Devices systems and methods for processing images
View Patent ↗Certain exemplary embodiments can comprise a method, which can comprise providing machine instructions adapted to render a final enhanced image of an object based upon a final enhanced set of data. The final enhanced set of data can be derived from a detected set of data obtained from an imaging device, such as a medical imaging device.
1. A method comprising:
receiving a first set of data from a magnetic resonance image device;
automatically applying a minimization algorithm to a pre-formulated functional that comprises a norm of a sparse representation of a trial image derived from said first set of data and a data fidelity term, said minimization algorithm comprising a conjugate gradient sub-algorithm and a Bregman iteration;
at each iteration of a plurality of iterations of said Bregman iteration:
updating said data fidelity term; and
repeating said conjugate gradient sub-algorithm to iteratively update said trial image, said conjugate gradient sub-algorithm comprising:
at each iteration of a plurality of iterations, finding an update direction of said trial image via:
computing a sub-gradient of said norm of said sparse image representation and a gradient of said data fidelity term;
applying a nonuniform Fourier transform of said trial image;
subtracting said first set of data from said nonuniform Fourier transform;
using an adjoint of said nonuniform Fourier transform to obtain said gradient of said data fidelity term;
finding a step size in said update direction using a backtracking line search algorithm;
updating said trial image based upon said update direction and said step size; and
rendering a final image when said Bregman iteration is finished.
2. A method comprising:
providing a processor and a medium with machine instructions adapted to render a final enhanced image of an object based upon a final enhanced set of data, said final enhanced set of data derived from a detected set of data obtained from a magnetic resonance device, said final enhanced set determined via a minimization algorithm applied to a formulated functional, said formulated functional based upon a trial image derived from said detected set of data, said minimization algorithm adapted to iteratively utilize a conjugate gradient sub-algorithm and a Bregman iteration, said formulated functional comprising:
a norm of a sparse representation of said trial image derived from said detected set of data; and
a data fidelity term; and further comprising: at each iteration of a plurality of iterations of said conjugate gradient sub-algorithm: finding an update direction of said trial image via: computing a sub-gradient of said norm of said sparse image representation and a gradient of said data fidelity term; applying a nonuniform Fourier transform of said trial image; subtracting said first set of data from said nonuniform Fourier transform; using an adjoint of said nonuniform Fourier transform to obtain said gradient of said data fidelity term; finding a step size in said update direction using a backtracking line search algorithm; and updating said trial image based upon said update direction and said step size.
3. The method of claim 2 , further comprising:
obtaining said detected set of data from said magnetic resonance image device.
4. The method of claim 2 , further comprising:
applying said minimization algorithm to said formulated functional.
5. The method of claim 2 , further comprising:
determining said norm of said sparse representation of said trial image derived from said detected set of data.
6. The method of claim 2 , further comprising:
at each iteration of a plurality of iterations of said Bregman iteration:
updating said data fidelity term; and
repeating said conjugate gradient sub-algorithm to iteratively update said trial image.
7. The method of claim 2 , further comprising:
determining said sparse representation of said trial image derived from said detected set of data.
8. The method of claim 2 , wherein said minimization algorithm attempts to minimize an equation:
min
m
Ψ
(
m
)
1
+
λ
A
m
-
y
2
2
where:
m is a representation of said trial image;
ψ is a transformation that transforms said representation of said trial image into a sparse representation;
λ is a Lagrange multiplier;
∥ ∥ is a norm;
A is a non-uniform Fourier transform matrix; and
y is undersampled measured k-space data.
9. The method of claim 2 , wherein said formulated functional comprises a term:
m
BV
=
∑
i
,
j
∇
x
m
re
(
i
,
j
)
2
+
∇
x
m
im
(
i
,
j
)
2
+
∇
y
m
re
(
i
,
j
)
2
+
∇
y
m
im
(
i
,
j
)
2
where:
m is a representation of said trial image;
m re is a real component of said representation of said trial image;
m im is an imaginary component of said representation of said trial image;
ψ is a transformation that transforms said representation of said trial image into a sparse representation;
∥m∥ BV is a hounded variation norm;
∇ is a gradient symbol;
∥ is an absolute value of an expression;
x is a horizontal direction in said representation of said trial image in a predetermined coordinate system;
y is a vertical direction in said representation of said trial image in said predetermined coordinate system;
i is an index for pixels in a vertical direction in said predetermined coordinate system; and
j is an index for pixels in a horizontal direction in said predetermined coordinate system.
10. The method of claim 2 , wherein said conjugate gradient sub-algorithm comprises evaluating an equation:
∇∥ m∥ BV =ζ i-1,j x −ζ i,j x +ζ i,j-1 y −ζ i,j y ,
where:
ζ
i
,
j
x
=
∇
x
m
(
i
,
j
)
∇
x
m
re
(
i
,
j
)
2
+
∇
y
m
re
(
i
,
j
)
2
+
∇
x
m
im
(
i
,
j
)
2
+
∇
y
m
im
(
i
,
j
)
2
+
ɛ
;
ζ
i
,
j
y
=
∇
y
m
(
i
,
j
)
∇
x
m
re
(
i
,
j
)
2
+
∇
y
m
re
(
i
,
j
)
2
+
∇
x
m
im
(
i
,
j
)
2
+
∇
y
m
im
(
i
,
j
)
2
+
ɛ
;
m is a representation of said trial image;
m re is a real component of said representation of said trial image;
m in is an imaginary component of said representation of said trial image;
ε is a small constant;
∇ is a gradient symbol;
∥ is an absolute value of an expression;
x is a horizontal direction in said representation of said trial image in a predetermined coordinate system;
y is a vertical direction in said representation of said trial image in said predetermined coordinate system;
i is an index for pixels in a vertical direction in said predetermined coordinate system; and
j is an index for pixels in a horizontal direction in said predetermined coordinate system.
11. The method of claim 2 , wherein said formulated functional comprises a term:
m
=
∑
p
α
L
,
p
φ
L
,
p
+
∑
q
=
L
+
1
J
∑
p
α
q
,
p
ϕ
q
,
p
where:
m is a representation of said trial image;
J is related to a length of said signal by N=2 J ;
L is a lowest level (coarsest level) in a multiscale representation;
α p,q are transformed coefficients;
φ L,p are scaling functions;
φ q,p are oscillatory wavelets;
q is an index for different levels; and
p is an index for a p th element in said multiscale representation.
12. The method of claim 2 , wherein said formulated functional comprises a curvelet transform.
13. The method of claim 2 , wherein said Bregman iteration comprises attempting to solve an equation:
m
k
*
=
arg
min
m
{
J
(
m
)
+
λ
H
(
m
,
y
+
v
k
-
1
*
)
}
where:
m is a representation of said trial image;
k is an iteration count for said Bregman iteration;
J(m) is a regularization function;
H(m, y+v* k-1 ) is a data fidelity term; and
λ is a Lagrange multiplier.
14. The method of claim 7 , further comprising:
at each iteration of a plurality of iterations of said conjugate gradient sub-algorithm, determining a non-uniform Fourier transform of an intermediate enhanced image.
15. The method of claim 2 , further comprising:
at each iteration of a plurality of iterations of said conjugate gradient sub-algorithm, determining a non-uniform Fourier transform of an intermediate enhanced image; and
comparing said non-uniform Fourier transform to said detected set of data.
16. The method of claim 2 , wherein said minimization algorithm further comprises a backtracking line search algorithm.
17. The method of claim 2 , further comprising:
determining that said minimization algorithm is complete.
18. A machine-readable medium storing machine instructions implementing the steps of:
providing machine instructions adapted to render a final enhanced image of an object based upon a final enhanced set of data, said final enhanced set of data derived from a detected set of data obtained from a magnetic resonance device, said final enhanced set determined via a minimization algorithm applied to a formulated functional, said formulated functional based upon a trial image derived from said detected set of data, said minimization algorithm adapted to iteratively utilize a conjugate gradient sub-algorithm and a Bregman iteration, said formulated functional comprising:
a norm of a sparse representation of said trial image derived from said detected set of data; and
a data fidelity term; and further comprising: at each iteration of a plurality of iterations of said conjugate gradient sub-algorithm: finding an update direction of said trial image via: computing a sub-gradient of said norm of saidsparse image representation and a gradient of said data fidelity term; applying anonuniform Fourier transform of said trial image; subtracting said first set of data from said nonuniform Fourier transform; using an adjoint of said nonuniformFourier transform to obtain said gradient of said data fidelity term; finding a step size in said update direction using a backtracking line search algorithm; andupdating said trial image based upon said update direction and said step size.