基于小波变换和二代曲波变换的乳腺钼靶X片图像增强
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摘要
近年来,国内因乳腺癌而死亡的人数逐年增多,乳腺癌的死亡率已占癌症死亡率的第四位,因此早期发现,早期诊断,早期治疗是降低乳腺癌死亡率的关键。乳腺钼靶X线摄影是诊断乳腺疾病的首选及有效方式。但是对于0期、1期乳腺癌其钼靶X线片中癌变特征表现不清晰,很容易造成医生的漏诊,因此如何对钼靶X线图像进行增强并有效控制噪声,是检测早期乳腺癌的关键所在。
本文主要针对乳腺癌病变区域的不同特征采用了不同的数学分析工具加以刻画。首次提出了将小波变换和二代曲波变换应用于钼靶X线图像的增强处理上;使得乳腺癌变异常的主要特征——钙化征和肿块征凸显。其中针对细微、不均匀分散或成簇的钙化图像,由于其图像特征都是各向同性的故采用正交小波变换,选用滤波函数,在抑制噪声的同时对图像进行有效增强,提高了图像的对比度。而对于乳腺钼靶X线图像中的增生、结节、肿瘤、乳腺局部结构等采用二代曲波变换,因为其轮廓呈曲线特征,曲线边缘是各向异性的,根据二代曲波变换满足“最优”图像表示方法的多分辩,带通,方向性等特点,使曲线波沿着曲线自适应“跟踪”奇异曲线,使得不同表现形式的特征病变部位特征更清晰。由于东西方人在乳腺组成上的差异,致使在其基础上发展的方法,并不适合东方人,为此,创建了适用于亚洲人的乳腺数字数据库(ADDSM,Asian Digital DatabaseFor Screening Mammography);通过ROC曲线,反映了该算法的灵敏度与特异度的差异,利用“ROC曲线下面积”的Az值反应了该算法的系统的性能。
Recently, breast cancer has become a serious threat to women's health. The diagnosis and prevention of the breast cancer receives much attention in the medical area. Early detection, early diagnosis and early treatment are the important elements to cure the breast cancer. Regular medical exam and breast X-ray photograph are the important methods to detect breast cancer at the early stage. It is an important topic to study the breast X-ray image, which is the key to detect and decrease the death rate of breast cancer. According to the universally acknowledged TNM staging method, the symptoms of breast cancer at stage 0 and stage 1 are not obvious.Because of the reduced clarity in the breast X-ray projection image, it is possible to cause the missed diagnosis of the early breast cancer. We need to enhance the X-ray image and intensify the unclear image features. It is a new topic to computer medical image processing.
In this paper, we focus on calcification and masses, the main features of abnormal carcinogenesis in molybdenum target X-ray image. Then enhance the edges of calcification points and masses in the breast image by wavelet transform and the second generation fast curvelet transform. We realize the second curvelet transform based on USFFT on the computer. Finally we analyze the difference of sensibility and specificity curve by using ROC curves. Also we use the Az value of the area under ROC curve to reflect the accuracy of the system. Besides that we create Asian breast digital database ADDSM. The system shows that , The experimental result show the presented method works well to enhance molybdenum target X-ray image and make the edges of calcification points and masses very clear. It can reduce the waste of medical resource, provide the better basis for diagnosis and detect the early symptoms. It will be helpful to reach the final goal to detect early and treat early.
1. Wavelet transform for calcification
From the point of view of image processing, breast image is made up of high frequency calcification points, low frequency background and extremely high frequency noise. They are all featured as isotropy. So we can apply wavelet transform to deal with the transform function at different resolutions and decompose the original signal to be sub-band signals with different resolutions, different frequency features and direction features. This paper uses biorthogonal wavelet to decompose and enhance the image and then uses wavelet to reconstruct the target image. This method is suitable to molybdenum target image with calcification points. It can enhance the image while restraining the noise and improve the contract to show calcification points clearer.
For an image, there is a gray value f (x, y) corresponding to any point (x,y). In atwo-dimension signal, we segment it at the given scale j, that is to segment evenly by(?) density in x and y direction. The integral knots are numbered (m,n) and the basisfunctions corresponding to the integral knots are taken simply as the tensor bases of one-dimension basis functions:
The combination coefficients of integral knots are taken as c_(m,n)~j. Thus we can construct f~j (x, y) to approximate the image f(x, y),
In two-dimension MRA, the following decomposition relation is given as follows:
where W_j (x, y) is the complementary space of V_j (x, y).
The scale functionφ(t) is translation orthogonal, thenφ_(j,m) (x),φ_(j,n) (y),ψ_(j,m)(x),ψ_(j,n)(y) are both translation orthogonal. Using orthogonality, we have the expansion coefficients as follows:
The corresponding restoration algorithm is:
The relative positions between calcification points and background noise articles are complicated in molybdenum target X-ray images. The traditional FFT algorithm enhances the images as well as the noise article, which makes it difficult to distinguish calcification points from noise. The adaptive time frequency window in wavelet analysis can add filter functions at the output terminal and set different thresholds for calcification points and background noise. It can remove the noise effectively as well as enhance the calcification points. Even the minimal ones become very clear, which will ensure no missed diagnosis happens.
2. Ridgelet transform for linear features
Wavelet can reflect point singularity and that is why it can show calcification points very well. But it can't represent higher dimension features. However the edges of masses, fibrous tissues and blood vessels are in form of line or curve. The wavelet can't determine the edges of the masses. It is needed to find basic functions to express more powerfully than wavelet. This paper uses ridgelet transform to enhance the linear edges in the breast image.
For a given integrable function f(x) with two variables, if functionΨ:R→R satisfies the admissible condition:
Ψ(x)is the admissible nerve activation function, ridgelet functionψ_(a,b,θ) (x) is defined as:
Here a is the scale factor, b is the translation factor,θis direction angle, x = (x_1,x_2).
Its continuous ridgelet transform is given as follows:
The corresponding inverse transform function is:
It is easy to see, ridgelet function is a constant on the line x_1 cosθ+ x_2 sinθ= C and a wavelet function in the direction vertical to the line. The ridgelet features are very obvious. After transformation, the scale, space and direction domain are more accurate. Obviously the ridgelet transform is a linear operation as wavelet transform.
In the two-dimension space, points and lines can be related by Radon transform. Hence ridgelet transform can be related to wavelet transform by Radon transform.
The Radon transform of f(x) is:
Thus ridgelet transform is converted to be one-dimension wavelet transform of Radon transform section.
The corresponding wavelet transform is:
Let the ridgelet transform of f(x) is R(f)(r) =< f,Ψ_r >, the ridgelet coefficients of the target function f can be found by the following equation:
So ridgelet analysis is equivalent to wavelet analysis of the target function in Radon transform domain. Here t is the variable,θis the constant.
3. The second generation curvelet tanform for masses
Although ridgelet transform works greatly to detect image edgess with linear singularity, the natural images are not necessarily to have linear edgess. They are curve singularity. Ridgelet transform can't deal with curve singularity and thus can't meet the needs.
Many articles are using the first generation curvelet to deal with curve singularity, but it is hard to realize on the computer because of the complexity and tediousness of computation, In this paper, we provide a new curve approximation method, the second curvelet transform, based on the first generation curvelet. It keeps the superiority in approximation and has simple algorithm and theory to apply and understand.
For simplicity, let x be space location variable,ωbe frequency variable, r,θis the polar coordinate in frequency domain.
Definition 1: In advance, we define the radius window W(r) of the smooth, nonnegative, real function (the support interval is r∈(1/2,2) ) and angle window V(t) (the support interval is t∈[-1,1]) ). W(r) , V(t) both satisfy the admissible conditions:
The expression shows the energy sum is 1 for radius windows and angle windows for all specified parameter in support intervals.
Then for every scale j≥j_0, we introduce the frequency window U_j, whoseFourier frequency domain definition is:
Here [j/2] is the integralization of j/2 . According to the definition, the support region of U_j is a kind of "cuniform" window in polar coordinate system infrequency domain. Figure 1 shows the block figures of curvelet in frequency domain and spatial domain. The left figure represents frequency domain, in which curvelets is segmented to be "cuniforms" in the shaded parts at each scale. The right figure gives each scale and direction in Descartes space. Therefore curvelet transform meets the scale feature of anisotropy. To get the real value of the curvelet transform, we take the sum exrpession form U_j (r,θ) + U_j (r,θ+π).
Fig.1 The block diagram of Curvelet in the frequency domain and corresponding scale in time domain.
Definition 2 If the Fourier transform ofφ_j(x) satisfies (?) = U_j(ω) , defineφ_j(x) as Mother Curvelet (Here we use U_j(ω_1,ω_2) to represent the frequencydomain window ). Then all the curvelet in scale 2~(-j) can be obtained by moving and rotation.
We can define the curvelet with scale 2~(-j) , direction angleθ_l, position(?) as: Here R_θrepresents the rotation matrix byθradian, R_θ~(-1) is its inverse matrix.
Definition 3: according to definition 1, 2, the coefficients of curvelet transform is the inner product of f∈L~2(R~2) andφ_(j,l,k) (x):
Because digital curvelet transform is performed in frequency domain, thus it is useful to present the above expression as the coefficients of curvelet in frequency domain.
Like wavelet theory, curvelet transform includes coarse scale terms. Introduce lowpass window W_0 satisfying:
Then for k = (k_1,k_2)∈Z~2, we can define the curvelets in coarse scale as:
The above expressions show that curvelets in coarse scale are anisotropic. "Complete" curvelet transform is made up of direction element (φ_(j,l,k)_(j≥j_0,k,l) in fine scale and isotropic father wavelet (Φ_(j_0,k))_k in coarse scale.
we have the output discrete coefficients c~D (j, l, k):
By the definition of curvelet transform in continuous domain, the frequency window U_j is obtained by segmenting smoothly the frequency domain according tothe angles and radial rings. But such segmentation is not suitable to two-dimension Descartes coordinate system. Thus we use square regions with the same center to segment the frequency domain (as Fig.2 shows the comparison of the segmentation of continuous curvelet transform and discrete curvelet transform in frequency domain.)
Fig.2 The frequency segmentation diagram of continuous and discrete curvelet
The sheared block S_(θ_1)~(-T) (k_1×2~(-j) ,k_2×2~(-j/2)) is not standard rectangle. To use theclassical FFT algorithm, after reconstructing rectangle grid, we can rewrite as:
Assume we give the array f[t_1, t_2],0≤t_1, t_2 In the following, we denote the sampler as: By interpolation trigonometric polynomial, we can also define (?) as:
Assume P_j = {(n_1, n_2): n_(1,0)≤n_1< n_(1,0) + L_(1,j) , n_(2,0)≤n_2 Based on the above introduction, we now apply USFFT to perform the fast discrete curvelet transform on the image f(t_1,t_2).
Ⅰ. Apply two-dimension FFT on the image f(t_1,t_2) and obtain theseries (?)(n_1, n_2), -n/2≤n_1,n_2 < n/2 .
Ⅱ. For different scale j and direction l, we resample or interpolate (?)(n_1, n_2) to get (?)[n_1, n_2- n_1 tanθ_l], where (n_1, n_2)∈P_j.
Ⅲ. Multiply window function (?)[n_1, n_2] by (?)[n_1, n_2- n_1 tanθ_l] to get new series (?)[n_1, n_2]:
Ⅳ. Apply two-dimension inverse FFT to (?)[n_1, n_2] and get the discrete curvelet transform coefficient c~D(j,l,k) for scale j , direction l and position k = {k_1,k_2).
4. The realization of the early aided diagnosis system
Preprocess the molybdenum target X-ray image by filter function, perform wavelet transform and the second generation curvelet transform on the mean and standard deviation after normalization. We apply different wavelet filter and threshold filter to different scale coefficients to locate the lesion regions. We also create Asian breast digital database ADDSM. The ROC curve shows the effectiveness of the system. The experimental result shows the early aided diagnosis system of breast cancer can give Az value 0.9161. It has high clinical value and can improve the accuracy of the diagnosis.
本文主要针对乳腺癌病变区域的不同特征采用了不同的数学分析工具加以刻画。首次提出了将小波变换和二代曲波变换应用于钼靶X线图像的增强处理上;使得乳腺癌变异常的主要特征——钙化征和肿块征凸显。其中针对细微、不均匀分散或成簇的钙化图像,由于其图像特征都是各向同性的故采用正交小波变换,选用滤波函数,在抑制噪声的同时对图像进行有效增强,提高了图像的对比度。而对于乳腺钼靶X线图像中的增生、结节、肿瘤、乳腺局部结构等采用二代曲波变换,因为其轮廓呈曲线特征,曲线边缘是各向异性的,根据二代曲波变换满足“最优”图像表示方法的多分辩,带通,方向性等特点,使曲线波沿着曲线自适应“跟踪”奇异曲线,使得不同表现形式的特征病变部位特征更清晰。由于东西方人在乳腺组成上的差异,致使在其基础上发展的方法,并不适合东方人,为此,创建了适用于亚洲人的乳腺数字数据库(ADDSM,Asian Digital DatabaseFor Screening Mammography);通过ROC曲线,反映了该算法的灵敏度与特异度的差异,利用“ROC曲线下面积”的Az值反应了该算法的系统的性能。
Recently, breast cancer has become a serious threat to women's health. The diagnosis and prevention of the breast cancer receives much attention in the medical area. Early detection, early diagnosis and early treatment are the important elements to cure the breast cancer. Regular medical exam and breast X-ray photograph are the important methods to detect breast cancer at the early stage. It is an important topic to study the breast X-ray image, which is the key to detect and decrease the death rate of breast cancer. According to the universally acknowledged TNM staging method, the symptoms of breast cancer at stage 0 and stage 1 are not obvious.Because of the reduced clarity in the breast X-ray projection image, it is possible to cause the missed diagnosis of the early breast cancer. We need to enhance the X-ray image and intensify the unclear image features. It is a new topic to computer medical image processing.
In this paper, we focus on calcification and masses, the main features of abnormal carcinogenesis in molybdenum target X-ray image. Then enhance the edges of calcification points and masses in the breast image by wavelet transform and the second generation fast curvelet transform. We realize the second curvelet transform based on USFFT on the computer. Finally we analyze the difference of sensibility and specificity curve by using ROC curves. Also we use the Az value of the area under ROC curve to reflect the accuracy of the system. Besides that we create Asian breast digital database ADDSM. The system shows that , The experimental result show the presented method works well to enhance molybdenum target X-ray image and make the edges of calcification points and masses very clear. It can reduce the waste of medical resource, provide the better basis for diagnosis and detect the early symptoms. It will be helpful to reach the final goal to detect early and treat early.
1. Wavelet transform for calcification
From the point of view of image processing, breast image is made up of high frequency calcification points, low frequency background and extremely high frequency noise. They are all featured as isotropy. So we can apply wavelet transform to deal with the transform function at different resolutions and decompose the original signal to be sub-band signals with different resolutions, different frequency features and direction features. This paper uses biorthogonal wavelet to decompose and enhance the image and then uses wavelet to reconstruct the target image. This method is suitable to molybdenum target image with calcification points. It can enhance the image while restraining the noise and improve the contract to show calcification points clearer.
For an image, there is a gray value f (x, y) corresponding to any point (x,y). In atwo-dimension signal, we segment it at the given scale j, that is to segment evenly by(?) density in x and y direction. The integral knots are numbered (m,n) and the basisfunctions corresponding to the integral knots are taken simply as the tensor bases of one-dimension basis functions:
The combination coefficients of integral knots are taken as c_(m,n)~j. Thus we can construct f~j (x, y) to approximate the image f(x, y),
In two-dimension MRA, the following decomposition relation is given as follows:
where W_j (x, y) is the complementary space of V_j (x, y).
The scale functionφ(t) is translation orthogonal, thenφ_(j,m) (x),φ_(j,n) (y),ψ_(j,m)(x),ψ_(j,n)(y) are both translation orthogonal. Using orthogonality, we have the expansion coefficients as follows:
The corresponding restoration algorithm is:
The relative positions between calcification points and background noise articles are complicated in molybdenum target X-ray images. The traditional FFT algorithm enhances the images as well as the noise article, which makes it difficult to distinguish calcification points from noise. The adaptive time frequency window in wavelet analysis can add filter functions at the output terminal and set different thresholds for calcification points and background noise. It can remove the noise effectively as well as enhance the calcification points. Even the minimal ones become very clear, which will ensure no missed diagnosis happens.
2. Ridgelet transform for linear features
Wavelet can reflect point singularity and that is why it can show calcification points very well. But it can't represent higher dimension features. However the edges of masses, fibrous tissues and blood vessels are in form of line or curve. The wavelet can't determine the edges of the masses. It is needed to find basic functions to express more powerfully than wavelet. This paper uses ridgelet transform to enhance the linear edges in the breast image.
For a given integrable function f(x) with two variables, if functionΨ:R→R satisfies the admissible condition:
Ψ(x)is the admissible nerve activation function, ridgelet functionψ_(a,b,θ) (x) is defined as:
Here a is the scale factor, b is the translation factor,θis direction angle, x = (x_1,x_2).
Its continuous ridgelet transform is given as follows:
The corresponding inverse transform function is:
It is easy to see, ridgelet function is a constant on the line x_1 cosθ+ x_2 sinθ= C and a wavelet function in the direction vertical to the line. The ridgelet features are very obvious. After transformation, the scale, space and direction domain are more accurate. Obviously the ridgelet transform is a linear operation as wavelet transform.
In the two-dimension space, points and lines can be related by Radon transform. Hence ridgelet transform can be related to wavelet transform by Radon transform.
The Radon transform of f(x) is:
Thus ridgelet transform is converted to be one-dimension wavelet transform of Radon transform section.
The corresponding wavelet transform is:
Let the ridgelet transform of f(x) is R(f)(r) =< f,Ψ_r >, the ridgelet coefficients of the target function f can be found by the following equation:
So ridgelet analysis is equivalent to wavelet analysis of the target function in Radon transform domain. Here t is the variable,θis the constant.
3. The second generation curvelet tanform for masses
Although ridgelet transform works greatly to detect image edgess with linear singularity, the natural images are not necessarily to have linear edgess. They are curve singularity. Ridgelet transform can't deal with curve singularity and thus can't meet the needs.
Many articles are using the first generation curvelet to deal with curve singularity, but it is hard to realize on the computer because of the complexity and tediousness of computation, In this paper, we provide a new curve approximation method, the second curvelet transform, based on the first generation curvelet. It keeps the superiority in approximation and has simple algorithm and theory to apply and understand.
For simplicity, let x be space location variable,ωbe frequency variable, r,θis the polar coordinate in frequency domain.
Definition 1: In advance, we define the radius window W(r) of the smooth, nonnegative, real function (the support interval is r∈(1/2,2) ) and angle window V(t) (the support interval is t∈[-1,1]) ). W(r) , V(t) both satisfy the admissible conditions:
The expression shows the energy sum is 1 for radius windows and angle windows for all specified parameter in support intervals.
Then for every scale j≥j_0, we introduce the frequency window U_j, whoseFourier frequency domain definition is:
Here [j/2] is the integralization of j/2 . According to the definition, the support region of U_j is a kind of "cuniform" window in polar coordinate system infrequency domain. Figure 1 shows the block figures of curvelet in frequency domain and spatial domain. The left figure represents frequency domain, in which curvelets is segmented to be "cuniforms" in the shaded parts at each scale. The right figure gives each scale and direction in Descartes space. Therefore curvelet transform meets the scale feature of anisotropy. To get the real value of the curvelet transform, we take the sum exrpession form U_j (r,θ) + U_j (r,θ+π).
Fig.1 The block diagram of Curvelet in the frequency domain and corresponding scale in time domain.
Definition 2 If the Fourier transform ofφ_j(x) satisfies (?) = U_j(ω) , defineφ_j(x) as Mother Curvelet (Here we use U_j(ω_1,ω_2) to represent the frequencydomain window ). Then all the curvelet in scale 2~(-j) can be obtained by moving and rotation.
We can define the curvelet with scale 2~(-j) , direction angleθ_l, position(?) as: Here R_θrepresents the rotation matrix byθradian, R_θ~(-1) is its inverse matrix.
Definition 3: according to definition 1, 2, the coefficients of curvelet transform is the inner product of f∈L~2(R~2) andφ_(j,l,k) (x):
Because digital curvelet transform is performed in frequency domain, thus it is useful to present the above expression as the coefficients of curvelet in frequency domain.
Like wavelet theory, curvelet transform includes coarse scale terms. Introduce lowpass window W_0 satisfying:
Then for k = (k_1,k_2)∈Z~2, we can define the curvelets in coarse scale as:
The above expressions show that curvelets in coarse scale are anisotropic. "Complete" curvelet transform is made up of direction element (φ_(j,l,k)_(j≥j_0,k,l) in fine scale and isotropic father wavelet (Φ_(j_0,k))_k in coarse scale.
we have the output discrete coefficients c~D (j, l, k):
By the definition of curvelet transform in continuous domain, the frequency window U_j is obtained by segmenting smoothly the frequency domain according tothe angles and radial rings. But such segmentation is not suitable to two-dimension Descartes coordinate system. Thus we use square regions with the same center to segment the frequency domain (as Fig.2 shows the comparison of the segmentation of continuous curvelet transform and discrete curvelet transform in frequency domain.)
Fig.2 The frequency segmentation diagram of continuous and discrete curvelet
The sheared block S_(θ_1)~(-T) (k_1×2~(-j) ,k_2×2~(-j/2)) is not standard rectangle. To use theclassical FFT algorithm, after reconstructing rectangle grid, we can rewrite as:
Assume we give the array f[t_1, t_2],0≤t_1, t_2
Assume P_j = {(n_1, n_2): n_(1,0)≤n_1< n_(1,0) + L_(1,j) , n_(2,0)≤n_2
Ⅰ. Apply two-dimension FFT on the image f(t_1,t_2) and obtain theseries (?)(n_1, n_2), -n/2≤n_1,n_2 < n/2 .
Ⅱ. For different scale j and direction l, we resample or interpolate (?)(n_1, n_2) to get (?)[n_1, n_2- n_1 tanθ_l], where (n_1, n_2)∈P_j.
Ⅲ. Multiply window function (?)[n_1, n_2] by (?)[n_1, n_2- n_1 tanθ_l] to get new series (?)[n_1, n_2]:
Ⅳ. Apply two-dimension inverse FFT to (?)[n_1, n_2] and get the discrete curvelet transform coefficient c~D(j,l,k) for scale j , direction l and position k = {k_1,k_2).
4. The realization of the early aided diagnosis system
Preprocess the molybdenum target X-ray image by filter function, perform wavelet transform and the second generation curvelet transform on the mean and standard deviation after normalization. We apply different wavelet filter and threshold filter to different scale coefficients to locate the lesion regions. We also create Asian breast digital database ADDSM. The ROC curve shows the effectiveness of the system. The experimental result shows the early aided diagnosis system of breast cancer can give Az value 0.9161. It has high clinical value and can improve the accuracy of the diagnosis.
引文
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