KR101531654B1 - Improvement method of convergence speed using gradient total variation in compressed sensing for three dimensional iterative imaging reconstruction - Google Patents

Abstract

The present invention relates to a 3D image reconfiguration method. The purpose of the present invention is to provide a convergence speed improving method by improving a regularization term to promote sparsity in a compressed sensing algorithm of a 3D image reconfiguration. For achieving the purpose, the present invention provides a 3D repetitive image reconfiguration method which includes the steps of: obtaining a target function which includes a fidelity term and the regularization term to promote the sparsity; obtaining a gradient function to differentiate the target function according to a gradient perspective algorithm; and reconfiguring a 3D image by obtaining the final result value which is converged through a repetitive operation for minimizing a convex from the gradient function.

Description

[0001] The present invention relates to a method for improving the convergence speed of a three-dimensional iterative image reconstruction using gradient all-

The present invention relates to a 3D reconstruction method, and more particularly, to a 3D reconstruction method using a compressed sensing (CS) algorithm in an iterative reconstruction (IR) based on computed tomography (CT) To a method for improving convergence speed of iterative reconstruction.

BACKGROUND ART For the past several decades, medical or industrial computed tomography (CT) techniques have been actively studied in which three-dimensional imaging is performed through the body part or object of a patient using X-ray, Is gradually expanding.

As is well known, CT is a technique of capturing a projected image of an object from various angles using rays such as X-rays, reconstructing it into a three-dimensional image through calculation, and obtaining a cross-sectional image (two-dimensional slide image) .

In recent trends of CT technology, low dose is the key keyword for minimizing the dose received by the patient when three-dimensional imaging is performed through the patient.

Methods for minimizing the dose include a method of reducing the number of CT scans (number of acquisitions), a method of lowering the tube current (mA) of the X-ray source, and a total generation time of the X- ms).

In the Feldkamp, Davis, and Kress (FDK) methods, which are widely used in reconstruction of 3D images, aliasing artifacts ) And the like.

The decrease of the tube current (mA) and the generation time (ms) causes a problem of increasing the noise in the reconstructed image.

In particular, since the problems in the FDK are not desirable for diagnosing or diagnosing a lesion of a patient, a method of reconstructing an image is demanded.

Compressed sensing (CS) theory has been widely studied because it is proved that 3D image reconstruction is possible with high image quality even with small number of image acquisition and noisy data.

Particularly, the total variation (TV) method utilizes the minimum variation of the x-ray attenuation that occurs when penetrating the human tissue, and thus is not only beneficial to reconstruction of the CT image, You can provide solutions.

An important problem of compression sensing with full variation is to solve the problem of constrained convex optimization described below.

는 놈(norm) 2이고,

는 최종 구하고자 하는 CT 볼륨 영상의 값, A는 라돈 전달연산자, b는 측정된 투시 데이터, λ는 규칙상수, 그리고

는 규칙항에서 전체변이(TV)를 나타낸다. here

Is the norm 2,

A is the radon transfer operator, b is the measured perspective data, λ is the rule constant, and

Represents the total variation (TV) in the rule term.

(i,j,k)는 (1×N) 차원 행렬로 벡터화되고, 전체변이(TV)는 하기 수학식 2와 같이 나타낼 수 있다.In the above equation (1), a three-dimensional CT volume

(i, j, k) is vectorized into a (1 x N) dimensional matrix, and the total variation (TV) can be expressed by the following equation (2).

The first fidelity term in Equation 1 is a term for enhancing the accuracy from the measured perspective data and the second regular term is the sparsity inherent in the human X- ).

FIG. 1 is a two-dimensional slice image obtained by capturing a head phantom by CT and reconstructing it in three dimensions.

FIG. 2 shows that the entire transition (TV) of Equation (2) is obtained in FIG. 1. In FIG. 2, most of the values are represented by black with 0 and are represented by white having a large value only at a part of the edge of the image Loses.

As a widely used method for solving the problem of Equation (1), a gradient projection algorithm which finds a solution repeatedly in a projected slope direction while a solution is forced to be non-negative is used.

의 경사(gradient) 함수

은 하기 수학식 3과 같이 정의된다.In Equation (1)

The gradient function of

Is defined by the following equation (3).

의 역투시로 설명되는 라돈 전치행렬 A의 전달연산자이고, n은 컨벡스 최소화를 위한 반복횟수를 나타내며,

는 수학식 2의 전체변이를

에 대해서 미분한 것이다(후술하는 수학식 7 참조). In Equation (3), T is physically

Of a transfer operator of radon transposed matrix A will be described as when yeoktu, n represents the number of iterations for minimizing convex,

≪ / RTI >< RTI ID = 0.0 >

(See Equation (7) to be described later).

At this time, the oblique perspective algorithm can be repeatedly used as follows to solve the problem of Equation (1).

은 아래의 수학식 5와 같이 정의될 수 있다.In Equation (4)

Can be defined as Equation (5) below.

은 n 반복에서 조정치(step size)이고, l은 복셀 위치, 그리고

은

에서

의 투시된 경사를 나타낸다. In the above equations (4) and (5)

Is the step size in n iteration, l is the voxel position, and

silver

in

Lt; / RTI >

의 선택에 따라서 수렴속도는 아주 종속적인데, 큰 값을 선택할 경우 수렴은 빠르지만 지역 최소치(local minimum)에 빠지는 문제가 발생할 수 있고, 너무 작은 값을 선택할 경우 수렴속도가 늦기 때문에 많은 계산시간이 요구되는 단점이 있다. Therefore, in equation (4)

The convergence speed is very fast depending on the selection of a large value. However, when the large value is selected, the convergence is fast, but the problem may fall into the local minimum value. If the value is too small, .

Another method for obtaining fast convergence in Equation (4) is to improve the convergence rate by improving the rule term that promotes the scarcity of Equation (1).

SUMMARY OF THE INVENTION Accordingly, the present invention has been made in view of the above-mentioned problems, and it is an object of the present invention to provide a method for improving a convergence rate by improving a rule term for promoting scarcity.

(i,j,k)를 구하기 위하여, 투시 데이터로부터 정확도를 강화시키기 위한 정확도 항(fidelity term)과, 희소성(sparsity)을 촉진시키는 규칙항(regularization term)을 포함하는 목적 함수를 구하는 단계; 경사 투시 알고리즘에 따라 상기 목적 함수를 미분한 경사 함수를 구하는 단계; 및 상기 경사 함수로부터 컨벡스 최소화를 위한 반복적 연산을 통하여 수렴되는 최종 결과값

(i,j,k)을 구하여 3차원 영상을 재구성하는 단계;를 포함하는 3차원 반복적 영상 재구성 과정에서, 상기 목적 함수는 규칙항에 전체변이(total variation)와, 전체변이를

에 대해 미분한 경사 전체변이(gradient total variation)를 함께 적용하여 구해지는 것을 특징으로 하는 3차원 반복적 영상 재구성의 수렴속도 개선방법을 제공한다.
According to an aspect of the present invention, there is provided a method of reconstructing a three-dimensionally reconstructed CT (computed tomography) volume image from a perspective data using a compression sensing technique,

obtaining an objective function including a fidelity term for enhancing the accuracy from the perspective data and a regularization term for promoting sparsity to obtain (i, j, k); Obtaining a slope function that differentiates the objective function according to a slope perspective algorithm; And a final result value converged through an iterative operation for minimizing convex from the slope function

(i, j, k), and reconstructing the three-dimensional image. In the three-dimensional iterative image reconstruction process, the objective function includes a total variation and a total variation

And the gradient total variation is differentiated with respect to the total variation of the three-dimensional repetitive image reconstruction.

Accordingly, in the present invention, by applying GTV to the iterative method for solving the limited Kenbex optimization problem in the compression-sensing (CS) algorithm of the three-dimensional image reconstruction and using the rule constant appropriately according to the situation The convergence speed can be improved, and the total computation time for repeated reconstruction of the three-dimensional image can be reduced.

FIG. 1 is a diagram showing an example in which a head phantom is CT-reconstructed in a three-dimensional manner, again using a two-dimensional slice image.
Fig. 2 is an exemplary diagram showing the total variation (TV) obtained in Fig.
FIG. 3 is an exemplary view showing an inclination total variation (GTV) obtained by differentiating the entire variation of FIG.
FIG. 4 is a flowchart illustrating a process of a 3D image repetitive reconstruction according to the present invention.
FIG. 5 is a diagram of a comparative example, showing an example of a reconstruction result of a FDK method, which is most widely used for 3D medical image reconstruction, as a two-dimensional slice image.
6 is a diagram of a comparative example, in which a result obtained by reconstructing a three-dimensional image using all the displacements (TV) in compression sensing is shown as a two-dimensional slice image.
FIG. 7 is a diagram of a comparative example, in which a result obtained by reconstructing a three-dimensional image using only the gradient total transition (GTV) is shown as a two-dimensional slice image.
FIG. 8 is an example of a 2D slice image obtained by reconstructing a three-dimensional image by further using an inclination total variation (GTV) on the entire side TV.

Hereinafter, exemplary embodiments of the present invention will be described in detail with reference to the accompanying drawings, which will be readily apparent to those skilled in the art to which the present invention pertains.

The present invention provides a method for improving the convergence rate by improving the rule for promoting scarcity. The present invention provides a method for increasing the convergence speed by adding a gradient total variation (GTV) sparsity.

In that approach, to solve the constrained convex optimization problem of Equation (1), we add the slope total variation (GTV) for the purpose of promoting sparsity in the second term as follows.

는 압축센싱 기법에서 목적 함수를,

은 목적 함수

를

에 대하여 미분한 함수, 즉 경사 투시 알고리즘의 미분에 의한 경사 함수를 나타낸다.Hereinafter,

Is an objective function in the compression sensing technique,

Is an objective function

To

Represents a function of differentiation with respect to the gradient function, that is, a slope function by the derivative of the slope perspective algorithm.

를 최소화하는 식은 하기 수학식 6과 같다.first,

Is expressed by Equation (6). &Quot; (6) "

는 놈(norm) 2이고,

는 최종 구하고자 하는 CT 볼륨 영상의 값, A는 라돈 전달연산자, b는 측정된 투시 데이터를 나타내며, λ는 규칙상수, 그리고

는 규칙항에서 전체변이(TV),

는 경사 전체변이(GTV)를 나타낸다. here

Is the norm 2,

Where A is the radon transfer operator, b is the measured perspective data, λ is the rule constant, and

Is the total variation (TV) in the rule term,

(GTV).

는 수학식 2의 전체변이를

에 대해서 하기와 같이 미분한 것이다.In the above Equation 6,

≪ / RTI >< RTI ID = 0.0 >

Is differentiated as follows.

의 규칙항과

의 규칙항의 합으로 나타내어진다.In the present invention, in the compression sensing for the three-dimensional image repetitive reconstruction, as shown in Equation (6), the inclination total variation is additionally calculated to reflect the rule term, and therefore, the rule term for promoting the scarcity

And

Is the sum of the rule terms of

)에 적용되는 규칙상수와 경사 전체변이(GTV:

)에 적용되는 규칙상수는 동일 값(λ)이거나 다른 값(λ₁,λ₂)이 될 수 있다.At this time, in the rule for solving the convex optimization problem,

(GTV: constant)

) Can be the same value (?) Or different values (? ₁ ,? ₂ ).

이 되지만, 다른 값일 경우 규칙항은

이 된다(여기서 λ₁≠λ₂ 임).If it is the same value, the rule term is expressed by Equation 6

, But for other values, the rule term is

(Where l ₁ ? ₂ ).

FIG. 3 is a graph showing a value obtained by calculating an inclination total variation (GTV) by applying Equation (7) in Equation (6). Most of the values have a value close to 0, which is black, and only a part thereof is shown in white.

The approach for minimizing Equation (6) uses a gradient projection algorithm that repeatedly finds the solution in the direction of the oblique slope as shown in the flow chart of FIG.

FIG. 4 is a flowchart illustrating a process of a 3D image repetitive reconstruction according to an embodiment of the present invention. Referring to FIG. 4, the process of the 3D image repetitive reconstruction includes the steps of calling the captured perspective data (CT parameters) Setting the setting values used for reconstruction, initializing the reconstruction by the back propagation method, and obtaining the radon transfer operator A and its transpose matrix A ^T , and additionally, the entire mutation and the inclination total mutation are compressed And applying it to sensing to repeatedly search for solutions.

를 구하기 위하여, 투시 데이터로부터 정확도를 강화시키기 위한 정확도 항(fidelity term)과, 희소성(sparsity)을 촉진시키는 규칙항(regularization term)을 포함하는 함수

의 규칙항에 전체변이(TV)와 경사 전체변이(GTV)를 함께 적용한 후, 경사 투시 알고리즘의 미분에 의한 경사 함수

으로부터 컨벡스 최소화를 위한 반복적 연산을 통하여 수렴되는 결과값

를 구함으로써 3차원 영상을 재구성하게 된다. In other words, in the present invention, a value of an image reconstructed three-dimensionally using a compression sensing technique

A function that includes a fidelity term for enhancing accuracy from the perspective data and a regularization term that promotes sparsity,

(TV) and GTV (GTV) are applied to the rule terms of the oblique perspective algorithm,

Which is converged through iterative operations for minimizing the convexes,

The 3D image is reconstructed.

를 최소값으로 만들어주기 위해 정확도 항에 더해지는 항이 된다.Here, the accuracy term is a term indicating an error as in the conventional case,

Is added to the term of accuracy to make it the minimum value.

의 경사(gradient) 함수

은 하기 수학식 8과 같이 나타내어진다. In the oblique perspective algorithm, Equation (6)

The gradient function of

Is expressed by the following equation (8).

의 역투시로 설명되는 라돈 전치행렬 A의 전달연산자이고,

는 상기 수학식 7을

에 대해서 미분한 결과이다. In Equation (8), T is physically

Is the forward operator of the radon transpose matrix A ,

(7)

As shown in Fig.

을 미분한 식

은 경사 전체변이

과 2차 경사 전체변이

에 각각 규칙상수(λ)를 적용한 항들의 합을 포함하는 식이 된다.Referring to Equation 8,

Lt; / RTI >

All slopes

And secondary slope total mutation

And the sum of the terms applying the rule constant ([lambda]).

이 되지만, 만약 다른 값일 경우

이 된다.If the rule constant is the same value,

, But if it is another value

.

이 최소화될 수 있는

은

을 0으로 하여 구해지는 수렴 값으로서 아래와 같이 구해질 수 있다. Also, the minimization problem of Equation (6) can be solved by repeatedly using the oblique perspective algorithm as shown in Equation (9)

This can be minimized

silver

Can be obtained as follows.

은 아래의 수학식 10과 같이 정의될 수 있다.In Equation (9)

Can be defined as Equation (10) below.

은 n 반복에서 조정치(step size)이고, l은 복셀 위치, 그리고

은

에서 함수

의 투시된 경사를 나타낸다.In the equations (9) and (10)

Is the step size in n iteration, l is the voxel position, and

silver

Functions in

Lt; / RTI >

The addition of the slope total variation (GTV) to the second rule term in Equation (6) is intended to improve the convergence rate by improving the rules for promoting scarcity. The known total variation (TV) promotes sparsity, Repetitive reconstruction tends to reach the optimal solution because it does not acquire perfect perfection at one time.

Therefore, by introducing the gradient total transition (GTV), the effect of further promoting the scarcity is obtained.

In order to confirm the effect of the present invention, the present inventors have applied the above-described methods of the present invention and reconstructed a 3D medical image using a chest phantom (3D reconstruction), and then conducted an image quality comparison experiment .

In the comparative experiment, when the FDK method most widely used in 3D medical images is applied, when only the total variation (TV) is applied as in Equation (1), only GTV of the second term in Equation (6) (TV + GTV) in which the total variation (TV) and the inclined total variable (GTV) are applied together in the second term in Equation (6) as shown in the present invention.

FIG. 5 shows an image reconstructed by applying the FDK method, FIG. 6 shows an image reconstructed by applying only the entire variation, FIG. 7 shows an image reconstructed by applying only the slope whole mutation, and FIG. We show reconstructed images by applying the inclination whole mutation together.

In the case of TV, GTV, and TV + GTV in the iterative reconstruction method, the number of repetition times is 10, 16, 22, 27, Performance was compared.

The experimental result of the same number of projection images shows that the experimental result of the analytical method FDK is as shown in FIG. 6, FIG. 7, and FIG. 8 when the iterative method of FIG. 5 is applied. And the FDK method using 361 of the perspective images is similar to the result of using 181 of the number of perspective images in the other iterative methods in terms of image quality, have.

After reconstructing the 3D medical image using chest phantom, we performed the image quality comparison experiment.

In the comparative experiment, when the FDK method most widely used in 3D medical images is applied, when only the total variation (TV) is applied as in Equation (1), only GTV of the second term in Equation (6) (TV + GTV) in which the total variation (TV) and the inclined total variable (GTV) are applied together in the second term in Equation (6) as shown in the present invention.

FIG. 5 shows an image reconstructed by applying the FDK method, FIG. 6 shows an image reconstructed by applying only the entire variation, FIG. 7 shows an image reconstructed by applying only the slope whole mutation, and FIG. We show reconstructed images by applying the inclination whole mutation together.

In the case of TV, GTV, and TV + GTV in the iterative reconstruction method, the number of repetition times is 10, 16, 22, 27, Performance was compared.

The experimental result of the same number of projection images shows that the experimental result of the analytical method FDK is as shown in FIG. 6, FIG. 7, and FIG. 8 when the iterative method of FIG. 5 is applied. And the FDK method using 361 of the perspective images is similar to the result of using 181 of the number of perspective images in the other iterative methods in terms of image quality, have.

Also, in the case of the iterative method, the overall variation method of FIG. 6, which is an existing method, showed better results than the inclination total variation method of FIG.

However, the TV + GTV method shown in FIG. 8 is superior to the TV method shown in FIG. 6 and the GTV method shown in FIG. 7. In particular, the TV + GTV method has more than 10 repetitions The results showed good results in terms of convergence speed.

While the present invention has been particularly shown and described with reference to exemplary embodiments thereof, it is to be understood that the scope of the present invention is not limited thereto. Various modifications and improvements And are included in the scope of the invention.

Claims (6)

The value of the CT (computed tomography) volume image reconstructed three-dimensionally from the perspective data using the compression sensing technique

obtaining an objective function including a fidelity term for enhancing the accuracy from the perspective data and a regularization term for promoting sparsity to obtain (i, j, k);
According to the oblique perspective algorithm,

Obtaining a differentiated slope function with respect to the slope function; And
A final result value converged through an iterative operation for minimizing convex from the slope function

(i, j, k), and reconstructing the three-dimensional image. In the three-dimensional iterative image reconstruction process, the objective function includes a total variation and a total variation

Wherein the gradient total variation is obtained by applying a derivative gradient total variation to the three-dimensional iterative image reconstruction.
The method according to claim 1,
Wherein the objective function is obtained by the following equation (E1).
E1:

here

Is an objective function,

Is the norm 2, A is the radon transfer operator, b is the measured perspective data, λ is the rule constant,

The total variation,

Is the total variation of slope.
The method according to claim 1,
Wherein the objective function is obtained by the following equation (E2).
E2:

here

Is an objective function,

Is the norm 2, A is the radon transfer operator, b is the measured perspective data, λ ₁ and λ ₂ (λ ₁ ≠ λ ₂ ) are the rule constants,

The total variation,

Is the total variation of slope.
The method of claim 2,
Wherein the gradient function is obtained by the following equation (E3).
E3:

here

Is the slope function, T is physically

The number of times of the transmission operator of radon transposed matrix A will be described as when yeoktu, n is repeated for the convex minimized,

All slope variants

To

Lt; / RTI >
The method of claim 3,
Wherein the gradient function is obtained by the following equation (E4).
E4:

here

Is the slope function, T is physically

The number of times of the transmission operator of radon transposed matrix A will be described as when yeoktu, n is repeated for the convex minimized,

All slope variants

To

Lt; / RTI >
The method according to any one of claims 2 to 5,
The entire mutation

And inclined whole mutation

Are defined by the following equations (E5) and (E6), respectively.
E5:

E6:

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