CN111833412B - Tikhonov regularized image reconstruction method based on fractional filter framework - Google Patents
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Abstract
The invention belongs to the field of medical image processing, and particularly relates to a Tikhonov regularized photoacoustic imaging reconstruction method based on a fractional filter frame, which comprises the following steps of: s1, constructing a system matrix by Matlab simulation to obtain a forward model of photoacoustic imaging; s2, obtaining a functional of ax=b through Tikhonov regularization, and selecting a fractional order on a (0, 1) interval; s3, obtaining lambda by adopting a Morozov deviation principle; s4, obtaining a reconstruction solution and a filtering function; s5, if the reconstruction solution and the filter function obtained in the S4 can maximize the SNR or CNR value of the reconstructed image, obtaining the most suitable regularization parameter value and the fractional order, otherwise, repeating the steps S2-S5, and further reducing the search interval. The fractional order Tikhonov regularization of the invention has higher quality than the reconstructed image regularized by the traditional standard Tikhonov. The superior performance of the fractional filter framework is due to the inclusion of fractional orders that control smoothness by increasing the norms of the reconstruction solution, rather than pure integer order regularization. The invention is used for reconstruction of photoacoustic imaging.
Description
Technical Field
The invention belongs to the field of medical image processing, and particularly relates to a Tikhonov regularized photoacoustic imaging reconstruction method based on a fractional filter frame.
Background
Medical imaging, as an auxiliary medical tool, can help humans detect, determine, recognize and study disease. Existing medical imaging modalities are inadequate to aid in human exploration and research of disease. Photoacoustic imaging (Photoacoustic tomography, PAT) is used as a novel lossless biomedical imaging technology, combines the advantages of pure optical imaging and pure ultrasonic imaging, and has the advantages of strong contrast, high sensitivity and deep imaging depth.
The photoacoustic imaging is to irradiate a target tissue with modulated laser, heat the tissue to excite ultrasonic waves, and reconstruct light absorption distribution reflecting the internal structure of the target tissue from the collected ultrasonic waves. The reconstruction algorithm affects the imaging quality of the image and its choice is therefore very important. The existing reconstruction algorithms include Filtered Back Projection (FBP), time reversal and Fourier transform-based reconstruction algorithms. These algorithms are based on a spherical Radon transform model, which has an inherent limitation that a large number of data points are required around the target object to accurately estimate the initial pressure distribution. Restoration of the initial pressure rising distribution based on the model is an uncomfortable problem due to the limited boundary data and the unavoidable noise mixed in the measured photoacoustic signal during the actual measurement.
Regularization is widely applied as a technology for solving the problem of pathological inversion, and has a plurality of classical regularization algorithms, such as Tikhonov regularization, which can enable the reconstructed image to have small gradient, and Laplacian has good reconstruction effect in an image smooth area due to isotropic evolution characteristics. However, due to the discontinuity of the image signal, in some unsmooth regions such as edges, the adopted L2 norm can cause excessive punishment of the image edges, and it is difficult to generate an accurate reconstructed image.
Disclosure of Invention
Aiming at the technical problem that the discontinuity of the traditional Tikhonov regularized image signal is difficult to generate an accurate reconstructed image, the invention provides the Tikhonov regularized image reconstruction method based on the fractional filter frame, which has high performance, high accuracy and small calculation amount.
In order to solve the technical problems, the invention adopts the following technical scheme:
a Tikhonov regularized image reconstruction method based on a fractional filter framework is characterized in that: comprises the following steps:
s1, constructing a system matrix A by Matlab simulation to obtain a forward model of photoacoustic imaging: ax=b, wherein x is a pressure initial value of each pixel in the imaging area, and b is actual measurement acoustic data on a boundary;
s2, obtaining a functional of ax=b through Tikhonov regularization, and selecting a fractional order alpha on a (0, 1) interval;
s3, calculating a parameter lambda by adopting a Morozov deviation principle, so that the residual norm is equal to the priori upper limit delta;
s4, obtaining a reconstruction solution x λ,α ＝VF(U T b) And a filter function f=diag (σ) α /(σ α+1 +λ))；
S5, the reconstruction solution and the filter function obtained in S4 can further establish the relation between the fractional order alpha and the noise level, wherein the fractional order alpha is reduced along with the increase of noise, and the value of SNR or CNR of the reconstructed image is maximized by changing the value of SNR or CNR of the acquired PA data to obtain the most suitable regularization parameter value lambda opt And fractional order alpha opt Otherwise, repeating S2-S5 to further narrow the search interval.
The method for constructing the system matrix A in the S1 comprises the following steps: the basic equation of the photoacoustic imaging is obtained through the thermal equation, the motion equation and the diffusion equation of the photoacoustic imaging:said->Is a three-dimensional space position vector, t is time, p is sound pressure, C is sound velocity, beta is an isobaric expansion coefficient, and C p Let I be the delta function for specific heat, and define the pulse emission time as the time zero, i.e. I (t) =delta (t), the solution of the basic equation of photoacoustic imaging is approximated by the Green function to get a photoacoustic signal: />The process of collecting power amplifier data at sensor locations by simulating photoacoustic signals through k-Wave is represented as a time-varying causal system, and a system matrix A of the time-varying causal system is constructed by using a tool box of the k-Wave, wherein the system matrix A comprises impulse responses of pixels in an imaging area as a system matrix AColumns, the imaging grid being n pixels, the imaging grid converting all columns stacked together into a size n 2 X 1 high column vector, the initial pressure rise of the pixels of the imaging grid is x, the system matrix a has m x n 2 To derive a forward model of photoacoustic imaging: ax=b.
The functional of ax=b in S2 is:the weighted normsThe W is a symmetrical semi-positive definite matrix, the +.>The alpha is a fractional order of alpha > 0; when α=1, matrix w=i, where I is the identity matrix, and the weighted norm allows selection of a fractional order α that takes a suitable value in the (0, 1) interval that maximizes the value of SNR or CNR of the reconstructed image, which reduces the disadvantage of too smooth a standard regularization solution.
The method for solving the parameter lambda by adopting the Morozov deviation principle in the S3 comprises the following steps: let e be the observation error and have an upper bound epsilon, satisfy the condition that E is less than epsilon,
define δ=ηε, then solve x λ,α Satisfy b-Ax λ,α ||＝δ，
Let G (λ) = |b-Ax λ,α || 2 -δ 2 ＝0，
When α is given, the G (λ) = |b-Ax λ,α || 2 -δ 2 =0 is a nonlinear implicit equation for λ, which is iteratively solved using the Newton method, i.e
The saidFor the iterative step size, the ∈>The expression of G' (λ) is:
the saidFrom the functional Γ Tikh Minimizing x relative to x Tikh ＝(A T A+λI) -1 A T b deriving lambda, i.e
Residual norms combined with standard Tikhonov algorithm:
the expression of the pushable step length is
Wherein the method comprises the steps of
Constructing a residual margin sequence of least squares solution from the residual norms of the standard Tikhonov algorithm, i.e
Based on min|ρ (i) - δ given δ 2 The corresponding sequence number i when the expression takes the minimum value can be known, so that the iteration initial value lambda 0 ＝σ i 。
The S4 obtains a reconstruction solution x λ,α ＝VF(U T b) And filter functionThe number f=diag (σ) α /(σ α+1 +λ)) is:
by the principle of deviation b-Ax λ || 2 The expression =δ can be given as a function of λ, where F is substituted for a as an n×m matrix whose diagonal elements are filter factors in fractional order Tikhonov format, i.e.
Obtaining
Consider k=min (m, n 2 ) Then And then differentiating, i.e. delta (lambda) can be obtained as an inverse function of lambda (delta),then calculate the reconstruction solution x λ,α ＝VF(U T b) And a filter function f=diag (σ) α /(σ α+1 +λ))。
Compared with the prior art, the invention has the beneficial effects that:
the fractional order Tikhonov regularization of the invention has higher quality than the reconstructed image regularized by the traditional standard Tikhonov. The superior performance of the fractional filtering framework is due to the inclusion of fractional order α, which controls smoothness by increasing the norm of the reconstruction solution, rather than pure integer order regularization. Wherein the regularization parameters are automatically selected by the bias principle. The method for reconstructing the image has small calculated amount, provides a better balance between noise suppression and feature maintenance, and can obtain a more reasonable reconstructed image.
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FIG. 1 is a process for automatically selecting fractional order and regularization parameters in accordance with the present invention.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
A Tikhonov regularized image reconstruction method based on a fractional filter frame comprises the steps that when electromagnetic wave pulses are irradiated on biological tissues to be detected, the tissues absorb electromagnetic energy to generate heat, and a thermal equation of photoacoustic imaging can be given at the moment:
wherein the method comprises the steps ofIs a three-dimensional spatial position vector, T is time, function H represents the heat absorbed by the tissue, function T represents the temperature at which the temperature rises, ρ is the density of the tissue (here assumed to be constant), C p Is specific heat and λ is thermal conductivity. In general, the duration of the electromagnetic wave pulse is much shorter than the thermal conduction time, so thermal conduction can be neglected, so (1) can be simplified as:
the tissue expands as a result of heating, thereby creating an initial sound field, and the sound field will change over time (i.e., sound waves are generated). The equation of motion and the equation of diffusion of the photoacoustic imaging sound field can be given at this time:
wherein the vector functionThe acoustic shift is represented, the function p represents the sound pressure, c is the sound velocity, and β is the isobaric expansion coefficient. The simultaneous equations (2) - (4) are written as the product of the light absorption distribution function a with respect to space and the irradiation pulse energy function I with respect to time, to obtain:
equation (5) can be used as a basic equation of photoacoustic imaging, and represents the relationship between the photoacoustic signal and the light absorption characteristics of tissue, and is a forward model describing photoacoustic imaging. Let I be the delta function and define the pulse transmit time as the time zero, I (t) =delta (t); the solution of equation (5) can be approximated by a Green function:
the linear equation obtained by the discretization of equations (2) to (4) is ax=b, and the linear equation cannot be directly solved because of the limited boundary data and unavoidable noise mixed in the measured photoacoustic signal in the actual measurement process. To reconstruct the initial sound pressure distribution, as shown in FIG. 1, a Tikhonov regularization method is commonly used, whose functional expression is
Wherein A is a system matrix, which comprises impulse responses of pixels in an imaging area as columns, x is a pressure initial value of each pixel in the imaging area, and b is actual measurement acoustic data on a boundary (detector position). Lambda is a regularization parameter for balancing the remainder and period of the linear equation (first term on right)The desired initial pressure profile (x). Higher regularization tends to overcomplete the image, while smaller lambda values amplify noise in the image. Can push out the functional Γ Tikh Relative to x is minimized to
x Tikh ＝(A T A+λI) -1 A T b (8)
Wherein I is an identity matrix. Equation (8) is a conventional least squares solution when λ=0. Singular value decomposition a=u Σv for a T Substituting the value into (8) and taking the diagonal element of the sigma as a singular value sequence, and then solving x of the standard Tikhonov Tikh Can be expressed as
Wherein the filtering factor is
From equation (9), the solution norm and residual norm of the standard Tikhonov algorithm are respectively
Generally x Tikh It appears too smooth, i.e., the filter factor denoising process loses the data detail that the exact solution contains. In order to overcome the defect, the invention provides a Tikhonov regularization method based on a fractional filter framework, which utilizes AA T The fractional order of the Moore-Penrose pseudo-inverse of the matrix is used as a half-norm of the weighting matrix to measure the scheme of residual errors in Tikhonov regularization. The floodfunction in equation (7) is rewritten as
In the weighted normsWhere W is a symmetric semi-positive definite matrix, the matrix W is as follows
Where α represents a fractional order of α > 0. When α=1, i.e. matrix w=i, formula (13) is standard Tikhonov regularization. Equation (13) has a unique solution for all positive values of the regularization parameter λ. Half norm I.I. | W Allowing the selection of the parameter α to take appropriate values in the (0, 1) interval reduces the disadvantage of the standard regularized solution being too smooth, so that the reconstructed solution resulting from equation (13) has improved image quality. Differentiating the formula (13) with x and making it equal to zero, as a result
((A T A) α+1/2 +λI)x＝(A T A) α-1/2 A T b (15)
Equation (15) can be rewritten as,
(A T WA+λI)x＝A T Wb (16)
SVD of A in the formula (16) to obtain,
(VSU T US α-1 U T USV T +λI)x＝VSU T U S α-1 U T b (17)
(VS α+1 V T +λI)x＝VS α U T b (18)
the canonical solution expression of fractional Tikhonov is
Different regularization methods involve different phi (sigma i ). When sigma is i When gradually tending to 0, phi (sigma i ) Also tends to be 0, while phi (sigma i ) Too fast convergence is the reason that the solution vectors are too smooth, i.e. the small singular value correlation vectors are lost, and these vectors reflect the details of the reconstructed data. Fractional order Tikhonov filter factor phi frac,α Represented by (sigma) as
In the present invention, the fractional order (α) is automatically selected based on maximizing SNR/CNR of the reconstructed image, α taking a suitable value such that φ frac,α Compared with the standard Tikhonov method, the Tikhonov regularization method based on the fractional filter framework can obtain a solution with higher precision.
In the traditional Tikhonov regularization method, larger and smaller singular values are assigned the same weight and are similarly processed no matter how noisy the environment is. However, in a noisy environment, to obtain a good image reconstruction, the smaller singular values need less weight, i.e. in the weighted kernel norm, smaller singular values should be assigned smaller weights. Therefore, compared with the standard method based on Tikhonov, the method introduces the fractional order alpha into the weighting matrix, wherein the fractional order alpha can be changed according to the size of noise, so that the damping and smoothness in the reconstruction solution are controlled to realize more accurate reconstruction.
The singular vectors of smaller singular values typically contain high frequency oscillations. The filter factor of the analytical score Tikhonov method (given in equation (20)) is calculated for smaller singular values, i.e., σ < <1, using taylor series expansion equation (20), ignoring higher order terms, resulting,
phi in equation (21) t '(α)＝σ α Considered as a function of a, and differentiated with respect to a,
φ t '(α)＝σ α ln(σ) (22)
for this equation, consider σ <1, where φ t (alpha) is a decreasing function because phi when sigma <1 t ' (alpha) < 0. Thereafter, each term in equation (21) is a decreasing function of α. Therefore, decreasing the fractional order from α=1 (Tikhonov) increases Φ (α), which means that the high frequency component in the fractional order Tikhonov reconstruction solution increases, i.e., a decrease in the smoothness of the image is achieved.
For the problem of large-scale inversion of PAT image reconstruction, it is difficult to select a proper regularization parameter lambda so that the algorithm has a good noise suppression effect, and meanwhile, the problem of over-regularization cannot be caused. Regularization parameter lambda is selected to be too small, and the solution vector norm of the equation is x λ Too large, the solution is unstable; λ is too large, denoising is too strong, ||x λ The solution is too smooth with smaller l. When the disturbance delta is known to be observed, the regularization parameter automatic selection method based on the Morozov deviation principle is adopted. Let e be the observed error and have an upper bound ε, satisfy
||e||≤ε (23)
Define δ=ηε, then solve x λ,α Satisfy the following requirements
||b-Ax λ,α ||＝δ (24)
Order the
G(λ)＝||b-Ax λ,α || 2 -δ 2 ＝0 (25)
When α is given, equation (25) is a nonlinear implicit equation for λ, which is iteratively solved for λ using Newton's method, i.e
In the method, in the process of the invention,for iterative step length +.>The expression of G' (lambda) is calculated as
In the method, in the process of the invention,from formula (8) deriving lambda, i.e
In combination with (12), the expression of the pushable step is
Wherein the method comprises the steps of
Whether or not the Newton method iteration sequence converges depends on the initial value lambda 0 For this feature, the following process is adopted. Constructing a residual margin sequence of a least squares solution from equation (12), i.e
Based on min|ρ (i) - δ given δ 2 The corresponding sequence number i when the expression takes the minimum value can be known, so that the iteration initial value lambda 0 ＝σ i 。
The preferred embodiments of the present invention have been described in detail, but the present invention is not limited to the above embodiments, and various changes can be made within the knowledge of those skilled in the art without departing from the spirit of the present invention, and the various changes are included in the scope of the present invention.
Claims (1)
1. A Tikhonov regularized image reconstruction method based on a fractional filter framework is characterized in that: comprises the following steps:
s1, constructing a system matrix A by Matlab simulation to obtain a forward model of photoacoustic imaging: ax=b, wherein x is a pressure initial value of each pixel in the imaging area, and b is actual measurement acoustic data on a boundary;
s2, obtaining a functional of ax=b through Tikhonov regularization, and selecting a fractional order alpha on a (0, 1) interval;
s3, calculating a parameter lambda by adopting a Morozov deviation principle, so that the residual norm is equal to the priori upper limit delta;
s4, obtaining a reconstruction solution x λ,α ＝VF(U T b) And a filter function f=diag (σ) α /(σ α+1 +λ))；
S5, the reconstruction solution and the filter function obtained in S4 can further establish the relation between the fractional order alpha and the noise level, wherein the fractional order alpha is reduced along with the increase of noise, and the value of SNR or CNR of the reconstructed image is maximized by changing the value of SNR or CNR of the acquired PA data to obtain the most suitable regularization parameter value lambda opt And fractional order alpha opt Otherwise, repeating S2-S5 to further narrow the search interval;
the method for constructing the system matrix A in the S1 comprises the following steps: the basic equation of the photoacoustic imaging is obtained through the thermal equation, the motion equation and the diffusion equation of the photoacoustic imaging:said->Is a three-dimensional space position vector, t is time, p is sound pressure, C is sound velocity, beta is an isobaric expansion coefficient, and C p Let I be the delta function for specific heat, and define the pulse emission time as the time zero, I (t) =delta (t),the solution of the basic equation of the photoacoustic imaging is approximated by the Green function to obtain a photoacoustic signal: />The process of collecting power amplifier data at sensor locations by simulating photoacoustic signals through k-Wave is represented as a time-varying causal system, and a system matrix a of the time-varying causal system is constructed using a toolbox of k-Wave, the system matrix a comprising each pixel impulse response as a column in an imaging region, the imaging grid being n x n pixels, the imaging grid converting all columns stacked together into n in size 2 X 1 high column vector, the initial pressure rise of the pixels of the imaging grid is x, the system matrix a has m x n 2 To derive a forward model of photoacoustic imaging: ax=b;
the functional of ax=b in S2 is:weighted norm +.>The W is a symmetrical semi-positive definite matrix, the +.>The alpha is a fractional order of alpha > 0; when α=1, matrix w=i, where I is a unitary matrix, and the weighted norm allows selection of a fractional order α, where the fractional order α takes a suitable value in the (0, 1) interval, which is a value that maximizes the SNR or CNR value of the reconstructed image, to reduce the drawback of too smooth of the standard regularization solution;
the method for solving the parameter lambda by adopting the Morozov deviation principle in the S3 comprises the following steps: let e be the observation error and have an upper bound epsilon, satisfy the condition that E is less than epsilon,
define δ=ηε, then solve x λ,α Satisfy b-Ax λ,α ||＝δ，
Let G (λ) = |b-Ax λ,α || 2 -δ 2 ＝0，
When α is given, the G (λ) = |b-Ax λ,α || 2 -δ 2 =0 is a nonlinear implicit equation for λ, which is iteratively solved using the Newton method, i.e
The saidFor the iterative step size, the ∈>The expression of G' (λ) is:
the saidFrom the functional Γ Tikh Minimizing x relative to x Tikh ＝(A T A+λI) -1 A T b deriving lambda, i.e
Residual norms combined with standard Tikhonov algorithm:
the expression of the pushable step length is
Wherein the method comprises the steps of
Constructing a residual margin sequence of least squares solution from the residual norms of the standard Tikhonov algorithm, i.e
Based on min|ρ (i) - δ given δ 2 The corresponding sequence number i when the expression takes the minimum value can be known, so that the iteration initial value lambda 0 ＝σ i ；
The S4 obtains a reconstruction solution x λ,α ＝VF(U T b) And a filter function f=diag (σ) α /(σ α+1 +λ)) is: by the principle of deviation b-Ax λ || 2 The expression =δ can be given as a function of λ, where F is substituted for a as an n×m matrix whose diagonal elements are filter factors in fractional order Tikhonov format, i.e.
Obtaining
Consider k=min (m, n 2 ) Then And then differentiating, i.e. delta (lambda) can be obtained as an inverse function of lambda (delta),then calculate the reconstruction solution x λ,α ＝VF(U T b) And a filter function f=diag (σ) α /(σ α+1 +λ))。
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