USRE44981E1 - Method for super-resolution reconstruction using focal underdetermined system solver algorithm - Google Patents
Method for super-resolution reconstruction using focal underdetermined system solver algorithm Download PDFInfo
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- USRE44981E1 USRE44981E1 US13/738,397 US201313738397A USRE44981E US RE44981 E1 USRE44981 E1 US RE44981E1 US 201313738397 A US201313738397 A US 201313738397A US RE44981 E USRE44981 E US RE44981E
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
- G01R33/00—Arrangements or instruments for measuring magnetic variables
- G01R33/20—Arrangements or instruments for measuring magnetic variables involving magnetic resonance
- G01R33/44—Arrangements or instruments for measuring magnetic variables involving magnetic resonance using nuclear magnetic resonance [NMR]
- G01R33/48—NMR imaging systems
- G01R33/54—Signal processing systems, e.g. using pulse sequences ; Generation or control of pulse sequences; Operator console
- G01R33/56—Image enhancement or correction, e.g. subtraction or averaging techniques, e.g. improvement of signal-to-noise ratio and resolution
- G01R33/561—Image enhancement or correction, e.g. subtraction or averaging techniques, e.g. improvement of signal-to-noise ratio and resolution by reduction of the scanning time, i.e. fast acquiring systems, e.g. using echo-planar pulse sequences
-
- G—PHYSICS
- G01—MEASURING; TESTING
- G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
- G01R33/00—Arrangements or instruments for measuring magnetic variables
- G01R33/20—Arrangements or instruments for measuring magnetic variables involving magnetic resonance
- G01R33/44—Arrangements or instruments for measuring magnetic variables involving magnetic resonance using nuclear magnetic resonance [NMR]
- G01R33/48—NMR imaging systems
- G01R33/4818—MR characterised by data acquisition along a specific k-space trajectory or by the temporal order of k-space coverage, e.g. centric or segmented coverage of k-space
- G01R33/4824—MR characterised by data acquisition along a specific k-space trajectory or by the temporal order of k-space coverage, e.g. centric or segmented coverage of k-space using a non-Cartesian trajectory
-
- G—PHYSICS
- G01—MEASURING; TESTING
- G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
- G01R33/00—Arrangements or instruments for measuring magnetic variables
- G01R33/20—Arrangements or instruments for measuring magnetic variables involving magnetic resonance
- G01R33/44—Arrangements or instruments for measuring magnetic variables involving magnetic resonance using nuclear magnetic resonance [NMR]
- G01R33/48—NMR imaging systems
- G01R33/54—Signal processing systems, e.g. using pulse sequences ; Generation or control of pulse sequences; Operator console
- G01R33/56—Image enhancement or correction, e.g. subtraction or averaging techniques, e.g. improvement of signal-to-noise ratio and resolution
- G01R33/5608—Data processing and visualization specially adapted for MR, e.g. for feature analysis and pattern recognition on the basis of measured MR data, segmentation of measured MR data, edge contour detection on the basis of measured MR data, for enhancing measured MR data in terms of signal-to-noise ratio by means of noise filtering or apodization, for enhancing measured MR data in terms of resolution by means for deblurring, windowing, zero filling, or generation of gray-scaled images, colour-coded images or images displaying vectors instead of pixels
Definitions
- the present invention relates a method for obtaining high resolution images using a magnetic resonance imaging (MRI) device. More particularly, the present invention relates to a method for obtaining high resolution MRI images by using a sparse reconstruction algorithm.
- MRI magnetic resonance imaging
- atomic nuclei are located in a strong magnetic field so as to cause a precession of the atomic nuclei.
- a high-frequency signal is applied to the atomic nuclei which are magnetized by a magnetic field generated by the precession, the atomic nuclei are excited into a high energy state.
- the high-frequency signal is removed, the atomic nuclei emit high-frequency signals.
- the magnetic properties of the materials constituting human body are measured from the emitted high-frequency signals, and the materials are reconstructed, thereby making an image.
- dynamic MRI is a technology for obtaining a moving image by observing and measuring a temporally changing process, such as cerebral blood flow, heart beat, etc.
- the phase encoding gradient is not used, so that the echo time is short, photographing can be performed without being restrained by breathing and blood flow movement, and an aliasing artifact in data downsampled in a radial shape is generated in a line shape, thereby causing relatively less visual confusion.
- the present invention has been made to solve the above-mentioned problems of the prior art, and the present invention provides a method of forming a high resolution image using a sparse reconstruction algorithm, the focal underdetermined system solver (FOCUSS) algorithm.
- FOCUSS focal underdetermined system solver
- the present invention provides a method for forming a high-resolution image, the method comprising the steps of: (a) outputting data for an image of an object; (b) downsampling the outputted data; (c) transforming the downsampled data into low-resolution image frequency data; and (d) reconstructing a high-resolution image from the transformed low-resolution image frequency data by applying focal underdetermined system solver (FOCUSS) algorithm.
- the image of an object can be a still image, a moving image, or both.
- the step (c) may be performed by inverse Radon transformation.
- the step (c) may be performed by inverse Fourier transformation.
- the method when the image is a moving image, the method may be performed in k-t space.
- the step (b) may be performed by obtaining all data in a frequency encoding direction during a predetermined period in a time domain and random-pattern data in a phase encoding direction according to each period.
- the step (c) may be performed by two-dimensional Fourier transformation.
- the step (d) may further comprise the steps of: (1) calculating a weighting matrix from the low-resolution image frequency data; (2) calculating image data from the weighting matrix and the low-resolution image frequency data satisfying a predetermined condition; and (3) when the image data is converged the high-resolution image, performing inverse Fourier transformation along a time axis to reconstruct the high-resolution image; or when the image data is not converged, updating the weighting matrix by using a diagonal element of the image data and repeating the step (2) with the updated weighting matrix until the image data is converged to the high-resolution image.
- the low-resolution image frequency data satisfying a predetermined condition in the step (2) may be calculated by Lagrangian transformation.
- the FOCUSS algorithm when a Fourier transform transformed by the Lagrangian transformation is replaced by a Fourier transform applied along a time axis and Radon transformation, the FOCUSS algorithm may be applied with respect to radial data in k or k-t space.
- the radial data may correspond to downsampled data obtained at a uniform angle.
- the FOCUSS algorithm may be applied with respect to spiral data in k or k-t space.
- the spiral data may correspond to downsampled data obtained at a uniform angle.
- the updating of weighting matrix in the step (3) may be performed by applying a power factor to absolute value of the diagonal element.
- Preferable range of the power factor is 0.5 to 1.
- FIG. 1 is a flowchart schematically illustrating a still/moving image reconstruction method using the FOCUSS algorithm according to the present invention.
- FIG. 2 is a flowchart schematically illustrating the FOCUSS algorithm according to the present invention.
- FIG. 1 is a flowchart schematically illustrating a high-resolution image reconstruction method using the focal underdetermined system solver (FOCUSS) algorithm according to an exemplary embodiment of the present invention.
- FOCUSS focal underdetermined system solver
- the downsampled data is transformed into a low-resolution initial estimation for sparse data.
- an inverse Radon transformation is applied to transform the radial data into a low-resolution initial estimation
- an inverse Fourier transform is applied to transform the random or spiral data into a low-resolution initial estimation.
- a Fourier transform is applied along the time axis, thereby obtaining an initial estimation for sparser data.
- a high-resolution image is reconstructed by applying the FOCUSS algorithm to the low-resolution initial estimation, to which the inverse Radon transformation or inverse Fourier transform has been applied, wherein image data is calculated by multiplying the low-resolution initial estimation by a weighting matrix of a predetermined condition.
- the weighting matrix is updated with the diagonal values of a matrix which is obtained by applying to the absolute value of the estimation data matrix a power factor ranging from 0.5 to 1, the low-resolution estimation data is recalculated to be optimized, and the FOCUSS algorithm is repeatedly performed until the estimation data is converged to the optimized high-resolution image.
- equation 1 corresponds to a magnetic resonance image signal acquisition equation for a still image. In contrast, when different data values are obtained depending on “t,” equation 1 corresponds to a magnetic resonance image signal acquisition equation for a moving image.
- the reason why the sparse characteristic is important is that signals other than “0” of ⁇ (y,f) are not scattered in a sparse distribution but are concentrated on a position in order to apply a compressed sensing theory, and that it is possible to achieve complete reconstruction of sparse signals from much less samples than those required for the Nyquist sampling limit.
- F represents a transform for making sparse data, and may be a Fourier transform or Radon transformation, which is calculated for a solution having a sparse characteristic by using the FOCUSS algorithm.
- the FOCUSS algorithm is used to obtain a solution of a sparse form for an underdetermined linear equation having no determined solution.
- v F ⁇ Equation 3
- equation 1 has a large number of solutions.
- equation 3 when a solution of equation 3 is determined to minimize the norm of “v,” energy shows a tendency to spread, so that the sampling rate increases.
- an image reconstruction method using the FOCUSS algorithm according to an exemplary embodiment of the present invention is applied, which is approached as an optimization problem for solving equation 3 according to the FOCUSS algorithm.
- Equation 4 “ ⁇ ” represents sparse data, “W” represents a weighting matrix having only diagonal elements, and “q” represents an optimized solution for the weighting matrix “W.”
- an initial substitution value of the weighting matrix may be calculated through a pseudo inverse matrix, in which when the sampled data corresponds to radial projection data, an inverse Radon transformation is applied to transform the radial data into a low-resolution initial estimation, and when the output data corresponds to random or spiral data, an inverse Fourier transform is applied to transform the random or spiral data into a low-resolution initial estimation.
- the output data is subjected to an inverse Fourier transform of “k” in k space, and is then transformed into low-resolution image data, thereby being arranged in terms of time.
- image frequency data in the frequency domain is obtained.
- all data during one period in a frequency encoding direction is obtained in a k x direction, and data having different random patterns depending on periods in a phase encoding direction is obtained in a k y direction.
- DC data corresponding to a time frequency of “0” in a y-f domain calculated as above is set to be “0,” and initial data to be substituted into the weighting matrix “W” is calculated (step 31 ).
- step 31 diagonal elements of initial data calculated in step 31 are substituted into an initial weighting matrix “W,” and a resultant weighting matrix is multiplied by low-resolution image data “q” which is an optimal solution for a given weighting matrix, thereby calculating image data “ ⁇ ” (step 32 ).
- step 33 a step of checking whether or not the calculated image data “ ⁇ ” is converged to the high-resolution image is performed.
- the procedure ends.
- the weighting matrix “W” is updated with diagonal elements of a matrix which is obtained by raising the absolute value of the calculated image data “ ⁇ ” to the power of 0.5 (step 34 ), for a higher-resolution image is obtained as the iteration performance is repeated.
- the exponent of the absolute value of the calculated image data “ ⁇ ” has a value within a range of 0.5 to 1.
- the low-resolution image data “q” is recalculated to be an optimal solution (step 35 ).
- the procedure ends after the calculated image has been obtained, and if the procedure is to obtain a moving image, the calculated image is subjected to an inverse Fourier transform along the time axis so that a time-based image can be reconstructed and then the procedure ends.
- Equation 5 Equation 5
- the advantage of the FOCUSS algorithm according to the present invention is to converge calculated image data to high-resolution image data “ ⁇ ” by updating a weighting matrix “W,” in which when steps 32 to 35 in the FOCUSS algorithm according to the present invention are repeated (n ⁇ 1) times, image data “ ⁇ ” is expressed as follows.
- ⁇ n ⁇ 1 [ ⁇ n ⁇ 1 (1), ⁇ n ⁇ 1 (2), . . . , ⁇ n ⁇ 1 (N)] T Equation 7
- Equation 4 may be expressed as follows.
- the FOCUSS algorithm according to the present invention can obtain an accurate result based on the compressed sensing theory, by setting the exponent to be “0.5” and repeating steps 34 and 35 several times.
- equation 10 The solution of equation 10 is calculated as equation 11, in which even when multiple coils are used, a result approximating to a high-resolution image can be obtained by setting the exponent to be “0.5” and repeating the steps.
- the FOCUSS algorithm according to the present invention is applied to each coil, and results as many as coils yielding a result are obtained by means of the least squares method, thereby obtaining a final result, which is expressed as equation 12 below.
- the FOCUSS algorithm according to the present invention may be applied to various data, which can be achieved through the Fourier transform of equation 5.
- random sampling of downsampled data corresponds to Gaussian random sampling
- low-resolution image data “q” satisfying a predetermined condition in step 32 is calculated by means of the Lagrangian, wherein when the Fourier transform of a predetermined condition transformed by the Lagrangian is replaced by a Fourier transform applied along a time axis and a Radon transformation, the FOCUSS algorithm can be applied to radial data.
- the radial data must satisfy a condition that the radial data is downsampled at a uniform angle.
- the low-resolution image data satisfying a predetermined condition in step 32 is calculated by means of the Lagrangian, as described above, wherein when Fourier transform of a predetermined condition transformed by the Lagrangian is replaced by a Fourier transform applied along a time axis and a Radon transformation, the FOCUSS can be applied to spiral data.
- the spiral data must satisfy a condition that the spiral data is downsampled at a uniform angle.
- the FOCUSS algorithm according to the present invention can reconstruct even radial data and spiral data to a high-resolution image.
- the FOCUSS algorithm according to the present invention can reconstruct an image from a very small amount of data, thereby reducing the scan time period. This means that it is possible to improve the time resolution, which is very important for both still and moving images.
- the present invention provides a new algorithm which can overcome the limitation of the conventional magnetic resonance imaging (MRI), and the new algorithm is expected to provide a more correct image and to help to examine a patient.
- MRI magnetic resonance imaging
Abstract
Description
v(k,t)=∫∫ρ(y,f)e−j2π(ky+ft)dydf Equation 1
minimize ∥π∥1
subject to ∥v−Fπ∥2≦ε Equation 2
v=Fρ Equation 3
find ρ=Wq
min∥q∥2 subject to ∥v−FWq∥2≦ε Equation 4
C(q)=∥v−FWq∥2 2+λ∥q∥2 2 Equation 5
ρn−1=[ρn−1(1), ρn−1(2), . . . , ρn−1(N)]T Equation 7
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KR1020070005906A KR100816018B1 (en) | 2007-01-19 | 2007-01-19 | Method for super-resolution reconstruction using focal underdetermined system solver algorithm |
KR1020070007568A KR100816020B1 (en) | 2007-01-24 | 2007-01-24 | Method for super-resolution reconstruction using focal underdetermined system solver algorithm on k-t space |
KR10-2007-0007568 | 2007-01-24 | ||
US11/823,633 US7881511B2 (en) | 2007-01-19 | 2007-06-28 | Method for super-resolution reconstruction using focal underdetermined system solver algorithm |
US13/738,397 USRE44981E1 (en) | 2007-01-19 | 2013-01-10 | Method for super-resolution reconstruction using focal underdetermined system solver algorithm |
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US7881511B2 (en) | 2011-02-01 |
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