Principles of Doppler Tomography
Principles of Doppler Tomography
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多普勒断层扫描原理
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发表时间：
2005
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通讯作者：
P. Juhlin
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作者：
Tekniska Högskolan;I. Lund;Matematiska Institutionen;P. Juhlin
本文展示了如何使用Radon变换来确定矢量场。提出了一种利用连续多普勒信号测量来确定运动流体速度场的方案。当流动被限制在有界区域时，就像大多数应用中的情况一样，它可以唯一地分解成一个梯度部分和一个旋转部分。如果流体是不可压缩的和无源的，则前者为零，而如果区域是简单连通的，则后者可以用本文提出的方法完全重建。对圆形截面长圆柱形容器内的层流流动进行了研究。在这种情况下，流型变为抛物线，这使得在描述速度场旋转的图像中，血管可以识别为典型的“N”形图案。无论如何切面，血管都会产生相同的多普勒层析图像。在研究血管中的流动时，所提出的想法也应该适用，即使其中的流动轮廓不是完全抛物线的。这些差异只会使“N”字形有些扭曲。标量Radon变换、Radon变换及相关变换目前在应用数学领域有着广泛的应用。特别是，在过去的十年里，医学成像技术越来越多地转向断层成像。其中占主导地位的是X射线CT、发射CT、超声CT和核磁共振成像。以相对较低的成本获得的计算机能力的巨大增长可能是这一发展背后的唯一最重要的因素。电子学的进步和改进的计算机算法也起到了作用。然而，基本的数学原理与以前大体相同。通常情况下，问题是测量组织如何在施加的测试光束中散射或吸收能量。控制这一过程的方程通常可以写成=/oexp(-jf/I(R)≈Tf)，其中/是测试光束的强度，而L是光束所走的路径。目的是确定线性衰减系数(*(X))如何在空间中变化。考虑有限范围的三维体的横截面。引入一个正交坐标系，使截面位于XY平面内。这个平面上的每一条直线都是由这对直线(S，9岁)唯一确定的，如下图1所示。注意，-oo表示单位向量u>；=(cos 9，sin 0，0)。则u>；和9是可互换的，并且行L可以以L(S，9)={r：r e J=0r-u；=S}的形式hr*/Ritten。图1：Radon变换的参数定义。现在让我们将自己限制在平面2=0，并引入函数(i为fi(Sj)=log(y-)=//i(R)rf/。Vin/Jr-w=i，则fi是/x的二维Radon变换(或X射线变换)，我们也记为fi(A.O)=TZfi。通过测量测试光束沿直线衰减的情况，L(S，6岁)可以计算出FI(S.O)。当A*和9变化以生成XY平面中的任何直线时，可以获得整个SO平面的函数/i(s，6)。借助于Radon逆变换(i=Tz~ft)，就可以计算XY平面上任意点的线性衰减系数/i(x，y)。有关标量Radon变换的更多背景，请参阅[3]和[4]。运动的流体和多普勒频移如果考虑的区域包含运动的流体，则多普勒频移为Car。用来测量这一运动的速度。在不考虑相对论效应的情况下，角频率为w的发射信号中的多普勒频移Aw可用以下方程描述
This paper shows how the Radon Transform can be used to determine vector fields. A scheme to determine the velocity field of a moving fluid by measurements with a continuous Doppler signal is suggested. When the flow is confined to a bounded domain, as is the case in most applications, it can be uniquely decomposed into one gradiental and one rotational part. The former vanishes if the fluid is incompressible and source-free, and the latter can be completely reconstructed by the methods proposed in this paper if the domain is simply connected. Special attention is paid to laminar flow in a long cylindrical vessel with circular cross-section. Under such conditions the flow profile becomes parabolic, which makes the vessel recognizable as a typical "N-shaped" pattern in an image describing the rotation of the velocity field. The vessel yields the same Doppler tomographic pattern, no matter how it is sectioned. The ideas presented should be applicable also when studying the flow in blood vessels, even if the flow profile in these is not quite parabolic. The discrepancies only make the "N-shape" somewhat distorted. The scalar Radon Transform The Radon and related transforms are currently in wide use in the fields of applied mathematics. In particular, during the last decade medical imaging techniques have turned more and more towards tomography. X-ray Computed Tomography (CT), Emission CT, Ultrasound CT and Nuclear Magnetic Resonance Imaging are the dominant areas. The enormous increase in computer power available at a relatively low cost is probably the single most important factor behind this development. Advances in electronics and improved computer algorithms have also contributed. However, the underlying mathematics is much the same as before. Typically, the issue is to measure how a tissue scatters or absorbs the energy in an applied test beam. The equation governing this process can often be written as = /oexp(-jf/i(r)«tf), where / is the intensity of the test beam and L is the path that the beam takes. The object is to determine how the linear attenuation coefficient, (*(x), varies in space. Consider a transverse section of a three dimensional body of finite extent. Introduce an orthogonal coordinate system such that the section lies in the xy plane. Each straight line in this plane is uniquely determined by the pair (s, 9), defined in Figure 1 below. Note that —oo denote the unit vector u> = (cos 9, sin 0,0). Then u> and 9 are interchangeable, and the line L can hr* ./ritten in the form L(s,9) = { r : r e J = 0r r-u; = s}. Figure 1: Definition of the parameters for the Radon transform. Let us now confine ourselves to the plane 2 = 0, and introduce the function (i as fi(sj) = log (y-) = / /i(r)rf/. Vin/ Jr-w=i Then fi is the two dimensional Radon transform (or X-ray transform) of /x which we also write fi(a.O) = TZfi. Measuring how a test beam is attenuated along a straight line, L(s,6), one can evaluate fi(s.O). and as A* and 9 vary to generate any line in the xy plane, the function /i(.s, 6) can be obtained for the entire sO plane. With the aid of the inverse Radon transform, (i = TZ~ft, it is then possible to calculate the linear attenuation coefficient, /i(x,y), at any point in the xy plane. For further background on the scalar Radon transform, the reader is referred to [3] and [4]. Moving Fluids and Doppler Shift If the domain under consideration contains a moving fluid, then the Doppler shift car. be used to measure the velocity of this motion. Disregarding relativistic effects, the Doppler shift, Aw, in an emitted signal with angular frequency w is described by the equation