε k )  Eq. 2
 
     In Equation 2, k is the frequency index, x is the measured data (e.g. in act  101 ), y is the true data (or the MR data uncorrupted by measurement noise or aliasing artifacts), and ε is a white Gaussian measurement noise with standard deviation σ. 
     The expected value of the measured data may depend on the sampling probability and the true signal.
 
 E   m ( x )= py   Eq. 3
 
     In Equation 3, E m (x) represents the expected value of the MR k-space measurements applying the undersampling according to the sampling density m. To remove any bias in the k-space measurements x caused by undersampling, the measurements x can be divided by n. The noise present due to the aliasing artifacts may then be determined to be the difference between the measured data x and the true data y. 
     
       
         
           
             
               
                 
                   
                     
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                   4 
                 
               
             
           
         
       
     
     In Equation 4, E m (|p −1 x−y| 2 ) is a variance of the expected value of the measured data as compared to the true signal y. In some cases, because MR data is likely to reside in lower frequencies, the sampling probability p may be chosen based on the frequency. For example, the sampling probability may be high in the lower frequencies and lower in the higher frequencies. The variance may be determined for each frequency, thereby giving a frequency-specific estimate of the variance of the acquired data. Together, the frequency-specific estimates of the variance may be a vector of the variances. Because the variance corresponds to the variance of the aliasing as well, Equation 4 may serve as a frequency-specific model of the aliasing noise. 
     However, Equation 4 only provides the expected energy (or intensity) of the aliasing artifacts with respect to the mask. In other words, Equation 4 does not provide an aliasing level that can be used for one single scan. This may be overcome by looking in w-space, a space resolved both in space and frequency. The w-domain may be a wavelet domain, a pyramidal decomposition or another space-frequency domain. In the w-domain, however, the energy of the noise in the w-space is a linear combination of the energy of the aliasing at all the sampled frequencies, weighted by the power spectrum of the convolution filter, W, applied to go from image space to w-space. Due to the Central Limit Theorem, which states that weighted sum of a large number of independent random variables is close to its expectation with high probability, the aliasing energy in w-space can be determined for each scan. The aliasing energy in the w-space may be determined from the noise level in the w-space and the power spectrum of the applied filter. Because both the energy of the noise in the w-space and the sampling probability (used in the mask, see, e.g. Equation 1) may be known, then the energy of the aliasing may be determined from only known quantities. 
     
       
         
           
             
               
                 
                   
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     In Equation 5, F represents the Fourier transform, WF −1 y represents the true image in w-space, r represents an unbiased estimator of the true image using only the available data, and a is the residual error of that estimator due to aliasing and measurement noise. This model represents the aliasing noise present in each band, or across multiple frequencies, in the w-space. Having a model of the aliasing noise, parameters of the noise may be determined. For example, the variance of the aliasing noise generally or per each frequency band may be determined. Where the w-transform is applied to multiple sub-bands, the noise level for each sub-band may be estimated. Taken together, a band-specific estimate of the energy spectrum of the aliasing noise may be determined. 
     One or more denoising techniques may be used to reduce the residual error. For example, the w-space data may be denoised using thresholding or combinations of low pass and high pass filters. Parameters of the denoiser may be chosen or refined such that the denoiser removes noise without introducing significant error into the result (e.g. by removing the measured signal along with the noise). The error may be defined as the mean squared error between the output of the denoiser and the underlying signal—if the output of the denoiser is significantly different from the underlying signal, then the denoiser has introduced significant error. In practice, the underlying signal may not be known a priori because reconstructing uses k-space measurements that include the signal mixed gaussian noise and aliasing artifacts from the compressed sensing. 
     With the noise level (due to aliasing) modeled for each sub-band in the w-space, the denoiser may remove a band-specific amount of noise from each band. Rather than manually tuning parameters of the denoiser or used predetermined values, the denoising parameters (e.g. corresponding to how much noise to remove from each band) may be chosen and tuned in order to minimize the error introduced by the denoiser. 
     The error introduced by the denoiser may be estimated, for example, using Stein&#39;s Unbiased risk Estimator (SURE). Because the value of SURE corresponds to or is equal to the error produced by the denoiser, parameters of the denoiser may be chosen, adjusted, or updated to minimize SURE, thereby minimizing the error introduced by the denoiser. Because the parameters may be chosen, adjusted, or updated based on SURE, the denoiser may be automatically tuned based on the signal and noise of the k-space measurements without any knowledge of the imaging environment (e.g. including what imaging device was used, which coils were used, and/or which part of the body was scanned). 
     SURE views the k-space measurements as a signal corrupted with Gaussian noise with mean zero and known variance represented by a sum of the noise and the underlying signal.
 
 r=r   0 +ε,  Eq. 6
         where ε˜ (0, t)       

     In Equation 6, r represents the signal corrupted with gaussian noise, c, added to the underlying signal r 0 . The noise has a mean of zero and a known variance, t. Reconstruction attempts to recover r 0  by applying a denoiser, g( ), to the signal r based on parameters of the noise variance, and free parameters of the denoiser λ. The performance of the denoiser may be determined based on an estimate of the error, or difference, between the output of the denoiser and the original signal r 0 . 
     
       
         
           
             
               
                 
                   
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     In Equation 7, E(SURE) is the expected value of an estimate of the error using Stein&#39;s Unbiased Risk Estimator, as applied to the right half of the equation, the expected value of the mean squared error between the result of the denoiser, g(r, λ), and the original signal r 0 . N represents the number of measurements in the k-space. The result is that if the noise variance can be estimated, then SURE can be minimized to tune the denoising parameters because minimizing SURE corresponds to minimizing the mean squared error of the denoised signal. 
     For denoising via thresholding, all locations with an intensity lower than the threshold intensity are reduced to zero. The value of the threshold intensity may be chosen to minimize the expected value of SURE, as above. The value of the threshold may be subtracted from all other locations (e.g. those at or above the threshold). 
     For denoising using high pass and/or low pass filters, different denoising techniques may be applied per each frequency band in the w-space. For example, a Weiner filter may be applied to low pass bands, and a Garrote filter may be applied to the high pass bands. In some cases, the range of frequencies included in the high pass bands and the low pass bands may be determined based on an estimate of the error introduced by the denoiser. For example, Stein&#39;s Unbiased risk Estimator (SURE) may provide an estimate of the error, or mean square difference, between the output of the denoiser and the underlying signal (e.g. without noise) present in the k-space measurements. Choosing ranges of the high pass frequencies and the low pass frequencies that minimize SURE also minimize the error or difference between the output of the denoiser and the underlying signal present in the measured k-space data. 
     Regardless of whether thresholding, high/low-pass filtering, or another denoising technique is used, the denoising and tuning process may remain the same. A processor (e.g. the image processor  603  of  FIG. 6 , described below) may receive or use the w-space data. The processor may receive or determine a sampling density (e.g. from act  101  or the MR imager). In some cases, the sampling density may be a setting for the k-space measuring or otherwise a known parameter of the MR imaging. The processor may construct a model of the aliasing artifacts in the w-space data. For example, the processor may generate the model based on Equation 4 for each band of the w-space using a band-specific sampling density and a band-specific noise level of the w-space data. The processor may determine a variance of the aliasing artifacts based on the model of the aliasing artifacts. Once the model of the aliasing artifacts is generated, the denoiser may be applied to the w-space data to obtain denoised data. The denoising parameters may be tuned based on an error of the denoised image. For example, a processor may use Equation 7 and the variance of the aliasing artifacts to estimate the error of the denoised image. The processor may check the estimated noise level against a threshold or against a previous estimate of the error. When the estimated error exceeds a threshold error (or if the change between the current error estimate and the previous error estimate exceeds a threshold change) the processor may change one or more denoising parameters, denoise the w-space data using the changed denoising parameters, and check the error again. When the estimated error (or the change in error) is below the threshold, the processor may perform one or more of the acts of  FIG. 1 . 
     The result of denoising is that denoised w-space image data is obtained. Using SURE, the error of the denoised w-space image data may be determined. When the error is acceptable (e.g. at or below an error threshold), the reconstruction process may proceed. Where the error is unacceptable (e.g. at or above an error threshold), the denoising parameters may be adjusted or tuned, and denoising repeated with the adjusted denoising parameters. For example, for denoising using a thresholding technique that resulted in unacceptable error, the value of the threshold (overall or per-band) may be raised or lowered, and the denoising repeated with the raised or lowered threshold. Because the denoising has increased the number of pixels with a value of zero and reduced the number of pixels with a non-zero value, the thresholding has increased the sparsity of the data. 
     In act  109 , the denoised w-space data is corrected for any bias introduced during the denoising. The correction ensures that the next reconstruction iteration produces an unbiased estimator of the true image in w-space and prevents the propagation of bias throughout reconstruction. The correction may be known as an Onsager correction. In some cases, the correction may be performed in each band, e.g. applied to the pixels in each band of the w-space. For example, the correction may be performed using the same method as in the Vector Approximate Message Passing algorithm in each band. The corrected denoised band q may be a weighted sum of the band before and after denoising. The weights may be determined based on the degrees of freedom a of the denoiser, defined as the mean over all pixels in the band of the derivative of the output denoised pixel with respect to the input noisy pixel. The degrees of freedom may also be known as the divergence, in analogy a similar quantity used in fluid mechanics. Mathematically equivalent weights may also be computed as the inverse of the noise variances before and after denoising. As a result of the bias correction, corrected w-space data is obtained. 
     
       
         
           
             
               
                 
                   
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     In act  111 , the corrected w-space data (e.g. obtained from act  109 ) is transformed into the image domain. The inverse w-transform may be used to transform the w-space data. As a result of the transforming, a denoised image is obtained. The denoised image may have less noise in the image than the image generated by the Fourier transform in act  103 . However, the denoising may have altered the content of the image, so the denoised image is compared for consistency with the original measurements taken by the MR device. 
     In act  113 , the denoised image is transformed into the k-space. The inverse Fourier transform may be used to transform the denoised image into the k-space domain. As a result of the transforming, a k-space representation of the denoised image is obtained. By transforming into the k-space, the k-space representation of the denoised image may be compared to the original measurements taken by the MR device (e.g. in act  101 ) which are also in the k-space. 
     Since the denoised image was modified (e.g. during denoising in act  107 ), the k-space representation of the denoised image now contains information in all spatial frequencies. In other words, the k-space of the representation of the denoised image has a ‘complete k-space’ (all or substantially all locations in k-space have non-zero values), as opposed to the measured (subsampled) k-space (e.g. of act  101 ) which had only few non-zero values. 
     In act  115 , a mask or filter is applied to the k-space representation of the denoised image. Because the undersampling in compressed sensing reduces the overall signal level of the k-space measurements, the k-space representation of the denoised image must be compensated for the sampling density to obtain an unbiased estimate of the image. In some cases, the mask or filter multiplies the intensity of pixels or locations in the k-space by the inverse of the sampling density. For example, dividing both sides of Equation 2 by m will compensate for the sampling density and result in an unbiased estimate of the image once transformed out of k-space and into the image domain. In this way, the mask or filter may act as a high pass filter: because the sampling density may be higher in lower frequency bands and lower in the higher frequency bands, multiplying by the inverse of the sampling density will have little effect on the lower frequency bands and a greater effect on the higher frequency bands. In some other cases, the mask or filter may set a value of pixels in the k-space located outside of the region where the compressed sensing measurements were generated (e.g. in act  101 ) as equal to zero. The result of applying the filter or mask is a compensated k-space representation of the denoised image. Once filtered or masked to the region in the k-space sampled by the imaging, the filtered or masked k-space representation of the denoised image can be compared to the measured k-space data (e.g. from act  101 ). 
     In act  117 , a difference k-space is created by subtracting the compensated k-space representation of the denoised image (e.g. from act  115 ) from the measured k-space data (e.g. from act  101 ). The difference between the compensated k-space representation of the denoised image and the measured k-space data is the error, z, that the denoising added to the w-space data (e.g. in act  107 ). In some cases, the subtracting may be normalized by a difference between a noise level of the compensated k-space representation of the denoised image (or the noise level of the denoised image) and a noise level of the measured k-space data. As a result of the error present in the k-space representation of the denoised image, the denoised image may be less consistent with the information (e.g. features, structures) present in the measured k-space data. The error, difference, or residual between the masked or filtered k-space representation of the denoised image and the measured k-space data may be known as a difference k-space.
 
 z=x−mFW   −1   q   Eq. 9
 
     In act  119 , a Fourier transform is applied to the difference k-space (e.g. from act  117 ) multiplied by the inverse of the sampling density. As a result of applying the transform, a difference image is obtained. The difference image may represent the noise removed in the denoising step and may be used to correct the noise in the compressed sensing data. 
     In act  121 , the difference image is added to the image obtained earlier in the iteration (e.g. from act  109 ). The addition of the difference image and the previous image may be normalized by a difference between a noise level of the difference image and a noise level of the previous image. For example, where the previous image is an image estimate generated from measured k-space data (e.g. in acts  101  and  103 ), the noise level of the previous image may be a noise level of the measured k-space data. The combination of the difference image and the previous image may be referred to as an updated image. The updated image may be checked to see if the exit conditions are met, for example, in act  123 . 
     
       
         
           
             
               
                 
                   
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     In act  123 , the exit conditions for the reconstruction process are checked. The number of iterations may be one exit condition. For example, a predetermined number of iterations may be performed before the reconstruction process exits. One, five, ten, or another number of iterations may be performed. The number of iterations may depend on one or more factors. For example, where higher quality or lower noise images are required, the reconstructions process may have more iterations before exiting. 
     Another exit condition may be how different an updated image is from the image generated from just the k-space data. A small difference between the updated image and the original image estimate (e.g. before reconstruction and denoising), or between the current updated image and the last updated image created during the previous iteration, may indicate that the reconstruction process is not creating a large improvement in the image quality. A threshold difference may be predetermined. When the difference between the current updated image and the last updated image or the original image is below the threshold, the reconstruction may exit. 
     In act  125 , the updated image may be output if one, one or more, or all of the exit conditions are met. The updated image may be output to a computer, such as a workstation, a server, or a cloud, for later retrieval or viewing on a display. 
       FIG. 2  illustrates the compressed sensing data and the aliasing artifacts across four images  201 ,  203 ,  205 , and  207 . Image  201  shows a power spectrum of the ground truth magnetic resonance data. No undersampling or compressed sensing has been applied, and, therefore, no aliasing artifacts are present. 
     Image  203  shows a power spectrum of magnetic resonance data obtained by undersampling the ground truth of Image  201 . This may correspond to the state of the image data acquired in act  101  of  FIG. 1 , before the denoising or other acts of the reconstruction process have been performed. Though some of the ground truth is visible in Image  203 , the undersampling has added a haze or layer of noise over the ground truth. The layer of noise is the aliasing artifacts. Unlike white or random noise that would be spread over the image regardless of where the ground truth signal is located, the aliasing artifacts are concentrated in areas of the power spectrum where the ground truth signal is located. In this way, the aliasing artifacts look like colored noise and are different from white noise. 
     Image  205  shows the aliasing artifacts after subtracting the measured k-space from the denoised image (e.g. after act  117  of  FIG. 1 ). The unstructured aliasing due to compressed sensing remains in the image after subtracting. The aliasing looks like low pass colored noise because the aliasing is modulated by the signal spectrum. 
     Image  207  shows the product of the sampling density and the signal spectrum. Because the aliasing is modulated by the signal spectrum and the sampling density, the product of the signal spectrum and the sampling density gives a texture that describes the texture of the aliasing. By modeling the aliasing using the signal spectrum and the sampling density, the aliasing can be removed during reconstruction (e.g. in act  107  of  FIG. 1 ). For example, a threshold may be set with a threshold value chosen or determined to remove the aliasing as present in image  207 . 
       FIG. 3  shows a set of different reconstructions across four pairs of images  301 ,  303 ,  305 , and  307 . Pair  301  shows an image resulting from a reconstruction of fully sampled magnetic resonance imaging data. The magnetic resonance data was obtained using a single coil. Below the image is an undersampling pattern applied to the magnetic resonance data to form the undersampled compressed sensing data used as the basis for reconstructions of images  303 ,  305 , and  307 . The undersampling for  303 ,  305 , and  307  was based on point sampling with an undersampling rate of 4.8. 
     Pair  303  shows an image reconstructed from compressed sensing data using an approximate message passing technique. A map of the error present in the reconstructed image is shown below. The denoising function in the reconstruction of  303  is soft thresholding. As opposed to the automatic and iterative denoiser parameter tuning described in  FIG. 1 , the reconstruction of  303  was performed using a thresholding parameter tuned by hand. The signal to noise ratio (SNR) of  303  is 34.1. 
     Pair  305  shows an image reconstructed from compressed sensing data using the method of  FIG. 1 . A map of the error present in the reconstructed image is shown below. The denoising function in the reconstruction of  305  is soft thresholding. The reconstruction of  305  was performed using a thresholding parameter tuned automatically by minimizing the value of SURE, as described above for act  107  of  FIG. 1 . The SNR of  305  is 36.2. Though the reconstruction in  305  used the same denoiser as in  303 , a higher SNR was achieved by automatically tuning the thresholding parameter. 
     Pair  307  shows an image reconstructed from compressed sensing data using the method of  FIG. 1 . A map of the error present in the reconstructed image is shown below. The denoising function in the reconstruction of  307  is a Weiner filter applied on the low pass band and a Garrote filter applied on the high pass bands. The reconstruction of  305  was performed using a thresholding with a pass band demarcation parameter tuned automatically by minimizing the value of SURE, as described above for act  107  of  FIG. 1 . The SNR of  307  is 37.1. Though a different denoiser was applied in the reconstruction of  307 , a high SNR was still achieved. Minimizing the value of SURE may allow for nearly any kind of denoiser to be tuned for a good reconstruction result. 
       FIG. 4  shows a set of different reconstructions across four pairs of images  401 ,  403 ,  405 , and  407 . Pair  401  shows an image resulting from a reconstruction of fully sampled magnetic resonance imaging data. The magnetic resonance data was obtained using eight receive coils. Below the image is an undersampling pattern applied to the magnetic resonance data to form the undersampled compressed sensing data used as the basis for reconstructions  403 ,  405 , and  407 . The undersampling for  403 ,  405 , and  407  was based on cartesian sampling with an undersampling rate of 2.5. 
     Pair  403  shows an image reconstructed from compressed sensing data using an approximate message passing technique. A map of the error present in the reconstructed image is shown below. The denoising function in the reconstruction of  403  is soft thresholding. As opposed to the automatic and iterative denoiser parameter tuning shown in  FIG. 1 , the reconstruction of  403  was performed using a thresholding parameter tuned by hand. The signal to noise ratio (SNR) of  403  is 34.3. 
     Pair  405  shows an image reconstructed from compressed sensing data using the method of  FIG. 1 . A map of the error present in the reconstructed image is shown below. The denoising function in the reconstruction of  405  is soft thresholding. The reconstruction of  405  was performed using a thresholding parameter tuned automatically by minimizing the value of SURE, as described above for act  107  of  FIG. 1 . The SNR of  405  is 36.8. Though the reconstruction in  405  used the same denoiser as in  403 , a higher SNR was achieved by automatically tuning the thresholding parameter. 
     Pair  407  shows an image reconstructed from compressed sensing data using the method of  FIG. 1 . A map of the error present in the reconstructed image is shown below. The denoising function in the reconstruction of  407  is a Weiner filter applied on the low pass band and a Garrote filter applied on the high pass bands. The reconstruction of  305  was performed using a thresholding with a pass band demarcation parameter tuned automatically by minimizing the value of SURE, as described above for act  107  of  FIG. 1 . The SNR of  407  is 37.1. Though a different denoiser was applied in the reconstruction of  407 , a high SNR was still achieved. Minimizing the value of SURE may allow for nearly any kind of denoiser to be tuned for a good reconstruction result. Though the number of coils and sampling pattern differed between  307  and  407  ( FIGS. 3 and 4 ), a similar SNR was achieved in the end without manually tuning parameters. By using SURE to tune the denoising parameters, the reconstruction process of  FIG. 1  may be automatically adaptable to different imaging and reconstruction environments. 
       FIG. 5  shows a set of different reconstructions across four pairs of images  501 ,  503 ,  505 , and  507 . Pair  501  shows an image resulting from a reconstruction of fully sampled magnetic resonance imaging data. The magnetic resonance data was obtained using eight receive coils. Below the image is an undersampling pattern applied to the magnetic resonance data to form the undersampled compressed sensing data used as the basis for reconstructions  503 ,  505 , and  507 . The undersampling for  503 ,  505 , and  507  was based on cartesian sampling with an undersampling rate of 3.8. 
     Pair  503  shows an image reconstructed from compressed sensing data using an approximate message passing technique. A map of the error present in the reconstructed image is shown below. The denoising function in the reconstruction of  503  is soft thresholding. As opposed to the automatic and iterative denoiser parameter tuning shown in  FIG. 1 , the reconstruction of  503  was performed using a thresholding parameter tuned by hand. The signal to noise ratio (SNR) of  503  is 32.1. 
     Pair  505  shows an image reconstructed from compressed sensing data using the method of  FIG. 1 . A map of the error present in the reconstructed image is shown below. The denoising function in the reconstruction of  505  is soft thresholding. The reconstruction of  505  was performed using a thresholding parameter tuned automatically by minimizing the value of SURE, as described above for act  107  of  FIG. 1 . The SNR of  505  is 33.4. Though the reconstruction in  505  used the same denoiser as in  503 , a higher SNR was achieved by automatically tuning the thresholding parameter. 
     Pair  507  shows an image reconstructed from compressed sensing data using the method of  FIG. 1 . A map of the error present in the reconstructed image is shown below. The denoising function in the reconstruction of  507  is a Weiner filter applied on the low pass band and a Garrote filter applied on the high pass bands. The reconstruction of  305  was performed using a thresholding with a pass band demarcation (threshold) parameter tuned automatically by minimizing the value of SURE, as described above for act  107  of  FIG. 1 . The SNR of  507  is 34.0. Though a different denoiser was applied in the reconstruction of  507 , a high SNR was still achieved. Minimizing the value of SURE may allow for nearly any kind of denoiser to be tuned for a good reconstruction result. 
       FIG. 6  is a block diagram of one embodiment of a magnetic resonance compressed sensing image reconstruction system  601  for reconstructing magnetic resonance compressed sensing data. The computing system  601  may include an image processor  603  coupled with a memory  605  and in communication with a network connection  607 , a magnetic resonance imager  609 , and a display  611 . The computing system  601  performs the acts of  FIG. 1  or other acts. 
     The image processor  603  may be a general purpose or application specific processor. The image processor  603  may be configured to or may execute instructions that cause the image processor  603  to receive compressed sensing data generated by a magnetic resonance imager  609 . In some cases, the compressed sensing data may be received via the network connection  607 . For example, the image processor  603  may receive the compressed sensing data via the network connection  607  that was transmitted by a remote server or medical imager  609 . In some other cases, the compressed sensing data may be received from the magnetic resonance imager  609  without being routed through the network connection  607 . The image processor  603  may be part of the MR imager  609 . 
     The image processor  603  may be configured to or may execute instructions that cause the processor to reconstruct the compressed sensing data into a medical image by denoising the compressed sensing data based on a noise level of the compressed sensing data and a sampling density of the compressed sensing data, where a result of the denoising is denoised image data and by generating an updated image based on the compressed sensing data and the denoised image data. Based on the updated image, the processor  601  may be configured to output an output image. In some cases, the output image may be transmitted to the display  611 . In some other cases, the output image may be output to the memory  605 , the network connection  607 , and/or the magnetic resonance imager  609 . 
     The memory  605  may be a non-transitory computer readable storage medium. The memory  605  may be configured to store instructions that cause the processor to perform an operation. For example, the memory  605  may store instructions that, when executed by the processor  601 , cause the processor  601  to perform one or more acts of  FIG. 1  or other acts. The memory  605  may be configured to store compressed sensing data, noise models, denoisers, denoising parameters, and other information related to executing the acts of  FIG. 1  or other acts. The instructions for implementing the processes, methods, and/or techniques discussed herein are provided on non-transitory computer-readable storage media or memories, such as a cache, buffer, RAM, removable media, hard drive, or other computer readable storage media. Non-transitory computer readable storage media include various types of volatile and nonvolatile storage media. 
     The network connection  607  may receive compressed sensing data. The network connection  607  may be in communication with the processor  603 , the memory  605 , the magnetic resonance imager  609 , and to computers, processor, and imagers external to the computing system  601 . The network connection  607  may provide the compressed sensing data to the processor  603  and/or the memory  605  for storage, recall, and reconstruction. 
     The magnetic resonance imager  609  may generate compressed sensing data. For example, the magnetic resonance imager  609  may generate compressed sensing data of an object located in an imaging volume of the magnetic resonance imager  609 . The magnetic resonance imager may be configured to perform one or more acts of  FIG. 1 . For example, the imager may be configured to perform act  101 . Compressed sensing data generated by the magnetic resonance imager  609  may be stored in the memory  605  and reconstructed by the processor  603 . 
     The display  611  may be configured to accept user input and to display audiovisual information to the user. In some cases, the display  611  may include a screen configured to present the audiovisual information. For example, the display  611  may present the output image of the reconstruction or one or more intermediate images from a reconstruction iteration. The display  611  may include a user input device. In some cases, the user may input information relating to a particular denoiser to be used in reconstruction. 
     While the invention has been described above by reference to various embodiments, it should be understood that many changes and modifications can be made without departing from the scope of the invention. It is therefore intended that the foregoing detailed description be regarded as illustrative rather than limiting, and that it be understood that it is the following claims, including all equivalents, that are intended to define the spirit and scope of this invention.