Patent Description:
An image taken by an imaging system, such as a magnetic resonance imaging (MRI) system, may be represented either as an image data in the space domain or as an image-related data in the k-space, i.e. the frequency domain. Sharp transitions in the image, such as those near the boundary of an organ, may be demonstrated in the k-space using relatively high frequency components. Nevertheless, the limited time of sampling or poor signal to noise (SNR) ratio may lead to the under-sampling of image data in the k-space. It may result in the shortage of high-frequency components in the image data, thus causing the phenomenon of "ringing" in the reconstructed image. It is often referred to as the "Gibbs artifact.

Accordingly, it would be desirable to effectively and substantially reduce the ringing artifact in the reconstructed image while substantially maintaining the resolution and the signal-to-noise ratio of the final image without increasing the scan time,.

Methods for reducing Gibbs artifacts in magnetic resonance images are known for example from the following documents:.

According to the first aspect of the invention a method is provided as defined in claim <NUM>. According to the second aspect of the invention a non-transitory computer readable medium comprising executable instructions is provided as defined in claim <NUM>. According to the third aspect of the invention a system is provided as defined in claim <NUM>.

The present disclosure relates to image processing. Specifically, one aspect of the present disclosure relates to a method for reducing the Gibbs artifact and/or the effect of under-sampling of k-space data in the reconstructed image. According to some embodiments of the present disclosure, the method may be based on undersampled image-related data in the k-space. Specifically, the method may include filling a matrix of sampled data in the k-space. Some procedure of pre-processing, such as filtering and padding, may be performed first. For example, a low-pass filter may be applied on the matrix of sampled data to attenuate the high-frequency component in the matrix of data. Besides the matrix of sampled data in the k-space, the matrix of sampled data may be extended to an outer area enclosing the matrix of data in the k-space, on which a padding may be performed. The padding may be a zero-padding or a non-zero padding.

Another aspect of the present disclosure relates to a non-transitory computer readable medium including executable instructions. The instructions, when executed by at least one processor, may cause the at least one processor to effectuate a method for image processing. In some embodiments, the non-transitory computer readable medium may include instructions for causing a computer to implement the method.

A further aspect of the present disclosure relates to a system for image processing. The system may include a frequency setting block to specify a first part and a second part of k-space. In some embodiments, the first part may include a first region and a second region. In some embodiments, the second part may be outside of the first part. The system may further include a storage block configured to fill a matrix comprising data in the first part of the k-space. The system may further include a filter configured to act on the matrix in the first part to produce a filtered data matrix in the first part. The system may further include a padding block to fill the second part of the k-space. The system may further include a calculation block to perform, based on a constraint, a plurality of iterations of an objective function for a target array comprising data in image domain, wherein the objective function is based on a total variation of the target array. The system may further include an image construction block to reconstruct an image based on the target array of data.

In some embodiments, the method may include one or more of the following operations. The k-space comprising a first part and a second part may be set. A matrix comprising data may be filled in the first part of the k-space. A filter may be applied on the matrix in the first part to produce a filtered data matrix in the first part. The second part of the k-space may be padded. A plurality of iterations of an objective function for a target array comprising data in image domain may be performed based on a constraint, wherein the objective function is based on a total variation of the target array. An image may be reconstructed based on the target array of data.

In some embodiments, the objective function may be based on a first function based on the Fourier transform of the target array, the filtered data matrix in the first part, and the padded data in the second part of the k-space.

In some embodiments, the matrix comprising data may be undersampled.

In some embodiments, the second part may be outside of the first part.

In some embodiments, the filter may be based on multiple orthogonal filters.

In some embodiments, the padding may be zero padding.

In some embodiments, the total variation may be based on a first order derivative of the target array of data in image domain.

In some embodiments, the total variation may be based on a second order derivative of the target array of data in image domain.

In some embodiments, the first part may be divided into a first region and a second region.

In some embodiments, the constraint may be given by setting the filtered data in the second region of the first part of the k-space to be invariant.

In some embodiments, the first function may be based on an L-<NUM> norm function.

In some embodiments, the constraint may be given as a term of the objective function based on the Fourier transform of the target array and the filtered data in the second region of the first part of the k-space.

In some embodiments, the constraint may be based on an L-<NUM> norm function.

In some embodiments, the constraint strength may be tuned by setting a coefficient for the term in the objective function.

These embodiments are non-limiting examples, in which like reference numerals represent similar structures throughout the several views of the drawings, and wherein:.

In the following detailed description, numerous specific details are set forth by way of example in order to provide a thorough understanding of the relevant disclosure. However, it should be apparent to those skilled in the art that the present disclosure may be practiced without such details. In other instances, well known methods, procedures, systems, components, and/or circuitry have been described at a relatively high-level, without detail, in order to avoid unnecessarily obscuring aspects of the present disclosure. Various modifications to the disclosed embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be applied to other embodiments and applications without departing from the scope defined by the appended claims. Thus, the present disclosure is not limited to the embodiments shown, but to be accorded the widest scope consistent with the claims.

It will be understood that the term "system," "engine," "unit," "module," and/or "block" used herein are one method to distinguish different components, elements, parts, section or assembly of different level in ascending order. However, the terms may be displaced by other expression if they may achieve the same purpose.

It will be understood that when a unit, engine, module or block is referred to as being "on," "connected to," or "coupled to" another unit, engine, module, or block, it may be directly on, connected or coupled to, or communicate with the other unit, engine, module, or block, or an intervening unit, engine, module, or block may be present, unless the context clearly indicates otherwise.

The terminology used herein is for the purposes of describing particular examples and embodiments only, and is not intended to be limiting. It will be further understood that the terms "include," and/or "comprise," when used in this disclosure, specify the presence of integers, devices, behaviors, stated features, steps, elements, operations, and/or components, but do not exclude the presence or addition of one or more other integers, devices, behaviors, features, steps, elements, operations, components, and/or groups thereof. It will be further understood that the terms "constructed" and "reconstruct," when used in this disclosure, may represent a similar process that an image may be obtained based on image data.

In some embodiments, the present disclosure may be applicable to various modalities of imaging systems. Exemplary imaging modalities may include Digital Subtraction Angiography (DSA), Magnetic Resonance Imaging (MRI), Magnetic Resonance Angiography (MRA), Computed tomography (CT), Digital Radiography (DR), Computed Tomography Angiography (CTA), Ultrasound Scanning (US), Positron Emission Tomography (PET), Single-Photon Emission Computerized Tomography (SPECT), CT-MR, CT-PET, CE-SPECT, DSA-MR, PET-MR, PET-US, SPECT-US, TMS (transcranial magnetic stimulation)-MR, US-CT, US-MR, X-ray-CT, X-ray-MR, X-ray-portal, X-ray-US, Video-CT, Vide-US, or the like, or any combination thereof. This is understood that the following descriptions are provided in connection with medical image processing for illustration purposes and not intended to limit the scope of the present disclosure. However, the invention as defined in the appended claims is limited to Magnetic Resonance Imaging.

The image processing disclosed herein may be used for purposes other than medical treatment or diagnosis. For instance, the image processing may be used for purposes of detecting a fracture within a structure or its progression over time, a nonuniform portion within a piece of material, etc..

In some embodiments, the subject may be a human being, an animal, an organ, a texture, a region, an object, a lesion, a tumor, or the like, or any combination thereof. Merely by way for example, the object may include a head, a breast, a lung, a trachea, a pleura, a mediastinum, an abdomen, a long intestine, a small intestine, a bladder, a gallbladder, a triple warmer, a pelvic cavity, a backbone, extremities, a skeleton, a blood vessel, or the like, or any combination thereof. In some embodiments, the medical image may include a 2D image and/or a 3D image.

For illustration purposes, the following description is provided to help better understanding an image processing. It is understood that this is not intended to limit the scope of the present disclosure. For persons having ordinary skills in the art, a certain amount of variations, changes and/or modifications may be deducted under guidance of the present disclosure. However, those variations, changes and/or modifications do not depart from the scope of the present disclosure.

The present disclosure relates to image processing. Specifically, the present disclosure relates to a system and method for reducing the Gibbs artifact and/or the effect of under-sampling of k-space data in the reconstructed image. According to some embodiments of the present disclosure, the method may be based on undersampled image-related data in the k-space. The method may include performing iterations of an objective function. The objective function may be based on a total variation of a target array of data in the image domain. The objective function may be further based on a total variation of the target array of data and a function of the residual between the Fourier transform of the target array of data and the filtered image data being padded in the k-space. The iteration may be subject to a constraint given by, for example, setting the filtered image-related data in a region (e.g., an inner region) of the k-space to be invariant. The method may be based on undersampled data in the k-space. The method may provide an improved image with reduced Gibbs artifact and/or a reduced effect due to the under-sampling.

<FIG> illustrates an imaging system according to some embodiments of the present disclosure. The system may include an MRI (magnetic resonance imaging) scanner <NUM>, a plurality of MRI auxiliary devices <NUM>, a central controller <NUM>, an input/output device <NUM>, and a data processing system <NUM>. The MRI scanner <NUM> may scan a subject located within it and generate a plurality of data relating to the subject. The MRI scanner <NUM> may include a main magnetic field generator, a plurality of gradient coils, a radiofrequency (RF) transmitter, and/or a RF receiver. The main magnetic field generator may generate a static magnetic field (for example, a magnetic field B0 in the Z direction). The main magnetic field generator may be of various types including, for example, a permanent magnet, a superconducting electromagnet, a resistive electromagnet, etc. The gradient coils may include X-gradient coils, Y-gradient coils, and Z-gradient coils. The gradient coils may generate magnetic field gradients to the main magnetic field in the X, Y, and/or Z directions to encode the spatial information of the subject being scanned. In some embodiments, the X-gradient is configured to provide the X-position information, which may be known as frequency encoding. In some embodiments, the Y-gradient is configured to provide the Y-positon information, which may be known as phase encoding. The RF transmitter may include a plurality of RF coils. The RF transmitter may generate a RF magnetic field. Under the coordinated action of the static magnetic field, the gradient magnetic field and the RF magnetic field, MR signals relating to the subject being scanned may be generated. The RF receiver may receive the MR signals for image construction. The RF receiver may include a plurality of RF coils. In some embodiments, the function, size, type, geometry, position, amount, and/or magnitude of the main magnetic field generator, gradient coils, RF transmitter and receiver may be determined or changed according to one or more specific conditions. For example, according to the difference in function and size, the RF coils may be classified as volume coils and local coils. In some embodiments, the volume coils may include birdcage coils, transverse electromagnetic coils, surface coils, saddle coils, etc. In some embodiments, the local coils may include birdcage coils, solenoid coils, saddle coils, flexible coils, etc..

The MRI auxiliary devices <NUM> may coordinate with the MRI scanner <NUM> to generate a plurality of data relating to a subject. The MRI auxiliary devices <NUM> may include one or more gradient amplifiers, a RF amplifier, and a positioning device. The gradient amplifiers may be connected with the gradient coils in the MRI scanner <NUM>. The gradient amplifiers may include an X gradient amplifier, a Y gradient amplifier, and a Z gradient amplifier. One or more of the gradient amplifiers may be connected to a waveform generator (not shown in <FIG>). The waveform generator may generate various gradient waveforms that are applied to the gradient amplifiers. The waveforms (for example, currents or voltages) may be amplified by the gradient amplifiers and applied to the gradient coils to control the magnetic field strength and direction in the MRI scanner <NUM>. The RF amplifier may be connected with the RF transmitter. The RF amplifier may be connected to a waveform generator (not shown in <FIG>). The waveform generator may generate RF signals that are applied to the RF amplifier. The RF signals may be amplified by the RF amplifier and conveyed to the RF transmitter to generate a RF magnetic field. The positioning device may be configured to adjust the position of the subject in the FOV (field of view) of the MRI scanner <NUM>. The positioning device may include a table to be moved to a desired position for or during the scan.

The central controller <NUM> may control the MRI scanner <NUM>, the MRI auxiliary devices <NUM>, the input/output device <NUM>, and/or the data processing system <NUM>. The central controller <NUM> may receive information from or send information to the MRI scanner <NUM>, the MRI auxiliary devices <NUM>, the input/output device <NUM>, and/or the data processing system <NUM>. For example, the central controller <NUM> may receive commands from the input/output device <NUM> provided by a user; the central controller <NUM> may process data input by a user via the input/output unit <NUM> and transform the data into one or more commands; the central controller <NUM> may control the MRI scanner <NUM>, the MRI auxiliary devices <NUM>, and/or the data processing system <NUM> according to the received commands or transformed commands; the central controller <NUM> may receive MR signals or data related to a subject from the RF receiver of the MRI scanner <NUM>; the central controller <NUM> may send MR signals or data to the data processing system <NUM>; the central controller <NUM> may receive processed data or constructed image from the data processing system <NUM>; the central controller <NUM> may send processed data or constructed image to the input/output device <NUM> for displaying. In some embodiments, the central controller <NUM> may include a computer, a program, an algorithm, a software, a storage device, and a plurality of interfaces of the MRI scanner <NUM>, the MRI auxiliary devices <NUM>, the input/output device <NUM>, and/or the data processing system <NUM>.

The input/output device <NUM> may receive input and/or output information. The input and/or output information may include programs, software, algorithms, data, text, number, images, voices, or the like, or any combination thereof. For example, a user may input some initial parameters or conditions to initiate a scan. As another example, some information may be imported from an external resource including, for example, a floppy disk, a hard disk, a wired terminal, a wireless terminal, or the like, or any combination thereof. The output information may be transmitted to a display, a printer, a storage device, a computing device, or the like, or a combination thereof.

The data processing system <NUM> may process data relating to a subject and construct an image. In some embodiments, the data processing system <NUM> may be a program, an algorithm, and/or a software implemented on the central controller <NUM>. In some embodiments, the data processing system <NUM> may be an independent system, coordinated with the central controller <NUM>, including a processer, a controller, a memory, a display, a program, an algorithm, and/or a software. The data to be processed may be generated from the MRI scanner <NUM>, or acquired from other external sources. For example, the data may be raw data generated from the MRI scanner <NUM>; the data may be pre-treated by the central controller <NUM>; the data may be pre-stored in a storage device of or accessible from the central controller <NUM>; the data may be imported from an external resource including, for example, a floppy disk, a hard disk, a wired terminal, a wireless terminal, or the like, or any combination thereof. In some embodiments, the data to be processed and/or the image already constructed may include noise, artifacts, etc. The data processing system <NUM> may reduce or eliminate the noise, artifacts, etc., in the data or image. An exemplary artifact is Gibbs artifact, which may be also known as Gibbs effect/phenomenon, ringing artifact/effect, Gibbs ringing, truncation artifact, and/or spectral leakage artifact. Gibbs artifact may be caused by under-sampling in high spatial frequencies during data generation (i.e., data may be under sampled). As illustrated in a schematic drawing <NUM> with Gibbs artifact, alternating bright or dark lines/bands may appear parallel to and adjacent to the boundary of an area with abrupt change of signal intensity, the bright band in the schematic drawing <NUM>. The multiple lines/bands may be regularly spaced, and fade as the distance to the boundary increases. In some embodiments, the data processing system <NUM> may reduce or eliminate Gibbs artifact. As illustrated in the schematic drawing <NUM>, lines/bands may become less visible and the Gibbs artifact may be reduced or eliminated via data processing.

It should be noted that the above description of the imaging system is merely provided for the purposes of illustration, and not intended to limit the scope of the present disclosure. Many alternatives, modifications, and variations may be apparent to those skilled in the art. For example, the MRI scanner <NUM> and the MRI auxiliary devices <NUM> may be combined with a computed tomography (CT) scanner, or a positron emission tomography (PET) scanner. As another example, the function of the system may be varied or changed according to specific implementation scenarios. Merely by way of example, the data processing system <NUM> may include a noise removing unit, or other units.

<FIG> is a block diagram depicting an image processing system according to some embodiments of the present disclosure. The image processing system may acquire image data, process image data to reduce or remove artifact, and construct an image. The image processing system may include a data acquisition module <NUM>, a storage module <NUM>, a control module <NUM>, a processing module <NUM>, and an input/output module <NUM>. The data acquisition module <NUM> may acquire image data. The image data acquired may be data in the Fourier region (or referred to as the spatial frequency space, or the k-space), data in the image domain (or space domain), or the like, or any combination thereof. The data may be acquired from a scan of a subject, or an external resource including, for example, a floppy disk, a hard disk, a wired terminal, a wireless terminal, or the like, or any combination thereof. In some embodiments, the data acquisition module <NUM> may include or communicate with an imaging device including, for example, an MRI device (for example, the MRI scanner <NUM> and the MRI auxiliary devices <NUM>), a computed tomography (CT) device, a positron emission computed tomography (PET) device, etc. It should be noted that the imaging device may be a single-modality imaging device, or a multi-modality imaging device, or the like, or any combination thereof. Exemplary imaging devices may include a PET-MRI device, a CT-MRI device, a remote MRI device that may communicate with one or more of other modules via a wired or wireless connection, etc..

Data regarding a subject may be generated from the imaging device and acquired by the data acquisition module <NUM>. In some embodiments, the data acquired may be undersampled. In some embodiments, the data acquisition module <NUM> may be connected with a terminal via a wired or wireless connection. Data may be transmitted from the terminal and received by the data acquisition module <NUM>. In some embodiments, the data acquisition module <NUM> may include a data reading device to read data from a data storage medium including, for example, a floppy disk, a hard disk, an optical disk (e.g., a compact disc (CD), a digital versatile disc (DVD)), a flash memory, a universal serial bus (USB) flash disk, an secure digital (SD) card, a compact flash (CF) card, a memory stick, etc..

The storage module <NUM> may store data of the image processing system. The data stored may be a numerical value, a signal, an image, information of a subject, an instruction, an algorithm, or the like, or a combination thereof. The data stored may be in the Fourier region (or referred to as the spatial frequency space, or the k-space), or in the image domain. The data stored may be acquired by the data acquisition module <NUM>, imported via the input/output module <NUM>, generated in the processing module <NUM>, or pre-stored in the storage module <NUM> during system initialization or before an operation of data processing. The storage module <NUM> may include a system storage (e.g., a disk) that is provided integrally (i.e. substantially non-removable), or a storage that is removably connectable to the system via, for example, a port (e.g., a UBS port, a firewire port, etc.), a drive (a disk drive, etc.), etc. The storage module <NUM> may include, for example, a hard disk, a floppy disk, selectron storage, random access memory (RAM), dynamic random access memory (DRAM), static random access memory (SRAM), bubble memory, thin film memory, magnetic plated wire memory, phase change memory, flash memory, a cloud disk, or the like, or a combination thereof. The storage module <NUM> may be connected or communicate with one or more of the data acquisition module <NUM>, the control module <NUM>, the processing module <NUM>, and the display module <NUM>. In some embodiments, the storage module <NUM> may be operationally connected with one or more virtual storage resources (e.g., cloud storage, a virtual private network, other virtual storage resources, etc.).

The control module <NUM> may be configured to control operation of the image processing system. In some embodiments, the control module <NUM> may control the operation of the data acquisition module <NUM> in data acquisition. For example, the control module <NUM> may control: the parameters setting (e.g., the magnetic field intensity, the magnetic field gradient, etc.) of an imaging device (if any), the production of the waveforms of the waveform generator in the imaging device (if any), and the position of a subject (if any) to be scanned, etc..

In some embodiments, the control module <NUM> may control the input or output of data into or from the storage module <NUM>. In some embodiments, the control module <NUM> may control data transmitting among the data acquisition module <NUM>, the storage module <NUM>, the processing module <NUM>, and/or the input/output module <NUM>.

In some embodiments, the control module <NUM> may control the operation of the processing module <NUM> in data processing. For example, the control module <NUM> may control: the order of data calculation or processing, the adjustment of some parameters that may be used in the processing, the time or condition for triggering a data processing or ending a data processing, etc..

In some embodiments, the control module <NUM> may control the operation of the input/output module <NUM> for data display. For example, the control module <NUM> may control: the data display quality, image contrast, image resolution, image color, etc..

The control module <NUM> may perform system control according to some parameters, commands or instructions from other modules of the system. In some embodiments, the parameters, commands or instructions may be acquired from the storage module <NUM>. In some embodiments, the control module <NUM> may receive commands from the input/output module <NUM> provided by a user, process information provided by a user via the input/output module <NUM>, transform the information into specific commands, or the like, or a combination thereof. The control module <NUM> may be constructed based on an application-specific integrated circuit (ASIC), a microcontroller, a field programmable gate array (FPGA), an ARM, or the like, or any combination thereof.

The processing module <NUM> may process data and construct an image. The data may be acquired from the data acquisition module <NUM>, or the storage module <NUM>. The data to be processed may be in the Fourier region or in the image domain, or any combination thereof. Data in the image domain may be transformed to the Fourier region by way of Fourier transform; data in the Fourier region may be transformed to the image domain by way of inverse Fourier transform. In some embodiments, data transformation from the Fourier region to the image domain may be called as image reconstruction. In some embodiments, data processing may be performed in the Fourier region. Exemplary data processing in the Fourier region may include data filtering based on frequency, noise reduction, padding, interpolation, etc. In some embodiments, data processing may be performed in the image domain. Exemplary data processing in the image domain may include interpolation, logarithmic transforms, power law transforms, histogram equalization, etc. In some embodiments, data processing may be performed in both the Fourier region and image domain. For example, some data processing may be performed in the Fourier region and some data processing in the image domain. As another example, data processing in the Fourier region and the image domain may be performed alternately. In some embodiments, the processing module <NUM> may reduce or remove artifacts including Gibbs artifact, motion artifact, flow artifact, metal artifact, chemical shift artifact, partial volume artifact, wrap around artifact, or the like, or any combination thereof. For example, the processing module <NUM> may apply a plurality of algorithms (e.g., low-pass filtering in the Fourier region, interpolation in the Fourier region or image domain, total variation constrained data extrapolation, or the like, or any combination thereof) to reduce or remove Gibbs artifact.

The input/output module <NUM> may receive or output information. Merely by way of example, the input/output module <NUM> may provide data for display. The data displayed may include a value, a text, an image, and information of a subject. In some embodiments, the input/output module <NUM> may include a display for displaying data transmitted from the data acquisition module <NUM>, the storage module <NUM>, and/or the processing module <NUM>. In some embodiments, the input/output module <NUM> may include an input device (e.g., a keyboard, a mouse, a touchscreen) to receive information from a user, transform the information to other types (e.g., transform a text to a string, transform a text to a command, transform a value to a parameter type that may be recognized, etc.), or send the information to other modules of the system.

This description is intended to be illustrative, and not to limit the scope of the present disclosure. Many alternatives, modifications, and variations will be apparent to those skilled in the art. The features, structures, methods, and other characteristics of the exemplary embodiments described herein may be combined in various ways to obtain additional and/or alternative exemplary embodiments. For example, the storage module <NUM> may be integrated into the processing module <NUM>, or the input/output module <NUM> is unnecessary for the system.

<FIG> is a flowchart illustrating a process for an MR scan according to some embodiments of the present disclosure. The process may include scanning a subject <NUM>; receiving MR data <NUM>; processing MR data <NUM>, and producing an MR image <NUM>.

In step <NUM>, an examination may be performed on a subject. The subject may be a human being, an animal, or a portion thereof including, for example, an organ, a texture, a region, an object, a lesion, a tumor, or the like, or any combination thereof. Merely by way for example, the object may include a head, a breast, a lung, a trachea, a pleura, a mediastinum, an abdomen, a long intestine, a small intestine, a bladder, a gallbladder, a triple warmer, a pelvic cavity, a backbone, extremities, a skeleton, a blood vessel, or the like, or any combination thereof. In some embodiments, the examination may be an MR scan. In some embodiments, a number of settings may be used for scanning different types of objects, wherein each setting may include a plurality of parameters. Merely by way of example, the parameters may include the strength of main magnetic field, the strength of ladder magnetic field, the frequency of RF transmit signal, the scan mode, the gantry speed, or the like, or any combination thereof.

From the scan, the raw data corresponding to the subject may be acquired in step <NUM>. In some embodiments, the raw data may be stored in the storage module <NUM> as numerical values. In some embodiments, the raw data may be exported and visualized in input/output modules <NUM> as matrices of numerical values. In some embodiments, the raw data may be expressed as data values in the k-space (or referred to as the frequency domain).

In step <NUM>, the raw data obtained from step <NUM> may be processed by a plurality of procedures. In some embodiments, a rectifying procedure may be performed to correct or remove any unreliable and incorrect data values. In some embodiments, a noise filtering procedure may be performed to remove the noise produced during the scan. In some embodiments, the rectifying procedure and noise filtering procedure may either be performed before or after receiving the raw data. In some embodiments, the MR data may be analyzed and processed in the k-space. In some embodiments, a filter may be implemented to remove data values in unwanted frequency ranges. Merely by way of example, a low pass filter may be used to remove data values in high-frequency ranges. As used herein, the low pass filter may refer to a filter passing low-frequency signals while attenuating (reducing the amplitude of) signals with frequencies higher than a cut-off frequency. In some embodiment, the raw data may be padded (for example, zero-padded) to reduce mosaic effect according to some embodiments of present disclosure. In some embodiments, the data values may be classified into several regions based on their frequencies and different procedures may be performed on different ranges of frequencies.

In some embodiments, an objective function (also referred to as cost function) may be constructed based on the size of the raw data matrix, the order of the objective function, a kernel function of the objective function, the numerical range of the raw data, the significance of Gibbs artifact and/or mosaic effect, or the like, or any combination thereof. In some embodiment, in order to obtain a satisfactory solution of the objective function, at least some raw data values may be modified. The modification may be subject to a constraint. Merely by way of example, the raw data may be classified into several regions based on their frequencies, and an objective function may be constructed so that data in only some regions may be modified.

After the raw data is processed, the processed data may be reconstructed to generate an image in step <NUM>. Merely by way of example, the reconstruction algorithm of the image may include a Fourier Transform (FT), a Fast Fourier Transform (FFT), an Inverse Fourier Transform (IFT), an Inverse Fast Fourier Transform (IFFT), a 2d Fourier Transform, a Discrete Fourier Transform (DFT), an iterative reconstruction, a backward projection, or the like, or any combination thereof.

As shown in <FIG>, in some embodiments, an image may be updated based on previous images and other factors. In some embodiments, a process may be implemented to transform the image back to data values in the k-space and the data values in the preceding iteration (also referred to as old values) may be modified based on the data values obtained in said process (also referred to as new values). In some embodiments, the loop may include some parameters to assess the effect of the new values on the old values. For instance, a contribution factor may indicate the contribution or effect of the new values on the old values. In some embodiments, the contribution factor may depend on the frequency of data values. After the data values are modified, the new image may then be generated based on the updated data values. In some embodiments, an image in a next iteration may be based on an image in a current iteration, i.e. an image may be iteratively dependent on the image obtained in a previous iteration. In some embodiments, an image may be iteratively dependent on images in a plurality of previous iterations.

In some embodiments, the feedback loop may be achieved only by software. In some embodiments, the loop may be achieved by both electronic circuits and software.

This description is intended to be illustrative, and not to limit the scope of the claims. Many alternatives, modifications, and variations will be apparent to those skilled in the art. The features, structures, methods, and other characteristics of the exemplary embodiments described herein may be combined in various ways to obtain additional and/or alternative exemplary embodiments. It should be appreciated for those skilled in the art that the disclosed method may be used in a plurality of examinations including a digital subtraction angiography (DSA) system, a magnetic resonance imaging (MRI) system, a magnetic resonance angiography (MRA) system, a computed tomography (CT) system, a computed tomography angiography (CTA) system, an ultrasound scanning (US) system, a positron emission tomography (PET) system, a single-photon emission computerized tomography (SPECT) system, a CT-MR system, a CT-PET system, a CE-SPECT system, a DSA-MR system, a PET-MR system, a PET-US system, a SPECT-US system, a TMS (transcranial magnetic stimulation)-MR system, an US-CT system, an US-MR system, an X-ray-CT system, an X-ray-MR system, an X-ray-portal system, an X-ray-US system, a Video-CT system, a Vide-US system, or the like, or any combination thereof. However, the invention as defined in the appended claims is limited to Magnetic Resonance Imaging.

<FIG> illustrates a block diagram of the processing module according to some embodiments of the present disclosure. As shown in the figure, the processing module <NUM> may include a computing unit <NUM>, an image generation unit <NUM>, and a storage unit <NUM>. The computing unit <NUM> may calculate different kinds of information received from, for example, the control module <NUM>, the storage module <NUM>, and/or the input/output module <NUM>. The information from the control module <NUM> may include information about the MRI system <NUM>, the strength of gradient field, the RF sections, the subject position <NUM>, or the like, or any combination thereof. In some embodiments, the computing unit <NUM> may calculate the data values in the k-space. In some embodiments, the data values in the k-space may be acquired from a plurality of channels. The plurality of channels may be associated with a single RF receiver of a MRI system. In some embodiments, the plurality of channels may be associated with more than one RF receiver of an MRI system. The computing unit <NUM> may combine the data values in different channels. In some embodiments, the data values in the k-space may be acquired from a single channel. In some embodiments, the computing unit <NUM> may calculate coefficients of a filter generated based on at least one window function. In some embodiments, the computing unit <NUM> may calculate coefficients and/or formula of an objective function. In some embodiments, the computing unit <NUM> may calculate the values and convergence rate of the objective function.

The image generation unit <NUM> may process the data such as magnetic resonance (MR) signals acquired from a subject and reconstruct them into one or more MR image. The image generation unit <NUM> may employ different kinds of imaging reconstruction techniques for the image reconstruction procedure. The image reconstruction techniques may include a Fourier Transform, a Fast Fourier Transform, an Inverse Fourier Transform, a 2d Fourier Transform, a Discrete Fourier Transform (DFT), an iterative reconstruction, a backward projection, or the like, or any combination thereof.

In some embodiments, the image generation unit <NUM> may include an iterative reconstruction to update the image until a condition is satisfied. In some embodiments, the condition may relate to an objective function.

The storage unit <NUM> may store the information that may be used by the computing unit <NUM> and/or the image generation unit <NUM>. The information may include programs, software, algorithms, data, text, number, images, etc. These examples are provided here for illustration purposes, and not intended to limit the scope of the present disclosure. The storage unit <NUM> may store algorithms including, for example recursion, a nonlinear conjugate gradient method, a bisection method, an exhaustive search (or brute-force search), a greedy algorithm, a divide and conquer algorithm, a dynamic programming method, an iterative method, a branch-and-bound algorithm, a backtracking algorithm, or the like, or any combination thereof.

It should be noted that the above description of the processing module is merely provided for the purposes of illustration, and not intended to limit the scope of the present disclosure. For persons having ordinary skills in the art, multiple variations or modifications may be made under the teachings of the present disclosure. For example, the assembly and/or function of module unit may be varied or changed. In some embodiments, the computing unit <NUM> and the image generation unit <NUM> may share one storage unit <NUM>. In some embodiments, the computing unit <NUM> and the image generation unit <NUM> may each have their own storage units, respectively. However, those variations and modifications do not depart from the scope of the present disclosure.

<FIG> illustrates a block diagram of the image generation unit according to some embodiments of the present disclosure. As shown in the figure, the image generation unit may include a pre-process block <NUM>, a parameter setting block <NUM>, a filter <NUM>, an iterative reconstruction block <NUM>, and a storage block <NUM>. The pre-process block <NUM> may perform some pre-processing to the raw data. The pre-processing may include image normalization, image segmentation, image reconstruction, image smoothing, suppressing, weakening and/or removing a detail, a mutation, a noise, or the like, or any combination thereof.

The data treated by the pre-process block <NUM> may be sent to the parameter setting block <NUM>. The data may contain the image-related data in the k-space, as well as various types of data related to the subject. The image-related data in the k-space may be undersampled. For example, the image-related data in the k-space may occupy a part of the k-space. The parameter setting block <NUM> may set the values of various parameters used in the process of image generation. The parameters may relate to the subject, including but not limited to, the age, weight, height, heart rate, blood oxygen level, blood pressure. The parameters to be set by the parameter setting block <NUM> may relate to the part or region in the k-space. For example, the image-related data in the k-space may occupy a part D1 of the k-space. The part D1 may have a dimension of M1×N1. In some embodiments, M1 may be equal to or different from N1. The parameter setting block <NUM> may designate another region D2, of dimension M2×N2, inside the part D1. In some embodiments, M2 may be equal to or different from N2. In some embodiments, M1 may be equal to or larger than M2. In some embodiments, N1 may be equal to or larger than N2. The parameters setting block <NUM> may also designate a part D3, of dimension M3×N3, containing the region D1. In some embodiments, M3 may be equal to or different from N3. In some embodiments, M1 may be equal to or smaller than M3. In some embodiments, N1 may be equal to or smaller than N3. In some embodiments, the parameter setting block <NUM> may set the values of some parameters related to the filter <NUM>. For example, the width of the pass band (PB), the width of the transition band (TB), and the threshold value for the filter <NUM>.

The filter <NUM> may treat the image-related data in the k-space. In some embodiments, the filter <NUM> may be a low-pass filter. In some embodiments, the filter <NUM> may be a band pass filter (BPF). The filter <NUM> may be characterized by its pass band (PB), transient band (TB), and the transient function within the transient band.

An exemplary one-dimensional prototype low-pass filter may be demonstrated as in <FIG>, and the effect of an exemplary one-dimensional prototype low-pass filter may be demonstrated as in <FIG>. Here after normalization the strength of the filter inside the pass band is one, and the pass band together with the transient band may satisfy the following relation specified in Equation (<NUM>): <MAT>.

In some embodiments, a weak filtering effect may be employed. For example, TB may be no less than <NUM>. In some embodiments, TB may be no less than <NUM>. In some embodiments, TB may be no less than <NUM>. The transient function f(k) may be a combination of trigonometric functions, inverse trigonometric functions, exponential functions, logarithmic functions, polynomial functions, or the like, or a combination thereof. For example, the transient function may be given as a form of Hanning window. For another example, the transient function may be given as a form of a Turkey window.

The threshold value after normalization Δ r may indicate the effect of diminishing high-frequency components of the data. The threshold for high frequency component may be related to the pass band (PB) and/or the transition band (TB). For example, the threshold for high frequency component may be set as the sum of pass band and transition band. In some embodiments, a weak filtering effect may be employed. For example, Δ r may be equal to or larger than <NUM>. In some embodiments, Δ r may be equal to or larger than <NUM>. In some other embodiments, Δ r may be equal to or larger than <NUM>.

The filter <NUM> may be generated by expanding multiple one-dimensional prototype low-pass filters. In some embodiments, these multiple one-dimensional prototype low-pass filters may be mutually orthogonal to each other. In other words, a first one-dimensional prototype low-pass filter may be along a first direction in the k-space, a second one-dimensional prototype low-pass filter may be along a second direction orthogonal to the first direction. For example, a first one-dimensional prototype low-pass filter may be along the k_x direction in the k-space, whereas a second one-dimensional prototype low-pass filter may be along the k_y direction that is orthogonal to the k_x direction.

The filter <NUM> may also be generated by expanding the one-dimensional prototype low-pass filter in a rotation-invariant way. For example, all the points in the k-space with the same radial coordinate squared kx<NUM> + ky<NUM> may have the same filtering effect.

The filtered image-related data in the k-space by the filter <NUM> may be sent to the iterative reconstruction block <NUM>. The iterative reconstruction block <NUM> may be configured to generate an expanded image-related data in the k-space. The iterative reconstruction block <NUM> may be based on an objective function on the image data in the k-space. The reconstruction block <NUM> may utilize an algorithm in connection with the objective function. In some embodiments, the algorithm may operate to reduce or minimize the objective function. The algorithm may be of an iterative type. In some embodiments, the algorithm may be a non-linear conjugate gradient algorithm, a Powell method, a downhill simplex method, a gradient descent method, a descent simplex method, a deepest gradient descending method, a conjugate gradient method, a pseudo-Newton method, a quasi-Newton method, a least-squares and Gauss-Newton method, a Broyden-Fletcher-Goldfarb-Shannon (BFGS) method, a limited-memory Broyden-Fletcher-Goldfarb-Shannon (L-BFGS) method, a simulated annealing method, an ant colony optimization (ACO) method, a genetics method, a Levenberg-Marquardt optimization method, a geometric hashing method, a particle swarm optimization (PSO) method, a firefly algorithm (FA) method, or the like, or a combination thereof. The algorithm may also be of stochastic type. In some embodiments, the algorithm may be a Monte-Carlo (MC) method, a fast Monte-Carlo method, an analog Monte-Carlo method, a Non-analogue Monte-Carlo method, a resampling method, or the like, or a combination thereof. The objective function may relate to the total variation (TV) of the image data in the k-space. A total variation of a first order may be the sum of the modulus of jumps between neighboring pixels of a reconstructed image I(x, y). A total variation of a first order may be referred as: <MAT>.

Calculation of the TV value according to Equation (<NUM>) uses the first-order derivative of the image with respect to the x-direction and the y-direction. In some embodiments, the total variation may be based on second-order derivatives. A total variation of a second order may be referred as: <MAT> where σ ∈ [<NUM><NUM>] may be a weighting factor. The weighting factor may be used to tune the smoothness of an image. For instance, if σ decreases from one to zero, the image may allow for the intensity gradients in an image and yield more naturally looking solutions.

The objective function may relate to the energy of the image data in k-space. The energy of the image data in the k-space may be given as the L-p norm of the image data in a region of the k-space, where p ≥ <NUM>. The reduction or minimization of the objective function may be subject to a constraint. For example, the constraint may be such that the image data inside the part D2 to be unchanged during the iteration process. For another example, the constraint may be such that the image data inside the part D3 to have a fixed energy.

The storage block <NUM> may be configured to connect to the pre-process block <NUM>, the parameter setting block <NUM>, the filter <NUM>, and the iterative reconstruction block <NUM>. The storage block <NUM> may store various types of data, such as the image, the parameters used in the parameter setting block <NUM>, image-related data in the k-space, or the like, or a combination thereof. The storage block <NUM> may store data by the way of electric, magnetic, optical energy, or virtual storage resources, etc. The storage block <NUM> that store data by the way of electric energy may include Random Access Memory (RAM), Read Only Memory (ROM), flash memory, or the like, or any combination thereof. The block <NUM> that stores data by the way of magnetic energy may include a hard disk, a floppy disk, a magnetic tape, a magnetic core memory, a bubble memory, a USB flash drive, or the like, or any combination thereof. The storage block <NUM> that store data by the way of optical energy may include Compact Disk (CD), Video Compact Disk (VCD), or the like, or any combination thereof. The storage block <NUM> that stores data by the way of virtual storage resources may include cloud storage, a virtual private network, and/or other virtual storage resources. The method to store data may include sequential storage, link storage, hash storage, index storage, or the like, or any combination thereof.

It should be noted that the above descriptions about the image generation unit is merely an example, should not be understood as the only embodiment. Obviously, to those skilled in the art, after understanding the basic principles of the connection between different blocks, and connection between the blocks may be modified or varied without departing from the principles. The modifications and variations are still within the scope of the current disclosure described above. In some embodiments, these blocks may be independent, and in some embodiments, part of the blocks may be integrated into one block to work together.

<FIG> illustrates a block diagram of an iterative reconstruction block <NUM> according to some embodiments of the present disclosure. As shown in the figure, the iterative reconstruction block <NUM> may include a frequency setting block <NUM>, a padding block <NUM>, a calculation block <NUM>, and an image construction block <NUM>. For the filtered image-related data in the k-space, an area in the k-space, called the natural region of the data, may be specified. The natural region of the data may be the set of coordinates in the k-space where the data has been given. In some embodiments, the natural region of the data in the k-space may be a part of dimension M1×N1 (see part <NUM> as in <FIG> for an illustration). The frequency setting block <NUM> may be used to specify an inner region inside the part for the data. The inner region may be a region of dimension M2×N2, where M2 ≤M1, N2 ≤ N1. In some embodiments, the inner region may be a combination (e.g., disjoint union) of several smaller regions inside the natural region.

The padding block <NUM> may fill data into an area outside the natural region of the data in k-space. In some embodiments, zero padding may be implemented on an area outside the natural region. In some embodiments, a non-zero padding may be implemented on an area outside the natural region. For example, the padding block <NUM> may assign a nonzero constant to an area outside the natural region. The data given on the extended region may be sent to the calculation block <NUM> for further processing.

The calculation block <NUM> may perform various types of calculation on the data in image domain or an image-related data in the k-space. In some embodiments, the first order total variation of data in the image domain may be calculated by the calculation block <NUM>. For example, for an image data I(x, y), the first order total variation of the image I(x, y) may be defined in Equation (<NUM>).

In some embodiments, the second order total-variation of an image may be calculated by the calculation block <NUM>. For example, for an image I(x, y), the second order total variation of the image I(x, y) may be defined in Equation (<NUM>).

The calculation block <NUM> may also perform Fourier transform on an image I, denoted as F(I). In some embodiments, the Fourier transform may be implemented as a Fourier transform matrix M_F transforming an image data from the image space to the corresponding image-related data in the k-space.

The image construction block <NUM> may utilize the output of padding block <NUM> and the calculated quantities from calculation block <NUM> for use in a searching algorithm. The searching algorithm may be implemented by the image construction block <NUM> to reduce or minimize an objective function on the image data in the image domain. In some embodiments, the algorithm may be of an iterative type. In some embodiments, the algorithm may be a non-linear conjugate gradient algorithm, a Powell method, a downhill simplex method, a gradient descent method, a descent simplex method, a deepest gradient descending method, a conjugate gradient method, a pseudo-Newton method, a quasi-Newton method, a least-squares and Gauss-Newton method, a Broyden-Fletcher-Goldfarb-Shannon (BFGS) method, a limited-memory Broyden-Fletcher-Goldfarb-Shannon (L-BFGS) method, a simulated annealing method, an ant colony optimization (ACO) method, a genetics method, a Levenberg-Marquardt optimization method, a geometric hashing method, a particle swarm optimization (PSO) method, a firefly algorithm (FA) method, or the like, or a combination thereof. The algorithm may be of a stochastic type. In some embodiments, the algorithm may be a Monte-Carlo (MC) method, a fast Monte-Carlo method, an analog Monte-Carlo method, a Non-analogue Monte-Carlo method, a resampling method, or the like, or a combination thereof.

The objective function may relate to the total variation and/or the energy of the image data in the k-space. The minimization of the objective function may be subject to a constraint. The constraint may be imposed upon the targeted image data, so that the targeted image data has a specific behavior in certain region of k-space. For example, the constraint may be such that the image data inside the region D1 to be unchanged during the iteration process. For another example, the constraint may be such that the image data inside the region D3 to have a fixed energy.

In some embodiments, the following objective function G(ρ) on the image data ρ in the image space may be implemented: <MAT> where λ<NUM> and λ<NUM> may be real constants. In some embodiments, the values of λ<NUM> and/or λ<NUM> may be between zero and one. Merely by way of example, the values of λ<NUM> and λ<NUM> may be chosen so that λ<NUM> = <NUM>λ<NUM>. F denotes the Fourier Transform operator, Y denotes the image-related data in the natural region of the k-space.

P<NUM> may stand for a matrix used to choose data points in the k-space. In some embodiments, the data points chosen by P<NUM> may be allowed to change during the iterative searching for a solution of the objective function. Merely by way of example, the solution may be such that the value of the objective function may be minimal. As another example, the solution may be such that the change in the value of the objective function in the two or more consecutive iterations may be equal to or smaller than a threshold. As a further example, the solution may be obtained after a certain number of iterations. The iterations may terminate when the solution is obtained.

In some embodiments, the following constraint on the image data ρ in image space may be implemented <MAT> where P<NUM> may be a matrix used to choose a set of data points in the k-space. For example, the image-related data in the k-space may be within a part D1 of the k-space. The part D1 may have a dimension of M1×N1. The parameter setting block <NUM> may designate region D2, of dimension M2×N2, within the part D1. The parameters setting block <NUM> may also designate a part D3, of dimension M3×N3, containing the part D1. In some embodiments, P<NUM> may be used to specify that the data points inside D2 to be unchanged during the iterative steps of the searching algorithm.

A searching algorithm to minimize the objective function may be implemented in the image construction block <NUM> to generate an image in the image space. The image may be output to the patient, or to the doctor, or to anyone related to the image. The image may be sent to a server, a cloud server, a network, a database for storage. The image may be subject to further process such as normalization or denoising.

It should be noted that the above descriptions about the iterative reconstruction block is merely an example, should not be understood as the only embodiment. Obviously, to those skilled in the art, after understanding the basic principles of the connection between different blocks, and connection between the blocks may be modified or varied without departing from the principles. The modifications and variations are still within the scope of the current disclosure described above. In some embodiments, these blocks may be independent, and in some embodiments, part of the blocks may be integrated into one block to work together.

<FIG> is an exemplary flowchart illustrating a process for reducing or removing the Gibbs artifact from an image according to some embodiments of the present disclosure.

In step <NUM>, image data or an image-related data in the k-space may be preprocessed. The pre-processing may include image normalization, image segmentation, image reconstruction, image smoothing, suppressing, weakening and/or removing a detail, a mutation, a noise, or the like, or any combination thereof.

In step <NUM>, various parameters related to the image-related data in k-space may be set. The parameters may be related to the subject, including but not limited to, the age, weight, height, heart rate, blood oxygen level, blood pressure. The parameters to be set may also be related to the specification of the region in the k-space. For example, the natural region of the image-related data in the k-space may be a part D1 of dimension M1×N1. In some embodiments, parameters for an inner region D2 inside the part may be specified. For example, the inner region D2 may be a region of dimension M2×N2, where M2 ≤M1, N2 ≤ N1. In some embodiments, the inner region may be a disjoint union of several smaller regions inside the natural region. In some embodiments, parameters for an outer region D3, of dimension M3×N3, containing the natural region D1, may be designated as well.

An exemplary illustration of setting a larger region containing the natural region may be seen in <FIG> where the image-related data in the k-space occupy. The region <NUM> is the natural region where the image-related data in k-space may originally be specified. The region <NUM> may be the extended area outside the region <NUM> in k-space for providing further process of the image related data. For example, zero padding may be performed on the region <NUM>.

<FIG> shows an exemplary illustration of setting a larger region containing the natural region and setting a smaller region inside the natural region where the image-related data in the k-space occupy. The region <NUM> may be the natural region where the image-related data in k-space may originally be specified. The region <NUM> may be the extended area outside the region <NUM> in the k-space for providing further process of the image related data. For example, padding may be performed on the region <NUM>. The region <NUM> may be inner region inside the natural region <NUM>. The inner region <NUM> may be subject to a constraint during the iteration steps of the searching algorithm, which may be described below.

In step <NUM>, a filtering may be applied on the image-related data in k-space. In some embodiments, the filtering may be a low-pass filtering. Some parameters of the low-pass filtering may be specified in step <NUM>. For example, the width of the pass band (PB), the width of the transition band (TB), and the threshold value for the low-pass filtering may be specified in step <NUM>. In some embodiments, these parameters may be specified in step <NUM>.

In step <NUM>, a procedure of padding may be performed on the filtered image-related data in k-space. In some embodiments, the filtered image-related data lies in a natural region. A zero-padding may be performed on the filtered image-related data by setting zero value in a neighboring area of the natural region. For example, the filtered image-related data may lie in a natural region as <NUM> of <FIG>. A zero-padding may be performed on the region <NUM>. Alternatively, a nonzero-padding may be performed on the region <NUM>, i.e. a nonzero constant may be set on the region <NUM>. The padded image-related data in the k-space may be sent to step <NUM> for further processing, or it may be sent to step <NUM> for image construction.

In step <NUM>, a process for performing substantially the minimization of an objective function for the image-related data in the k-space may be implemented. Here the process of minimization may be realized if the difference between the achieved objective function values and theoretical minimal objective function values is below a threshold. The objective function may relate to the total variation of the image data in the k-space. The total variation may be a first order total variation, a second order total variation or the like, or any combination thereof. The first order total variation and the second order total variation may be shown in Equation (<NUM>) and Equation (<NUM>), respectively. The objective function may relate to the energy of the image data in the k-space. The energy of the image data in the k-space may be given as an L-p norm of the image data in the k-space, where p ≥ <NUM>. The minimization of the objective function may be subject to a constraint. For example, the constraint may be such that the image data inside the region D1 to be unchanged during the iteration process. For another example, the constraint may be such that the image data inside the region D3 to have a fixed energy.

In step <NUM>, an image based on the resulted data in k-space may be generated. The way of generating an image from the data in the k-space may be familiar to those skilled in the art. If the image based on the resulted data in k-space is not satisfactory, the data in k-space may be sent back to step <NUM> for further process. If the image based on the resulted data in k-space is satisfactory, then the image will be sent to step <NUM> for outputting.

It should be noted that the flowchart of a process for removing the Gibbs artifact from an image described above is provided for the purposes of illustration, and not intended to limit the scope of the present disclosure. For persons having ordinary skills in the art, various variations and modifications may be conduct under the teaching of the present disclosure. However, those variations and modifications may not depart from the protecting scope of the appended claims. For example, in step <NUM>, the image related data in the k-space may be a combination of data from a plurality of channels, which may be referred to as channel fusion. For another example, in step <NUM>, the image related data in the k-space may be from a single channel. In some embodiments, the process of channel fusion to obtain the image related data in the k-space via a plurality of channels may lose the phase information of the image, such as by using the method of Sum of Square (SoS). Then the step <NUM> of zero-padding may be performed before the process of channel fusion. Steps <NUM> through <NUM> may be performed sequentially at an order other than that described above in <FIG>. At least two steps of steps <NUM> through step <NUM> may be performed concurrently. Steps 701through <NUM> may be merged into a single step or divided into a number of steps. In addition, one or more other operations may be performed before/after or in performing steps <NUM> through <NUM>. In some embodiments, at least one of steps <NUM> through <NUM> may be unnecessary and may be omitted.

<FIG> illustrates an exemplary flowchart of a process for an image reconstruction according to some embodiments of the present disclosure. Data may be acquired in step <NUM> directly through a data scan process in step <NUM>, or it may be obtained through a plurality of processes after the data scan process in step <NUM>. The processes may include but not limit to low-pass filtering, padding, a frequency analysis, or the like, or a combination thereof. The data may be obtained from a storage medium of or accessible by the data processing system <NUM>.

In some embodiments, the image reconstruction described in the present disclosure may be implemented by one or more modules, units or blocks in the system of the present disclosure. In some embodiments, the image reconstruction may include a step of updating the image by changing the position of the scanned subject.

Based on the data acquired in step <NUM>, the parameters may be set in step <NUM>. In some embodiments, the parameters may relate to the performance of the image reconstruction. In some embodiments, the parameters may include one or more parameters or properties of an objective function. Merely by way of example, one or more parameters or properties of the objective function may include order, condition of convergence, kernel function, numerical range of the objective function, or the like, or any combination thereof.

During the first iteration, an initial image may be acquired in step <NUM> by the input relating to the initially estimated image in step <NUM>, while in any other iteration, the image in step <NUM> may be updated by a reconstructed image generated from the previous iteration.

After the image in step <NUM> is updated, the value of the objective function may be calculated in step <NUM>. The calculation may include substituting the magnitude of at least a portion of the updated image into the objective function. In some embodiments, the magnitude of the updated image may be determined by the grey level or intensity of a plurality of pixels inside the updated image. As shown in <FIG>, a determination may be made in step <NUM> to determine whether a condition is satisfied. Merely by way of example, the condition may be such that the value of the objective function may be minimal. As another example, the condition may be such that the change in the value of the objective function in the two or more consecutive iterations may be equal to or smaller than a threshold. As a further example, the condition may be satisfied after a certain number of iterations. If the condition is satisfied, the iteration may terminate and the current image may be stored as a target image in step <NUM> so that the image may be output as a final image. If the condition is not satisfied, the process may proceed back to step <NUM>. In some embodiments, the image obtained in an iteration may be updated based on either the image or the value of the objective function in the preceding iteration. After the image is updated in step <NUM> in the new iteration, the similar process may be performed until the condition is satisfied in step <NUM>. The image satisfying the condition may then be output as the final image.

In some embodiment, the objective function may take the following form: <MAT> subject to <MAT>.

In some embodiment, the objective function may take the following form: <MAT> where F may denote the Fourier Transform matrix, ρ may denote the estimated image, Y may denote the processed data values in the k-space obtained in step <NUM>, P<NUM> may stand for a matrix used to choose data points in the k-space. In some embodiments, the data points chosen by P<NUM> may be allowed to change during a search for a solution of the objective function. The search may proceed iteratively. P<NUM> may be a matrix used to choose a set of data points in the k-space. For example, the image-related data in the k-space may be within a region D1 of the k-space. The region D1 may have a dimension of M1×N1. The parameter setting block <NUM> may designate a region D2, of dimension M2×N2, within the region D1. In some embodiments, P<NUM> may be used to specify that the data points inside D2 to be unchanged during the searching process. TVn may denote an nth-order total variation reconstruction calculation. In some embodiments, λ<NUM>, λ<NUM> and λ<NUM> may be real numbers. Merely by way of example, λ<NUM> and λ<NUM> may be smaller or equal to <NUM>, while λ<NUM> may be greater <NUM>. The coefficient λ<NUM> may be referred to as the constraint strength.

Additionally, ∥x∥<NUM>, or denoted as L2-norm, may be defined as: <MAT>.

As shown in the formulae (<NUM>) and (<NUM>), the converging of the objective function may include two steps: setting the estimated image ρ in D2 to be the Fourier Transform of the processed data values Y of the corresponding region and keeping it unchanged; and updating the estimated image ρ in D1 and calculating the value of the objective function in formula (<NUM>) until a condition is satisfied. In some embodiments, the condition may be satisfied when the value of the objective function is smaller than a threshold. In some embodiments, the condition may be satisfied when a minimum value of the objective function is obtained. In some embodiments, the condition is satisfied when the change in the value of the objective function in a number of iterations (e.g., a number of successive iterations) is below a threshold. In some embodiments, the condition is satisfied when the difference between the value of the objective function and a target value is below a threshold. In some embodiments, the condition is satisfied when a specified number of iterations are performed.

Refer to formula (<NUM>), the converging of the objective function may include a step of updating the estimated image ρ in both D1 and D2 and calculating the value of the objective function in formula (<NUM>) until a condition is reached. In some embodiments, data values of estimated image ρ in D2 region may be changed to a less extent than the data values in D1 region.

It should be noted that the flowchart of a process for removing the Gibbs artifact from an image described above is provided for the purposes of illustration, and not intended to limit the scope of the present disclosure. For persons having ordinary skills in the art, various variations and modifications may be conduct under the teaching of the present disclosure. However, those variations and modifications may not depart from the protecting scope of the appended claims. For example, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM> and step <NUM> may be performed sequentially at an order other than that described above in <FIG>. At least two steps of step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM> and step <NUM> may be performed concurrently. Step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM> and step <NUM> may be merged into a single step or divided into a number of steps. In addition, one or more other operations may be performed before/after or in performing step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM> and step <NUM>. At least one of step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM> and step <NUM> may be unnecessary and may be omitted.

<FIG> illustrates another exemplary flowchart of a process for an image reconstruction according to some embodiments of the present disclosure. Step <NUM>, step <NUM>, step <NUM>, step <NUM>, and step <NUM> may be the same as step <NUM>, step <NUM>, step <NUM>, step <NUM>, and step <NUM> of <FIG>, respectively.

Before the first iteration, a current smallest value used in step <NUM> and step <NUM> may be set to zero or any appropriate values. The current smallest value may represent the smallest value of the objective function of all the values from first iteration to the current iteration. After the value of the objective function is calculated in step <NUM>, a determination may be made in step <NUM> to determine whether the value of the objective function is smaller than the current smallest value. If the value of objective function is smaller than the current smallest value, the current smallest value may be replaced and updated by the value of the objective function in step <NUM>. If the value of objective function is not smaller than the current smallest value, the process may proceed back to step <NUM>.

After at least a value of the objective function is smaller than the current smallest value, a determination in step <NUM> similar to that of <FIG> may be performed, and similar iterative process may follow as described elsewhere in present disclosure. Once the condition is satisfied, the current image may be saved as the target image in step <NUM>, the image may then be output as the final image.

In some embodiments, step <NUM> may be performed after the image is updated in step <NUM>. In some embodiments, the parameters in step <NUM> may be set and changed in every iteration. Merely by way of example, the order, condition of convergence, kernel function, and numerical range of the objective function, or the like, or any combination thereof may be changed in every iteration.

In some embodiments, the objective function in formula (<NUM>), formula (<NUM>), and formula (<NUM>) may be used in step <NUM> of <FIG>, so that the condition may be satisfied if and only if the current smallest value is smaller than a threshold.

It should be noted that the flowchart of a process for removing the Gibbs artifact from an image described above is provided for the purposes of illustration, and not intended to limit the scope of the present disclosure. For persons having ordinary skills in the art, various variations and modifications may be conduct under the teaching of the present disclosure. However, those variations and modifications may not depart from the protecting scope of the appended claims. For example, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM> and step <NUM> may be performed sequentially at an order other than that described above in <FIG>. At least two steps of step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM> and step <NUM> may be performed concurrently. Step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM> and step <NUM> may be merged into a single step or divided into a number of steps. In addition, one or more other operations may be performed before/after or in performing step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM> and step <NUM>. At least one of step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM>, step <NUM> and step <NUM> may be unnecessary and may be omitted.

<FIG> illustrates a further exemplary flowchart for a process of an image reconstruction according to some embodiments of the present disclosure. Step <NUM>, step <NUM>, step <NUM>, step <NUM>, and step <NUM> may be the same as those of <FIG>.

After the value of the objective function is calculated in step <NUM>, the value may be stored in the storage unit <NUM> in step <NUM>. In some embodiments, only a single value of the objective function may be stored so that the value in each iteration may replace the value stored in the previous iteration. In some embodiments, a plurality of values of the objective function from a plurality of iterations may be stored. A determination in step <NUM> may be performed as to whether the condition is satisfied as described before in <FIG> and <FIG>. If the condition is satisfied, the current image may be saved as the target image in step <NUM>, the image may then be output as the final image. If the condition is not satisfied, a contribution factor, or bias, may be calculated based on the previous value(s) stored in step <NUM>. The contribution factor may represent how the previous value(s) in one or more previous iteration contribute to the image in the current iteration. The contribution factor may be calculated from the value in the previous iteration or may be calculated from a plurality of values of the objective function from a plurality of iterations based on the values stored in step <NUM>.

In some embodiments, the contribution factor may be calculated by a machine learning method. According to the machine learning method, the algorithm in calculating the contribution factor may be updated based on the magnitude and/or amount of the values stored in step <NUM>.

After calculating the contribution factor, or bias, in step <NUM> the image may be updated based on the calculated contribution factor and the similar process may follow until the preset condition is satisfied. And the image satisfying the condition may then be output as the final image.

In some embodiments, the process may include applying an algorithm to perform the iteration process as described in <FIG>, <FIG>, and/or <FIG>. The algorithm may include but not limit to a recursion, a nonlinear conjugate gradient method, a bisection method, an exhaustive search (or brute-force search), a greedy algorithm, a divide and conquer algorithm, a dynamic programming method, an iterative method, a branch-and-bound algorithm, a backtracking algorithm, or the like, or any combination thereof.

It should be noted that the above descriptions of the image reconstruction processes are provided for the purposes of illustration, not intended to limit the scope of the present disclosure. For persons having ordinary skills in the art, various variations and modifications may be conducted under the teaching of the present disclosure. However, those variations and modifications may not depart the protecting scope of the appended claims. It should be appreciated for those skilled in the art that the process described in <FIG> may be combined to form a process which store both the current smallest value of the objective function and value in a plurality of previous iterations. Accordingly, the image may be updated by the contribution factor calculated from values in a plurality of previous iterations and the preset condition may be determined by the current smallest value.

In some embodiments, the present image reconstructing process may be used in data relating to various types of subjects including but not limiting to a head, a breast, a lung, a pleura, a mediastinum, an abdomen, a long intestine, a small intestine, a bladder, a gallbladder, a triple warmer, a pelvic cavity, a backbone, extremities, a skeleton, a blood vessel, or the like, or any combination thereof. In some embodiments, the image reconstructing process may include a plurality of configurations, wherein each configuration may specifically be used in processing an image of a single type of subject. In some embodiments, the configuration may be used in processing the images of a plurality of subjects.

As described elsewhere in the present disclosure, the image reconstructing process may include steps of low pass filtering, padding (e.g., zero padding), frequency analysis, and iterative image reconstruction. In some embodiments, the configuration described before may include a selection of low pass filters, a selection of size of the region where padding is applied, a division of data values for the application of a constraint in the k-space, a selection of objective function, or the like, or a combination thereof.

<FIG> illustrate two exemplary configurations in processing images of different subjects or different sections of the same subject in an image reconstructing process. In some embodiments, <FIG> illustrates an exemplary configuration with strong low pass filtering and strong iterative reconstruction. In some embodiments, <FIG> illustrates an exemplary configuration with a weak low pass filtering and a weak iterative reconstruction.

As shown in <FIG>, the low pass filter <NUM> and <NUM> may each include a passband (referred to as PB in the figure), a transition band (referred to as TB in the figure), and a threshold (referred to as Δr in the figure). In some embodiments, the passband may be the region of frequency that is not filtered and/or attenuated (e.g., subject to a filter of zero strength), the stopband (not shown in the figure) may be the region of frequency that is mostly filtered and/or attenuated (e.g., subject to a filter of the highest strength compared to the passband and the transition band), and transition band may be the region between the passband and the stopband subject to an intermediate filtering and/or attenuation (e.g., subject to a filter of an intermediate strength compared to the passband and the stopband). In some embodiments, the threshold may indicate the highest amplitude that the oscillatory tail in the stopband, referred to as a ripple in the stopband (also referred to as a stopband ripple) may reach.

The filter <NUM>, as illustrated in <FIG>, may include a narrow passband, a wide transition band, and a low threshold Δr. As used herein, a low threshold Δr may indicate that the threshold is less than a characteristic percentage of the normalized amplitude of the frequency characteristic curve. For example, the characteristic percentage may be <NUM>% or less. As another example, the characteristic value may be <NUM>% or less. As used herein, a narrow band may indicate that the band is less than or equal to a specific threshold. For example, the specific threshold may be <NUM>% or less of (TB + PB + SB), i.e., the summation of transition band, Pass band, and Stop band. As another example, the specific threshold may be <NUM>% or less of (TB + PB + SB). As used herein, a wide band may indicate that the band is larger than or equal to a threshold. For example, the threshold may be <NUM>% or more of (TB + PB + SB), i.e., the summation of transition band, Pass band, and Stop band. As another example, the threshold may be <NUM>% or more of (TB + PB + SB). This configuration may indicate that only a small range of frequencies may remain unfiltered in the PB, while a large range of frequencies may be filtered in a TB. It may also indicate that frequencies higher than the TB may be heavily filtered and attenuated to a low amplitude indicated by Δr.

Referring now to <FIG>, the filter <NUM> may include a wide passband, a narrow transition band and a threshold Δr of high range, for example <NUM>% to <NUM>%, of the normalized amplitude of the frequency characteristic curve. This configuration may indicate a large range of frequencies may remain unfiltered in the PB, while a small range of frequencies may be filtered in the TB. It may also indicate that frequencies higher than TB may be moderately filtered and attenuated to a medium amplitude indicated by Δr.

<FIG> also illustrate two exemplary configurations of the division of ranges of frequencies and the selection of objective functions in the k-space (also referred to as the frequency range). According to some embodiments of the present disclosure, the k-space may be divided into several ranges of frequencies.

In some embodiments, as shown in <FIG>, <NUM> may represent a padded region, region <NUM> and region <NUM> may each represent a region that may be modified during the iterative reconstruction, while region <NUM> and region <NUM> may each represent a region that either may not be modified or may only be slightly modified during the iterative reconstruction. In some embodiments, region <NUM> may correspond to region <NUM> in <FIG>. In some embodiments, region <NUM> or region <NUM> may correspond to region <NUM> in <FIG>, while region <NUM> or region <NUM> may correspond to region <NUM> in <FIG>.

In some embodiments, region <NUM> or region <NUM>, together with region <NUM> may be correspond to D1 in formula (<NUM>) and (<NUM>), while region <NUM> or region <NUM> may correspond to D2 in formula (<NUM>) and (<NUM>).

Referring back to <FIG>, in which region <NUM> is larger than region <NUM>, the data of a large region of frequencies may be modified during the iterative reconstruction (also referred to as strong iterative reconstruction).

As shown in the <FIG>, in which region <NUM> is smaller than region <NUM>, the data of a small region of frequency may be modified during the iterative reconstruction (also referred to as weak iterative reconstruction).

In some embodiments, the configuration described in <FIG> may be used in processing an image with high Gibbs artifacts. In some embodiments, the configuration described in <FIG> may be used in processing an image with low Gibbs artifacts.

This description is intended to be illustrative, and not to limit the scope of the claims. Many alternatives, modifications, and variations will be apparent to those skilled in the art. The features, structures, methods, and other characteristics of the exemplary embodiments described herein may be combined in various ways to obtain additional and/or alternative exemplary embodiments. It should be appreciated for those skilled in the art that the configuration described in <FIG> may be combined arbitrarily. Merely by way of example, it should be obvious to those skilled in the art to combine the weak filtering in <FIG> with the strong iterative reconstruction in <FIG>, vice versa. It should also be obvious to those skilled in the art to, for example, select or design a filter with narrow TB and wide PB as described in <FIG> but with a low threshold as described in <FIG>.

<FIG> illustrates three exemplary images generated from water phantom scanning and different processing steps according to some embodiments of the present disclosure. In some embodiments, water phantom with different water mold textures, shapes and other conditions were used for scanning, and water phantom scanning was performed for examining the effectiveness of the data processing system. Images <NUM>, <NUM>, and <NUM> were based on the same data array acquired from the same water phantom scanning.

The image <NUM> was generated through a data processing protocol excluding filtering. Specifically, the size of the data array in the k-space was doubled through zero-padding outside of the natural region of the data in the k-space. Then an image was constructed through an inverse Fourier transform of the data in both the natural region and the padded region in the k-space. An exemplary image constructed according to this zero-padding protocol is shown in the image <NUM>.

The image <NUM> was generated through a data processing protocol including filtering. Specifically, the data were filtered through a low-pass filter having a Hanning window in the transition band with a normalized transition bandwidth of <NUM> and a Δr of <NUM>. The size of the filtered data array in the k-space was doubled through zero-padding outside the natural region of the data in the k-space. Then an image was constructed through an inverse Fourier transform of the filtered data in the natural region and the data in the padded region. An exemplary image constructed according to this traditional low-pass filtering protocol is shown as the image <NUM>.

The image <NUM> was generated through a data processing protocol including low-pass filtering, zero-padding, and total variation reconstruction, as described in the present disclosure. Specifically, the data were filtered through a low-pass filter having a Hanning window in the transition band with a normalized transition bandwidth of <NUM> and a Δr of <NUM>. The size of the filtered data array in the k-space was doubled through zero-padding outside the natural region of the data in k-space. Then a total variation reconstruction was performed. In the total variation reconstruction, a function as shown in Equation (<NUM>) was used as the objective function and a function as shown in Equation (<NUM>) was used as the constraint. The parameter λ<NUM> was set as <NUM>, the parameter λ<NUM> was set as <NUM>, the parameter P<NUM> was a matrix used to choose a region D<NUM> in which the frequency of data were no larger than the highest frequency of the original data multiplied by <NUM>, and the parameter P<NUM> was a matrix used to choose the remaining region D<NUM> other than D<NUM>. The total variation reconstruction process included a plurality of iterations. During the iteration, steps <NUM> to <NUM> in <FIG>, <FIG> may be performed for minimizing the objective function. An exemplary target image representing the minimization of the objective function is shown in the image <NUM>.

In comparison with the images <NUM> and <NUM>, the Gibbs artifacts in image <NUM> are significant. For example, alternating bright and dark lines or bands (see arrows in <NUM>) may be present next to the border of an abrupt intensity change, regardless of the shape of the water phantom.

In image <NUM>, Gibbs artifacts may be reduced substantially through traditional low-pass filtering. For example, the numbers of the alternating bright and dark lines were reduced; the intensity of the artifact lines were weakened (see arrows in <NUM>). However, there were still some residual artifacts at the border of an abrupt intensity change, and the clarity of the image was deteriorated, especially for the detail structure at the borders. It may be indicated that the traditional low-pass filter may be effective in reducing Gibbs artifacts to some extent, but at the expense of lowered image clarity.

In image <NUM>, Gibbs artifacts were reduced through image reconstruction involving low-pass filtering and total variation. For example, the alternating bright and dark lines were greatly removed and became essentially invisible; the details of the structures at the border of an abrupt intensity change retained (see arrows in <NUM>). It may be indicated that the low-pass filtering in the present disclosure combined with the total variation reconstruction process may be effective in removing Gibbs artifacts and retaining the detailed structures or the clarity of the image.

<FIG> illustrates three exemplary images generated from cervical spine (C-spine) scanning and different processing steps according to some embodiments of the present disclosure. Images <NUM>, <NUM>, and <NUM> were based on the same data array acquired from the same C-spine scanning.

The image <NUM> was generated through a data processing protocol excluding filtering. The processing procedure and parameters used in the processing were the same with that of image <NUM>. For example, the size of the data array was doubled after zero-padding.

The image <NUM> was generated through a data processing protocol including filtering. The processing procedure and parameters used in the processing were the same with that of image <NUM>. For example, a low-pass filter which having a Hanning window in the transition band with a normalized transition bandwidth of <NUM> and a Δr of <NUM> was used.

The image <NUM> was generated through a data processing protocol including low-pass filtering, zero-padding, and total variation reconstruction, as described in the present disclosure. The processing procedure and parameters used in the processing were the same with that of image <NUM>. For example, a low-pass filter having a Hanning window in the transition band with a normalized transition bandwidth of <NUM> and a Δr of <NUM> was used, the parameter λ<NUM> was set as <NUM>, the parameter λ<NUM> was set as <NUM>, and the parameter P<NUM> was a matrix used to choose a region M<NUM> in which the frequency of data were no larger than the highest frequency of the original data multiplied by <NUM>.

Similarly with images <NUM>, <NUM>, and <NUM>, the image <NUM> has significant Gibbs artifacts (see arrows in <NUM>); the image <NUM> has substantially reduced Gibbs artifacts but with residual artifacts and blurred border of an abrupt intensity change (see arrows in <NUM>); the image <NUM> has significantly reduced Gibbs artifacts, and retained sharp and clear borders and detailed structures (see arrows in <NUM>), especially for the areas around the fatty tissue on the back of the neck. It may be determined that the low-pass filtering in the present disclosure combined with the total variation reconstruction process may be effective in removing Gibbs artifacts and retaining the detailed structures or the clarity of the image.

Having thus described the basic concepts, it may be rather apparent to those skilled in the art after reading this detailed disclosure that the foregoing detailed disclosure is intended to be presented by way of example only and is not limiting.

The program code may execute entirely on the operator's computer, partly on the operator's computer, as a stand-alone software package, partly on the operator's computer and partly on a remote computer or entirely on the remote computer or server. In the latter scenario, the remote computer may be connected to the operator's computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection may be made to an external computer (for example, through the Internet using an Internet Service Provider) or in a cloud computing environment or offered as a service such as a Software as a Service (SaaS).

Furthermore, the recited order of processing elements or sequences, or the use of numbers, letters, or other designations therefore, is not intended to limit the claimed processes and methods to any order except as is specified in the claims. For example, although the implementation of various components described above may be embodied in a hardware device, it may also be implemented as a software only solution-e.g., an installation on an existing server or mobile device.

Similarly, it should be appreciated that in the foregoing description of embodiments of the present disclosure, various features are sometimes grouped together in a single embodiment, figure, or description thereof for the purpose of streamlining the disclosure aiding in the understanding of one or more of the various inventive embodiments. This method of disclosure, however, is not to be interpreted as reflecting an intention that the claimed subject matter requires more features than are expressly recited in each claim. Rather, inventive embodiments lie within the scope defined by the appended claims.

In some embodiments, the numbers expressing quantities of ingredients, properties such as molecular weight, reaction conditions, and so forth, used to describe and claim certain embodiments of the application are to be understood as being modified in some instances by the term "about," "approximate," or "substantially. " For example, "about," "approximate," or "substantially" may indicate ±<NUM>% variation of the value it describes, unless otherwise stated. Accordingly, in some embodiments, the numerical parameters set forth in the written description and attached claims are approximations that may vary depending upon the desired properties sought to be obtained by a particular embodiment. In some embodiments, the numerical parameters should be construed in light of the number of reported significant digits and by applying ordinary rounding techniques. Notwithstanding that the numerical ranges and parameters setting forth the broad scope of some embodiments of the application are approximations, the numerical values set forth in the specific examples are reported as precisely as practicable.

Claim 1:
A method of image processing comprising:
setting k-space comprising a first part (<NUM>) and a second part (<NUM>, <NUM>); wherein the second part (<NUM>, <NUM>) is outside of the first part (<NUM>); filling, in the first part (<NUM>) of the k-space, a matrix comprising sampled magnetic resonance imaging data
applying a filter on the matrix in the first part (<NUM>) to produce a filtered data matrix in the first part;
padding the second part (<NUM>, <NUM>) of the k-space with padded data; reconstructing
an initial image based on the filtered data matrix in the first part (<NUM>) of the k-space and the padded data in the second part (<NUM>, <NUM>) of the k-space;
performing, based on a constraint, a plurality of iterations of an objective function for a target array related to the initial image in image domain, wherein the objective function is based on a total variation of the target array; and
reconstructing an image based on the target array, characterized in that the first part (<NUM>) is divided into a first region (<NUM>, <NUM>, <NUM>) and a second region (<NUM>, <NUM>, <NUM>), and the constraint is given by setting the filtered data in the second region (<NUM>, <NUM>, <NUM>) of the first part (<NUM>) of the k-space to be invariant.