Inverse Fourier Transform in the Gamma Coordinate System

Abstract

This paper provides auxiliary results for our general scheme of computed tomography. In 3D parallel-beam geometry, we first demonstrate that the inverse Fourier transform in different coordinate systems leads to different reconstruction formulas and explain why the Radon formula cannot directly work with truncated projection data. Also, we introduce a gamma coordinate system, analyze its properties, compute the Jacobian of the coordinate transform, and define weight functions for the inverse Fourier transform assuming a simple scanning model. Then, we generate Orlov's theorem and a weighted Radon formula from the inverse Fourier transform in the new system. Furthermore, we present the motion equation of the frequency plane and the conditions for sharp points of the instantaneous rotation axis. Our analysis on the motion of the frequency plane is related to the Frenet-Serret theorem in the differential geometry.

1. Introduction

In [1], we developed a general scheme for 2D and 3D, parallel- and divergent-beam computed tomography (CT). Different from traditional Radon's or Tuy's formulas starting with the inverse Fourier transform in the spherical coordinate system, our new framework is based on an instantaneous cylindrical coordinate system. However, in [1], the coordinate system was not formally defined, and the Jacobian of the transform was not explicitly calculated. Given the basic role of our general scheme in CT theory and some confusion in the literature [2–5], here, we treat the Γ coordinate system in a mathematically strict way.

The organization of this paper is as follows. In Section 2, we demonstrate how the inverse Fourier transform formula generates different reconstruction formulas, with a special attention on different properties of projection truncation. In Section 3, we define the Γ coordinate system and deduce its Jacobian factor. In Section 4, we study the motion of the frequency plane and present the condition for sharp points of the instantaneous rotation axis. In Section 5, a simplified model for helical CT reconstruction is given. In the appendixes, we present a weighted Λ reconstruction formula, a weighted Radon's formula, a deduction of Orlov's formula, and an alternative way to study the motion of the frequency plane. For convenience, we use the same notations as in [1].

2. Inverse Fourier Transform in Commonly Used Coordinate Systems

In this section, we write the inverse Fourier transform in commonly used coordinate systems, derive corresponding reconstruction formulas, and demonstrate their difference on truncation of projection data.

As shown in Figure 1, in the 3D space 𝑅³,Ψ(𝑟)⃗ is an object function to be reconstructed, whose Fourier transform is Ψ(𝑘⃗). According to Fourier analysis, we have Ψ𝑟⃗=^∞−∞^∞−∞^∞−∞Ψ𝑘⃗exp2𝜋𝑖𝑘⃗⋅𝑟⃗𝑑𝑘₁𝑑𝑘₂𝑑𝑘₃,(1) where 𝑟⃗=(𝑥,𝑦,𝑧) and 𝑘⃗=(𝑘₁,𝑘₂,𝑘₃) are 3D vectors in the real space and Fourier domain, respectively. The unit vectors along the three axes are i,⃗j,⃗ and k⃗. Note that the frequency vector 𝑘⃗ and the unit vector k⃗ are different. This formula states that to reconstruct an 3D object Ψ(𝑟)⃗, we need its Fourier transform Ψ(𝑘⃗) in the whole 3D space. In X-ray CT, the Fourier transform Ψ(𝑘⃗) can be calculated when sufficiently many parallel projections are measured.

(a)

(b)

(c)

Figure 1 

3D reconstruction problem in parallel-beam geometry and the cylindrical coordinate system. (a) A parallel-beam scans an object along a semicircle locus on the unit sphere; (b) In the frequency domain, when 𝜙0 increases from 0 to 𝜋, the frequency plane 𝜙=𝜙0 scans every point exactly once except for the points on the 𝑘3 axis, and the end point of the normal vector 𝑂𝑆 moves along the equator from (0,−1,0) to (0,1,0); (c) For simplicity, we suppose that the detector panel passes through the origin.

Now, let us specify the simplest 3D reconstruction problem. In Figure 1(a), 𝑆(𝜙)=𝑆(sin𝜙,−cos𝜙,0) with 𝜙∈[0,𝜋] is a point on a great semicircle of the unit sphere. In the 3D space, we define the unit vectors 𝑒⃗=𝑂𝑆=(sin𝜙,−cos𝜙,0) and 𝑒^⟂=(cos𝜙,sin𝜙,0). Then, 𝑒,⃗𝑒^⟂ and k⃗ are another set of orthogonal unit vectors. As shown in Figure 1(c), the new components of the vector 𝑟⃗ can be denoted as 𝑢=𝑟⃗⋅𝑒,⃗𝜌=𝑟⃗⋅𝑒^⟂,𝑧=𝑟⃗⋅k.⃗(2) The projection at point 𝑆 is defined by 𝑃_𝑆(𝜌,𝑧)=𝑃_𝜙(𝜌,𝑧)=^∞−∞Ψ𝑢𝑒⃗+𝜌𝑒^⟂+𝑧k⃗𝑑𝑢.(3) Our purpose is to reconstruct Ψ(𝑟)⃗ when 𝑃_𝜙(𝜌,𝑧) is known for all 𝜙∈[0,𝜋]. Let us see how the selection of a coordinate system will determine the reconstruction formula, paying an attention to longitudinal data truncation.

2.1. Signed Cylindrical Coordinate System

In a cylindrical coordinate system, the inverse Fourier transform can be rewritten as Ψ𝑟⃗=^𝜋0^∞−∞^∞−∞Ψ𝑘⃗exp2𝜋𝑖𝑘⃗⋅𝑟⃗||𝑘_𝜌||𝑑𝑘_𝜌𝑑𝑘₃𝑑𝜙,(4) where (𝑘_𝜌,𝑘₃,𝜙) with −∞<𝑘_𝜌<∞,−∞<𝑘₃<∞,0≤𝜙<𝜋 is the signed cylindrical coordinates of 𝑘⃗. As shown in Figure 1(b), we have 𝑘⃗=𝑘⃗𝑘_𝜌,𝑘₃,𝜙=𝑘_𝜌𝑒^⟂(𝜙)+𝑘₃k.⃗(5) When 𝜙₀ increases from 0 to π, the plane 𝜙=𝜙₀ passes through every frequency point exactly once (except for the points on the axis 𝑂𝑘₃ which have a zero measure), and the normal vector 𝑂𝑆 of the plane 𝜙=𝜙₀ moves along a great semicircle on the unit sphere with the starting point A(0,−1,0) and the end point B(0,1,0). Note that the coordinate transform from (𝑘₁,𝑘₂,𝑘₃) to (𝑘_𝜌,𝑘₃,𝜙) is one to one (except for the points on the axis 𝑂𝑘₃ with zero measure), and the absolute value of Jacobian is |𝑘_𝜌|.

From the parallel projection 𝑃_𝜙(𝜌,𝑧), we can calculate the Fourier transform of the object Ψ𝑘_𝜌,𝑘₃,𝜙==𝑃_{𝜙∞−∞}𝑘_𝜌,𝑘₃^∞−∞𝑃_𝜙(𝜌,𝑧)exp−2𝜋𝑖𝑘_𝜌𝜌+𝑘₃𝑧𝑑𝜌𝑑𝑧.(6) Substituting (6) into (4), we have Ψ𝑟⃗=^𝜋0^∞−∞^∞−∞𝑃_𝜙𝑘_𝜌,𝑘₃exp2𝜋𝑖𝑘⃗⋅𝑟⃗||𝑘_𝜌||𝑑𝑘_𝜌𝑑𝑘₃𝑑𝜙.(7) The reconstruction scheme behind this formula can be denoted as 𝑃_𝜙(𝜌,𝑧)⟶Ψ𝑘⃗⟶Ψ𝑟⃗,(8) which indicates that we first calculate all the Fourier components from projections and then reconstruct the object using the inverse Fourier transform. In this process, the object is reconstructed as a whole body.

In fact, Formula (7) can be written in the filtered backprojection form as Ψ𝑟⃗=^𝜋0𝑃_𝜙(𝜌,𝑧)𝑑𝜙,(9) where the filtered projection is given by 𝑃_𝜙=(𝜌,𝑧)_∞−∞^∞−∞𝑃_𝜙𝑘_𝜌,𝑘₃exp2𝜋𝑖𝑘⃗⋅𝑟⃗||𝑘_𝜌||𝑑𝑘_𝜌𝑑𝑘₃=^∞−∞^∞∞𝑃_𝜙𝑘_𝜌,𝑘₃exp2𝜋𝑖𝑘_𝜌𝜌exp2𝜋𝑖𝑘₃𝑧||𝑘_𝜌||𝑑𝑘_𝜌𝑑𝑘₃=𝑃_𝜙(𝜌,𝑧)∗1−2𝜋²𝜌².(10) The symbol ∗ denotes the 1D convolution operation about variable 𝜌. Since the filtration is along the horizontal direction only, the projection can be longitudinally truncated, that is, a plane with a specified 𝑧 value can be reconstructed independently. The reconstruction scheme behind (9) with (10) can be denoted by Ψ𝑘⃗⟶𝑃_𝜙(𝜌,𝑧)⟶Ψ𝑟⃗,(11) which states that certain frequency components form the projection, and the object can be reconstructed directly from projections.

When projections are longitudinally truncated, the Fourier transform cannot be calculated. How and why can we reconstruct a part of the object?

The answer is as follows. What we want to reconstruct is the object Ψ(𝑟)⃗ instead of the Fourier transform Ψ(𝑘⃗). Though we cannot calculate every single frequency component, we can calculate the contribution of the frequency components on the whole frequency plane (as well as its neighborhood) to a point to be reconstructed, see (10). In the aforementioned case, projections contain as all frequency components as the object does, but the ratio of the frequency components is not suitable. Then, the only thing we need to do is to perform a 1D filtration to adjust this ratio. An analog is that the Fourier components can be considered as raw material, projections a semifinished product, and the reconstructed object the final product. If we use the semifinished product to make a product, we do not need change it back to the raw material. The role of Fourier analysis is to tell us how far from the semifinished product to the final product. Simply speaking, the essence of image reconstruction from projections is filtration. Therefore, a suitable coordinate system should show the character of the filtration in an easy way.

2.2. Signed Spherical Coordinate System

In the signed spherical coordinate system shown in Figure 2, the inverse Fourier transform can be rewritten as Ψ𝑟⃗=^𝜋0^{𝜋/2−𝜋/2}^∞−∞Ψ𝑘⃗exp2𝜋𝑖𝑘⃗⋅𝑟⃗𝑘²cos𝜗𝑑𝑘𝑑𝜗𝑑𝜙,(12) where (𝑘,𝜗,𝜙) with −∞<𝑘<∞,−𝜋/2<𝜗<𝜋/2,0≤𝜙<𝜋 is the signed spherical coordinates of 𝑘⃗.

(a)

(b)

Figure 2 

The sphere coordinate system and the detector plane. (a) The sphere coordinate system in the Fourier domain, and (b) the relationship between projections and the Radon transform.

For convenience, we denote 𝑘⃗=𝑘⃗(𝑘,𝜗,𝜙)=𝑘𝑛⃗, where 𝑛⃗=𝑛(⃗𝜗,𝜙)=𝑒^⟂(𝜙)cos𝜗+k⃗sin𝜗 is a unit vector.

For a unit vector 𝑛⃗=𝑛(⃗𝜗,𝜙), and a real number 𝑙∈(−∞,∞), the Radon transform of the object is defined as an integral 𝑅Ψ𝑙,𝑛⃗=𝑅Ψ(𝑙,𝜗,𝜙)=_{⃗𝑟⋅⃗𝑛=𝑙}Ψ𝑟⃗𝑑²𝑟,⃗(13) on the plane described by 𝑟⃗⋅𝑛⃗=𝑙.

The Fourier transform can be calculated from the Radon transform by Ψ𝑘⃗=Ψ𝑘𝑛⃗=^∞−∞𝑅Ψ𝑙,𝑛⃗exp(−2𝜋𝑖𝑘𝑙)𝑑𝑙.(14) Thus, the inverse Fourier transform (12) becomes Radon's formulaΨ𝑟⃗=−14𝜋²^𝜋0^{𝜋/2−𝜋/2}𝜕²𝜕𝑙²||||_{𝑙=⃗𝑟⋅⃗𝑛}=−1𝑅Ψ𝑙,𝑛⃗cos𝜗𝑑𝜗𝑑𝜙4𝜋²∇²^𝜋0^{𝜋/2−𝜋/2}=−1𝑅Ψ𝑙,𝑛⃗||_{𝑙=⃗𝑟⋅⃗𝑛}cos𝜗𝑑𝜗𝑑𝜙8𝜋²∇²_Ω𝑅Ψ𝑙,𝑛⃗||_{𝑙=⃗𝑟⋅⃗𝑛}𝑑𝑛,⃗(15) where ∇² is the Laplace operator, and Ω is the unit sphere.

Radon's formula tells us that to reconstruct the object at a point, what we need is the Radon transform of the planes through the point and its neighborhood. However, what we measure by the detector is line integrals along X-rays instead of the Radon transform [6] but it can be calculated from projections by 𝑅Ψ𝑙,𝑛⃗==𝑅Ψ(𝑙,𝜗,𝜙)_∞−∞𝑃_𝜙(𝑙cos𝜗−𝑣sin𝜗,𝑙sin𝜗+𝑣cos𝜗)𝑑𝑣.(16) We call (16) the first relation between the Radon transform and parallel projection.

Formulas (15) and (16) form a scheme to reconstruct the object from its parallel projections. Here, the projection is not allowed to be longitudinally truncated. From Section 2.1, however, we have seen that the object can be reconstructed slice by slice from parallel projections. Why is the truncation not allowed when using the Radon formula?

For comparison, we rewrite Radon’s formula in the filtered backprojection format Ψ𝑟⃗=^𝜋0𝑃_𝜙(𝜌,𝑧)𝑑𝜙.(17) with 𝑃_𝜙(𝜌,𝑧)=^∞−∞^{𝜋/2−𝜋/2}=1Ψ𝑘⃗exp2𝜋𝑖𝑘⃗⋅𝑟⃗𝑘²cos𝜗𝑑𝑘𝑑𝜗−4𝜋²^{𝜋/2−𝜋/2}𝜕²𝜕𝑙²||||_{𝑙=⃗𝑟⋅⃗𝑛=𝜌cos𝜗+𝑧sin𝜗}𝑅Ψ𝑙,𝑛⃗cos𝜗𝑑𝜗.(18) Comparing (18) with (10), we have the following relation between the Radon transform and parallel-beam projections 1−4𝜋²^{𝜋/2−𝜋/2}𝜕²𝜕𝑙²||||_{𝑙=⃗𝑟⋅⃗𝑛=𝜌cos𝜗+𝑧sin𝜗}=𝑃_𝜙𝑅Ψ𝑙,𝑛⃗(𝜗,𝜙)cos𝜗𝑑𝜗(𝜌,𝑧)∗1−2𝜋²𝜌².(19) We call this the second relation between the Radon transform and parallel-beam projections. Based on the relationship between fan-beam and parallel-beam projections [1], the right hand side of (19) can be expressed in terms of cone-beam projections as well.

Substituting (16) into (19), we get an identity of parallel projections 1−4𝜋²^{𝜋/2−𝜋/2}𝜕²𝜕𝑙²||||_{𝑙=𝜌cos𝜗+𝑧sin𝜗}=×𝑃_𝜙^∞−∞𝑃_𝜙(𝑙cos𝜗−𝑣sin𝜗,𝑙sin𝜗+𝑣cos𝜗)𝑑𝑣cos𝜗𝑑𝜗(𝜌,𝑧)^∗1−2𝜋²𝜌².(20) Now, we see that reconstruction of an object from parallel-beam projections does allow truncation of projection data but to verify the identity (20) needs the information on the whole projection domain. Reconstruction from parallel-beam projections using the Radon formula means numerically verifying this identity. We point out that if the projection 𝑃_𝜙(𝜌,𝑧) is replaced by a general function with two variables, this identity holds true as well; see Appendix A.

Similarly, in cone-beam CT, reconstruction from cone-beam projections using the Radon formula, that is, the Grangeat's framework, means numerically verifying the cone-beam version of this identity. This is the reason why the Grangeat framework cannot deal with the long object problem.

We see that the Radon formula comes from the inverse Fourier transform in the signed spherical coordinate system. In this simplest reconstruction problem, there are two shortcomings with the Radon formula: (1) what we measure with X-ray is line integrals rather than planar integrals, (2) the projection cannot be longitudinally truncated.

2.3. Standard Spherical Coordinate System

In a standard spherical coordinate system, the inverse Fourier transform is written as Ψ𝑟⃗=^02𝜋^{𝜋/2−𝜋/2}^∞0Ψ𝑘⃗exp2𝜋𝑖𝑘⃗⋅𝑟⃗𝑘²cos𝜗𝑑𝑘𝑑𝜗𝑑𝜙,(21) where (𝑘,𝜗,𝜙) with  0≤𝑘<∞,−𝜋/2<𝜗<𝜋/2,0≤𝜙<2𝜋 is the standard spherical coordinates of 𝑘⃗, as shown in Figure 2.

From the well-known generalized function relations [7] 𝑖𝛿(𝑙)=^∞−∞exp(2𝜋𝑖𝑘𝑙)𝑑𝑘,𝜋𝑙=^∞−∞sgn(𝑘)exp(2𝜋𝑖𝑘𝑙)𝑑𝑘,(22) we have 12𝛿(𝑙)+2𝜋𝑖𝑙=^∞01exp(2𝜋𝑖𝑘𝑙)𝑑𝑘,2𝛿^(𝑙)+𝑖𝜋𝑙³=−4𝜋²^∞0𝑘²exp(2𝜋𝑖𝑘𝑙)𝑑𝑘.(23) Based on the convolution theorem, we have ^∞0Ψ=𝑘⃗exp2𝜋𝑖𝑘⃗⋅𝑟⃗𝑘²𝑑𝑘_∞0=−1Ψ𝑘𝑛⃗exp(2𝜋𝑖𝑘𝑙)𝑘²𝑑𝑘8𝜋²𝜕²𝜕²𝑙𝑅Ψ𝑙,𝑛⃗−4𝜋𝑖³𝑙³∗𝑅Ψ𝑙,𝑛⃗.(24) Therefore, the inverse Fourier transform becomes Ψ𝑟⃗=−18𝜋²^02𝜋^{𝜋/2−𝜋/2}𝜕²𝜕𝑙²−𝑖𝑅Ψ𝑙,𝑛⃗||||_{𝑙=⃗𝑟⋅⃗𝑛}cos𝜗𝑑𝜗𝑑𝜙4𝜋³^02𝜋^{𝜋/2−𝜋/2}𝑅Ψ𝑙,𝑛⃗∗𝑙1³||||_{𝑙=⃗𝑟⋅⃗𝑛}cos𝜗𝑑𝜗𝑑𝜙.(25) Based on the odd and even symmetry, we have the Radon formula again [6] Ψ𝑟⃗=−14𝜋²^𝜋0^{𝜋/2−𝜋/2}𝜕²𝜕𝑙²𝑅Ψ𝑙,𝑛⃗||||_{𝑙=⃗𝑟⋅⃗𝑛}cos𝜃𝑑𝜃𝑑𝜙.(26) The difference between the signed and standard spherical coordinate systems is little, and the signed spherical coordinate system leads to Radon's formula more easily.

It is important to underline that the inverse Fourier transform may yield various reconstruct formulas in different coordinate systems and allow different degrees of data truncation. Among the commonly used coordinate systems, the signed cylindrical coordinate system is the most convenient one to solve the simplest 3D reconstruction problem. To handle the general 3D reconstruction problem, in the following we will introduce a variant of the signed cylindrical coordinate system, which is referred to as the Γ coordinate system.

3. Γ Coordinate System and Its Jacobian Factor

What plays an important role in the 3D reconstruction field is a variant of the cylindrical coordinate system, which can be named the Γ coordinate system.

As shown in Figure 3(b),Γ_AB is a three-times differentiable curve connecting points A(0,−1,0) and B(0,1,0) on the unit sphere, whose length is 𝜃₀≥𝜋.𝑆(𝜃) is a point on Γ_AB parameterized by 𝜃∈[0,𝜃₀] the length of the segment AS. Let us introduce three orthogonal unit vectors 𝑒𝑒₁⃗₃⃗=𝑂𝑆,=𝑑𝑑𝜃𝑒³⃗𝑒₂⃗,=𝑒₃⃗×𝑒₁⃗,(27) where 𝑒₁⃗ represents the tangential direction, 𝑒₂⃗ the instantaneous rotation axis of the frequency plane. Note that they are functions of 𝜃=∫^𝑆𝐴|𝑑𝑒₃⃗| (see Figures 3(a) and 3(b)).

(a)

(b)

(c)

Figure 3 

Nonuniqueness of the representation of a point in the Γ coordinates. (a) The Γ coordinate system, (b) a general curve ΓAB on the unit sphere, and (c) the intersections of ΓAB and the great circle orthogonal to a given vector 𝑘⃗. When 𝑆 moves from A to B ΓAB, some region may be scanned more than once.

In the Fourier domain, we call the plane through the origin 𝑂 and orthogonal to the vector 𝑂𝑆 the frequency plane, denoted as ∏(𝑂𝑆). Every point 𝑘⃗∈∏(𝑂𝑆) can be expressed as 𝑘⃗=𝜔₁𝑒₁⃗+𝜔₂𝑒₂⃗,(28) with 𝜔₁=𝑘⃗⋅𝑒₁⃗,𝜔₂=𝑘⃗⋅𝑒₂⃗.

More generally, for every given triplet (𝜔₁,𝜔₂,𝜃) with 𝜔₁∈(−∞,∞),𝜔₂∈(−∞,∞),𝜃∈[0,𝜃₀], we can define a 3D vector 𝑘⃗=𝑘⃗𝜔₁,𝜔₂,𝜃=𝜔₁𝑒₁⃗(𝜃)+𝜔₂𝑒₂⃗(𝜃),(29) and the triplet (𝜔₁,𝜔₂,𝜃) is a set of Γ coordinates of the point 𝑘⃗.

However, a given point 𝑘⃗∈𝑅³ may be represented by several sets of Γ coordinates, because when 𝑆(𝜃) moves along Γ_AB, the rotation axis 𝑒₂⃗ keeps changing, and hence the frequency plane ∏(𝑂𝑆) may scan the given point 𝑘⃗ more than once. The number of times of the point being scanned, denoted as 𝐽(𝑘⃗), equals the number of intersections between the curve Γ_AB and the great circle orthogonal to the vector 𝑘⃗ (Figure 3(c)). To find the all intersections, one can solve the equation 𝑘⃗⋅𝑒₃⃗(𝜃)=0,(30) to obtain the solutions ¹𝜃,²𝜃,…,^{⃗𝐽(𝑘)}𝜃. For every solution, such as ¹𝜃, one can further have ¹𝜔₁=𝑘⃗⋅𝑒₁⃗(¹𝜃),¹𝜔₂=𝑘⃗⋅𝑒₂⃗(¹𝜃). Thus, one gets the 𝐽(𝑘⃗) groups of coordinates 𝑘⃗==⋯𝑘⃗¹=𝜔𝑘₁,⃗¹𝜔₂,¹_{𝐽⃗𝑘}𝜃=𝑘⃗²𝜔₁,²𝜔₂,²𝜃𝜔₁,^{𝐽⃗𝑘}𝜔₂,^{𝐽⃗𝑘}𝜃.(31) We call this new system the Γ coordinate system, since the system is based on the curve Γ_AB. This coordinate system is a variant of the cylindrical system. For a differential arc 𝑆𝑆^, the new system is similar to a cylindrical coordinate system if the scanned volume is considered. When Γ_𝐴𝐵 is a great semicircle, the Γ coordinate system becomes the cylindrical coordinate system.

In the Γ coordinate system, the inverse Fourier transform can be expressed as Ψ𝑟⃗=^𝜃0₀^∞−∞^∞−∞×Ψ𝑊𝑘⃗𝜔||₁𝜔,𝜔₂,𝜃exp2𝜋𝑖𝑘⃗⋅𝑟₁⃗||𝑑𝜔₁𝑑𝜔₂𝑑𝜃,(32) with 𝑘⃗=𝜔₁𝑒₁⃗(𝜃)+𝜔₂𝑒₂⃗(𝜃). In [1], we obtained the absolute value of the Jacobian, |𝜔₁| for the coordinate transform from (𝑘₁,𝑘₂,𝑘₃) to (𝜔₁,𝜔₂,𝜃), by similarity between the Γ coordinate system and the cylindrical system.

Based on (29) in the Γ coordinate system, the absolute value of the Jacobian factor can be calculated as follows: ||||𝜕𝑘⃗𝜕𝜔₁×𝜕𝑘⃗𝜕𝜔₂⋅𝜕𝜕𝑘𝜃⃗||||=|||𝑒¹⃗×𝑒₂⃗⋅𝜔₁𝑑𝑑𝜃𝑒¹⃗+𝜔₂𝑑=|||𝑒₃⃗𝑑𝜃𝑒²⃗|||⋅𝜔₁𝑑𝑑𝜃𝑒¹⃗+𝜔₂𝑑=||𝜔₁𝑑𝜃𝑒²⃗|||||,(33) where we have used the relations discussed in the next section𝑒₃⃗⋅𝑑𝑑𝜃𝑒¹⃗=−1,𝑒₃⃗⋅𝑑𝑑𝜃𝑒²⃗=0.(34) Then, a weight function 𝑊(𝜔₁,𝜔₂,𝜃) is defined for all the coordinates (𝜔₁,𝜔₂,𝜃), such that the summation of the weight function on the 𝐽(𝑘⃗) groups of coordinates of the same 𝑘⃗∈𝑅³ remains one, that is, ^𝐽^{⃗𝑘}_𝑗=1𝑊^𝑗𝜔₁,^𝑗𝜔₂,^𝑗𝜃=1.(35) An example of the weight functions is [1, 8] 𝑊𝜔₁,𝜔₂,𝜃=sgn𝑘₂sgn𝜔₁,(36) with 𝑘₂=𝑘⃗⋅𝑗⃗=[𝜔₁𝑒₁⃗(𝜃)+𝜔₂𝑒₂⃗(𝜃)]⋅𝑗.⃗

With this weight function, (32) becomes a parallel-beam reconstruction formula from the Fourier slice theorem, and further Ye-Wang’s cone-beam formula [8] via the relationship between parallel-beam and cone-beam projections [1]. With the weight function, the filtration is no longer along the tangential direction generally.

Clearly, the weight function is not unique. For instance, for frequency points with 𝐽(𝑘⃗)=3, in addition to the weight function defined by (36), (1−1+1), we can also choose (1/3+ 1/3+1/3), (0+1+0), or a more complicated one, (1+1−1) for some points and (−1+1+1) for the rest points, leading to different parallel-beam reconstruction formulas. Furthermore, using the approach described in [1], we can arrive at Katsevich’s helical cone-beam formulas [9–11], and so forth.

If one takes the weight function 𝑊(𝜔₁,𝜔₂,𝜃)=1 on the right side of (32), what we will reconstruct is actually another function Φ𝑟⃗=^𝜃0₀^∞−∞^∞−∞=exp2𝜋𝑖𝑘⃗⋅𝑟⃗Ψ𝑘⃗||𝜔₁||𝑑𝜔₁𝑑𝜔₂𝑑𝜃_𝑅3exp2𝜋𝑖𝑘⃗⋅𝑟⃗Ψ𝑘⃗𝐽𝑘⃗𝑑³𝑘⃗.(37) From (37), we observe that the reconstructed image Φ(𝑟)⃗ equals the true object Ψ(𝑟)⃗ if and only if 𝐽(𝑘⃗)=1, that is, the trajectory Γ_AB is a great semicircle. The condition for Φ(𝑟)⃗ to be an approximation of the object function, that is, Φ(𝑟)⃗≈Ψ(𝑟)⃗, is that 𝐽(𝑘⃗)=1 in a major portion of the Fourier domain and 𝐽(𝑘⃗) is close to 1 in the rest region. Based on the relation between 𝐽(𝑘⃗) and the curve length of Γ_AB, a necessary condition is 𝜃₀≈𝜋, or equivalently 𝜀=𝜃₀/𝜋−1≈0. Otherwise, Φ(𝑟)⃗ and Ψ(𝑟)⃗will have a great difference. Unfortunately this formula was once thought as exact, which was essentially Theorem 3 in [2] and Theorem 4.3 in [3]. The details can be seen in [1].

Generally speaking, the criterion to identify exact formulas from approximate ones is to evaluate if every frequency component contributes exactly once to the final reconstruction, and the difference between exact formulas comes from different forms of weight functions.

4. Motion of the Frequency Plane and Condition for Sharp Points

In this section, we will study motion of the frequency plane, present the condition for sharp points of the locus of the rotation axis, and introduce the Euler angles. Let us simplify notations 𝑒₃̇⃗=(𝑑/𝑑𝜃)𝑒₃⃗,𝑒₃̈⃗=(𝑑²/𝑑𝜃²)𝑒₃⃗, and so on. In this section, we assume that Γ is a three times differentiable curve on the unit ball, which may be not complete.

Obviously we have the following.

Proposition 1. For any point 𝑆(𝜃)∈Γ, 𝑒₁⃗,𝑒₂⃗,𝑒₃⃗ defined by (27) are orthogonal unit vectors.

Definition 1. For every point 𝑆(𝜃)∈Γ, let us define 𝜎=𝑒₃⃗⋅𝑒₃̇⃗×𝑒₃̈⃗,(38) which is the signed volume spanned by the three vectors 𝑒₃⃗,𝑒₃̇⃗,𝑒₃̈⃗. If we consider the curve length 𝜃 as time, 𝑒₃⃗,𝑒₃̇⃗,𝑒₃̈⃗ are the position, velocity, and the acceleration of point S.

Proposition 2. The motion of 𝑒₁⃗,𝑒₂⃗,𝑒₃⃗ can be described by the following differential equations 𝑑𝑑𝜃𝑒³⃗𝑑=𝑒₁⃗,𝑑𝜃𝑒¹⃗𝑑=𝜎𝑒₂⃗−𝑒₃⃗,𝑑𝜃𝑒²⃗=−𝜎𝑒₁⃗.(39)

Proof. The first equation is the definition. Now, let us prove the other two. Since 𝑒₁⃗,𝑒₂⃗,𝑒₃⃗ are orthogonal unit vectors, we can write 𝑒𝑒₁₁⃗⃗⋅𝑒₁⃗=1,𝑒₂⃗⋅𝑒₂⃗=1,𝑒₃⃗⋅𝑒₃⃗=1,⋅𝑒₂⃗=0,𝑒₂⃗⋅𝑒₃⃗=0,𝑒₃⃗⋅𝑒₁⃗=0.(40) Applying 𝑑/𝑑𝜃 on the above equations, one has 𝑒₁̇⃗𝑒₁̇⃗⋅𝑒₁⃗=0,𝑒₂̇⃗⋅𝑒₂⃗=0,𝑒₃̇⃗⋅𝑒₃⃗=0,⋅𝑒₂⃗+𝑒₁⃗⋅𝑒₂̇⃗=0,𝑒₂̇⃗⋅𝑒₃⃗=0,𝑒₃⃗⋅e₁̇⃗=−1.(41) Using the definition of 𝜎, we have 𝑒₁̇⃗⋅𝑒₂⃗=𝑒₁̇⃗⋅𝑒₃⃗×𝑒₁⃗𝑒₁⃗=𝑒₃⃗⋅𝑒₃̇⃗×𝑒₃̈⃗=𝜎,⋅𝑒₂̇⃗=−𝜎.(42) Using the orthogonal unit vectors 𝑒₁⃗,𝑒₂⃗,𝑒₃⃗, 𝑒₁̇⃗ and 𝑒₂̇⃗ can be expressed as 𝑑𝑑𝜃𝑒¹⃗𝑑=𝜎𝑒₂⃗−𝑒₃⃗,𝑑𝜃𝑒²⃗=−𝜎𝑒₁⃗,(43) which completes the proof.

From these differential equations, we recognize that the motion of 𝑒₁⃗,𝑒₂⃗,𝑒₃⃗ is a rotation in the 3D space.

Now, we consider the frequency plane Π(𝑂𝑆) fixed with 𝑒₁⃗,𝑒₂⃗, 𝑒₃⃗ as a rigid body. When 𝑆(𝜃) moves along the curve Γ, the rigid body rotates in the 3D space.

Let us define Ω⃗=𝑒₂⃗+𝜎𝑒₃⃗.(44) We will see that Ω⃗ is the angular velocity of the rigid body by the following proposition.

Proposition 3. The motion of the vectors 𝑒₁⃗,𝑒₂⃗,𝑒₃⃗ can be expressed as 𝑑𝑑𝜃𝑒³⃗𝑑=Ω⃗×𝑒₃⃗,𝑑𝜃𝑒¹⃗𝑑=Ω⃗×𝑒₁⃗,𝑑𝜃𝑒²⃗=Ω⃗×𝑒₂⃗.(45)

Proof. It is easy to verify the three equations one by one with Proposition 2 and the definition of Ω⃗.

Proposition 4. The angular acceleration of the rigid body is Ω̇⃗=𝜎̇𝑒₃⃗(𝜃).(46)

Proof. In fact, we have Ω̇⃗=𝑑𝑑𝜃𝑒²⃗=𝑒₂̇⃗+𝜎𝑒₃⃗+𝜎̇𝑒₃⃗+𝜎𝑒₃̇⃗=−𝜎𝑒₁⃗+𝜎̇𝑒₃⃗+𝜎𝑒₁⃗=𝜎̇𝑒₃⃗.(47)

Proposition 5. For any point 𝑆(𝜃)∈Γ, we have 𝜎𝜎̇==00⟺⟺𝑒𝑒⃛₃₃̈⃗⃗=−𝑒₃⃗,(48)=−𝜎²+1𝑒₃̇⃗.(49)

Proof. Let us first prove (48).
If 𝜎=0, by Proposition 2 we have 𝑒₃̈⃗=−𝑒₃⃗. On the other hand, if 𝑒₃̈⃗=−𝑒₃⃗, we have 𝜎=𝑒₃⃗⋅[𝑒₃̇⃗×𝑒₃̈⃗]=0.
Now, let us prove (49).
By Proposition 2, we have 𝑒₁̇⃗=𝜎𝑒₂⃗−𝑒₃⃗.(50) Applying 𝑑/𝑑𝜃 to both sides of (50) and using 𝜎̇=0, we have 𝑒₁̈⃗=𝜎̇𝑒₂⃗+𝜎𝑒₂̇⃗−𝑒₃̇⃗=𝜎−𝜎𝑒₁⃗−𝑒₁⃗=−𝜎²+1𝑒₁⃗.(51) On the other hand, if 𝑒⃛₃⃗=−(𝜎²+1)𝑒₃̇⃗, we have 𝜎̇=𝑑𝑑𝜃𝑒³⃗=𝑒₃̇⃗⋅𝑒₃̇⃗×𝑒₃̈⃗=𝑒₃⃗⋅𝑒₃̇⃗×𝑒₃̈⃗+𝑒₃⃗⋅𝑒₃̈⃗×𝑒₃̈⃗+𝑒₃⃗⋅𝑒₃̇⃗×𝑒⃛₃⃗⋅𝑒₃̇⃗×𝑒⃛₃⃗=0.(52) This completes the proof. Physically, 𝑒⃛₃⃗ is the change rate of the acceleration 𝑒₃̈⃗.

Proposition 6. 𝜎=0 for every point 𝑆(𝜃)∈Γ, if and only if Γ is a great circle or a great arc.

Proof. Since 𝜎=0 for every point 𝑆(𝜃)∈Γ, the motion equation of the three unit vectors, (39), is simplified as 𝑑𝑑𝜃𝑒³⃗𝑑=𝑒₁⃗,𝑑𝜃𝑒¹⃗𝑑=−𝑒₃⃗,𝑑𝜃𝑒²⃗=0.(53) Since (𝑑/𝑑𝜃)𝑒₂⃗=0 for every point, 𝑒₂⃗ is a fixed unit vector. Since (𝑑/𝑑𝜃)𝑒₃⃗=𝑒₁⃗=𝑒₂⃗×𝑒₃⃗, 𝑒₃⃗ rotates about the fixed axis 𝑒₂⃗. Since 𝑒₃⃗⟂𝑒₂⃗, 𝑆, the endpoint of 𝑒₃⃗, draws a great circle or a great arc; see Figure 4(a).
On the other hand, if Γ is a great circle or a great arc, one can verify that 𝑒₃̈⃗=−𝑒₃⃗. Therefore, we have 𝜎=𝑒₃⃗⋅(𝑒₃̇⃗×𝑒₃̈⃗)=0. This completes the proof.

(a)

(b)

(c)

Figure 4 

The circles on unit sphere and a helix in 3D space. (a) The illustration of 𝜎=0 and𝜎̇=0, (b) a circular locus on the unit sphere and the motion of the frequency plane, and (c) the helix in 3D space and the motion of its TNB frame. The figure is not drawn to scale.

Proposition 7. 𝜎̇=0 for every point 𝑆(𝜃)∈Γ, if and only if Γ is a circle or a circular arc.

Proof. Since 𝜎̇=0, by Proposition 4 we have Ω̇⃗=𝜎̇𝑒₃⃗(𝜃)=0. We know that Ω⃗ is a fixed vector. By Proposition 3, we have 𝑑𝑑𝜃𝑒³⃗=Ω⃗×𝑒₃⃗.(54) Therefore, 𝑒₃⃗ rotates about the fixed axis Ω⃗, and 𝑆, the endpoint of 𝑒₃⃗, draws a circular arc. Furthermore, the radius of the circle is 1/√1+𝜎²; see Figure 4(b).
On the other hand, if Γ is a circular arc, the line connecting the circle center and the origin 𝑂 is the rotation axis. We know that 𝑒₃̈⃗ is pointing to the axis. By circular symmetry, 𝜎=𝑒₃⃗⋅(𝑒₃̇⃗×𝑒₃̈⃗) is constant at every point of the circular arc. Therefore, 𝜎̇=0 on the circular arc. This completes the proof.

Clearly, Proposition 6 is a special case of Proposition 7; see Figure 4(a).

One important case is that 𝜎=𝑒₃⃗⋅(𝑒₃̇⃗×𝑒₃̈⃗)=0, for a certain point 𝑆(𝜃)∈Γ. At such a moment, we have (𝑑/𝑑𝜃)𝑒₂⃗=−𝜎𝑒₁⃗=0, the rotation axis of the frequency plane does not move, the point 𝑆 moves along a great arc, and the Γ coordinate system is more similar to a cylindrical coordinate system. We say that 𝜃 is a stationary point of 𝑒₂⃗ if 𝜎(𝜃)=0. If 𝜎 has different signs before and after 𝜎=0, the stationary point will become a sharp point, since in this case 𝑒₂⃗ goes forward, stops and goes back according to (𝑑/𝑑𝜃)𝑒₂⃗=−𝜎𝑒₁⃗. Therefore, we have the following straightforward proposition.

Proposition 8. A necessary condition for the curve 𝑒₂⃗(𝜃) to have a sharp point at 𝜃₁∈(0,𝜃₀) is 𝜎(𝜃₁)=0; a sufficient condition for the curve 𝑒₂⃗(𝜃) to have a sharp point at 𝜃₁∈(0,𝜃₀) is that 𝜎(𝜃) has different signs for 𝜃<𝜃₁ and for 𝜃>𝜃₁.

Based on the definition of 𝜎 and the equation of the osculating plane, 𝜎(𝜃₁)=0 means that the osculating plane at 𝑆(𝜃₁)∈Γ passes through the origin since (𝑒₃⃗−0)⋅(𝑒₃̇⃗×𝑒₃̈⃗)=0.𝜎(𝜃) changes its sign before and after 𝜃₁ means that the osculating plane sweeps the origin, or relatively speaking, the origin goes from one side of the osculating plane to the other. An example can be seen in the next section. Sharp points serve as the landmarks for defining weight functions.

Proposition 9. For any given point on the frequency plane 𝑘⃗=𝜔₁𝑒₁⃗+𝜔₂𝑒₂⃗, the motion equation is 𝑑𝑑𝜃𝑘⃗==−Ω⃗𝜎×𝜔𝑘₂⃗𝑒₁⃗(𝜃)+𝜎𝜔₁𝑒₂⃗(𝜃)−𝜔₁𝑒₃⃗(𝜃).(55)

Proof. It is straightforward from Propositions 2 or 3.
We call (55) the motion equation of the frequency plane.
There are two applications of Proposition 9. First, it can be used to compute the Jacobian of the Γ coordinate system ||||𝜕𝑘⃗𝜕𝜔₁×𝜕𝑘⃗𝜕𝜔₂⋅𝜕𝜕𝑘𝜃⃗||||=||||𝑒³⃗⋅𝜕𝑘⃗𝜕𝜃||||=||−𝜔¹||=||𝜔₁||.(56) Second, since 𝑒₁⃗⋅𝜕𝜕𝜃𝑘⃗=−𝜎𝜔²𝑒₂⃗,⋅𝜕𝜕𝜃𝑘⃗=𝜎𝜔¹,(57) we know that among the three vectors 𝜕𝑘⃗/𝜕𝜔₁,𝜕𝑘⃗/𝜕𝜔₂,𝜕𝑘⃗/𝜕𝜃, the last one is not orthogonal to the first two. Hence, a general Γ coordinate system is not orthogonal. Notice that 𝜕𝑘⃗/𝜕𝜃=𝜕𝑘⃗/𝜕𝜃|_𝜔1,𝜔2=𝑑𝑘⃗/𝑑𝜃
Note that the frequency plane rotates about the axis 𝑒₂⃗ with the unit angular velocity while the vectors 𝑒₁⃗ and 𝑒₂⃗ themselves rotate about the axis 𝑒₃⃗ with the angular velocity 𝜎; see Figure 5(a). We see that the Γ coordinate system is different from the cylindrical coordinate system in the motion process. However, since the spin about the axis 𝑒₃⃗ of the frequency plane does not have an effect on the scanned volume, their Jacobians have the same expression. Figure 5(b) shows the shape of a differential volume of the Γ coordinate system.

(a)

(b)

Figure 5 

Differential motion of the frequency plane. (a) Two decomposed rotations and (b) one combined rotation.

Proposition 10. When S moves along a curve Γ on the unit sphere, the scanned volume in the unit ball by the frequency plane is 𝑉=4𝜃⁰3=_{||⃗𝑘||<1}𝐽𝑘⃗𝑑³𝑘⃗,(58) where 𝜃₀ is the length of Γ, and 𝐽(𝑘⃗) is the number of times point 𝑘⃗ is scanned.

Proof. In fact, the scanned volume can be calculated in both the Γ and the Cartesian coordinate systems as follows: 𝑉=^𝜃0₀_{𝑘2=𝜔21+𝜔22<1}=||𝜔₁||𝑑𝜔₁𝑑𝜔₂𝑑𝜃_𝜃00^1𝑘=0^2𝜋𝜙=0=4𝑘²||cos𝜙||𝑑𝑘𝑑𝜙𝑑𝜃3^𝜃0₀𝑑𝜃=43𝜃⁰=_{𝑘21+𝑘22+𝑘23<1}𝐽𝑘⃗𝑑³𝑘⃗.(59) This completes the proof.

This concise formula provides a basic relation between the curve length, the scanned volume by the frequency plane, and the J function.

When we study the rotation problem, Euler angles (𝜗,𝜙,𝛿) are helpful. As shown in Figure 6(a), (𝜗,𝜙) with 𝜗∈(−𝜋/2,𝜋/2),𝜙∈(−∞,∞) are the spherical coordinate of 𝑆=𝑆(𝜗,𝜙),𝛿 is the angle between 𝑒_𝜗⃗ and 𝑒₂⃗. 𝑒_𝜙, and 𝑒_𝜗⃗ are the latitude and longitude unit vectors at point 𝑆 in the spherical coordinate system.

(a)

(b)

Figure 6 

Euler angles. (a) The Euler angles of the frequency plane and (b) the rotation angle of frequency plane relative to the detector plane.

The Euler angles are functions of the curve length 𝜙𝜗𝛿===𝜙𝜗𝛿(((𝜃𝜃𝜃))).,,(60) The three unit vectors can be represented as follows: 𝑒₁⃗𝑒₃⃗=𝑒₃⃗(𝜗,𝜙),𝑒₂⃗=sin𝛿𝑒_𝜗⃗+cos𝛿𝑒_𝜙=𝜗̇𝑒_𝜗⃗+𝜙̇cos𝜗𝑒_𝜙,=−cos𝛿𝑒_𝜗⃗+sin𝛿𝑒_𝜙=−𝜙̇cos𝜗𝑒_𝜗⃗+𝜗̇𝑒_𝜙.(61) Furthermore, one can verify the motion equations of the three unit vectors in Proposition 2.

Based on the motion theory of rigid body, the angular velocity of a rigid body can be written as Ω⃗=−𝜗̇𝑒_𝜙+𝜙̇𝑘⃗+𝛿̇𝑒₃⃗=−𝜗̇𝑒_𝜙+𝜙̇cos𝜗𝑒_𝜗⃗+𝜙̇sin𝜗+𝛿̇𝑒₃⃗=𝑒₂⃗+𝜎𝑒₃⃗.(62) Therefore, we have the following relationships:𝑒₂⃗=𝜎−=𝜗̇𝑒𝜙_𝜙̇s+in𝜙𝜗̇c+os𝛿𝜗̇=𝑒_𝜗⃗Ω=⃗₀Ω⃗₀−Ω⃗₀⋅𝑒₃⃗𝑒³⃗,𝛿̇=−Ω̇⃗₀⋅𝑒₃⃗−Ω̇⃗₀⋅𝑒₁⃗,⋅𝑒₁⃗=Ω⃗₀⋅𝑒₁̇⃗,(63) where we have used a new notation Ω⃗₀=−𝜗̇𝑒_𝜙+𝜙̇𝑘⃗,(64) which is the first two terms of the total angular velocity Ω⃗ of (62).

In fact, Ω⃗₀ is the angular velocity of the detector with the unit vectors 𝑒_𝜙,𝑒_𝜗⃗ when 𝑆 moves along Γ. To be specific, we consider the detector for parallel-beam projection as a rectangle of finite width and length through the origin orthogonal to direction 𝑒₃⃗=𝑂𝑆. During any motion, one side of the detector always keeps horizontal, that is, parallel to the direction 𝑒_𝜙. The pose of the detector is completely determined by the point 𝑆=𝑆(𝜗,𝜙), or two Euler's angles (𝜗,𝜙). The relative motion of the frequency plane to the detector is described by the angle 𝛿; see Figure 6(b).

In summary, there are three sets of orthogonal vectors (beside i,⃗j,⃗k⃗, the unit vectors of the Cartesian coordinate system) and three important planes. The orthogonal unit vectors 𝑒_𝜙,𝑒_𝜗⃗, 𝑒₃⃗ are related to the movement of the detector and 𝑒₃⃗, 𝑒₃̇⃗=𝑒₁⃗, 𝑒₂⃗=𝑒₃⃗×𝑒₁⃗ are related to the movement of the frequency plane, see Figure 6(b). It is easy to verify that Ω⃗=𝑒₂⃗+𝜎𝑒₃⃗=𝑒₁⃗×𝑒₁̇⃗ and hence the vectors 𝑒₁⃗, 𝑒₁̇⃗, Ω⃗ are the third set of orthogonal vectors. The direction of Ω⃗ is the normal of the osculating plane and its magnitude, √1+𝜎², is the reciprocal of the radius of the osculating circle. Here the osculating plane is spanned by 𝑒₁⃗, 𝑒₁̇⃗ [15] and the osculating circle is referred to as the intersection between the osculating plane and the unit ball. When S moves along a curve Γ on the unit ball, it can be viewed that the point S moves along an osculating circle, whose orientation and radius are described by Ω⃗. From Ω⃗=𝑒₂⃗+𝜎𝑒₃⃗, we can see that 𝜎 is an angular velocity of the spin of the frequency plane. The locus lengths of the endpoints of 𝑒₃⃗,𝑒₁⃗, 𝑒₂⃗ are 𝜃₀, ∫^𝜃0₀√1+𝜎²𝑑𝜃, ∫^𝜃0₀|𝜎|𝑑𝜃, respectively, according to Proposition 2.

5. Example of a Curve with 𝐽(𝑘⃗)=1,3

Here, we discuss the reconstruction based on the simplified model of the 3D helical cone beam reconstruction, that is, 3D helical parallel-beam reconstruction.

The curve 𝐶 in 𝑅³ defined by 𝐶=𝑆^∈𝑅³∶𝑂𝑆^=𝑦⃗(𝜑)=𝑟cos𝜑,𝑟sin𝜑,𝜑ℎ𝜑∈−𝜋2𝜋,2,𝜋2(65) is a segment of a helix with radius 𝑟 and pitch ℎ, which is shown in Figure 7 for 𝑟=1,ℎ=3.

Figure 7 

Motion process of the frequency plane. The red curve is a segment of a helical scanning trajectory. The blue curve is its projection on the unit sphere. The black curve is the great circle through 𝑆1,𝑆3, and 𝑆5. Five positions of the frequency plane are color coded. The points inside the triangle are scanned 3 times while the rest points are scanned only once.

In fact, it is formed by a combination of a uniform circular motion in the XOY plane and a uniform straight motion along the 𝑍 direction.

The projection of the curve 𝐶 on the unit sphere Ω is Γ=𝑆∈Ω∶𝑂𝑆=𝑒₃⃗(𝜑)=𝑦⃗(𝜑)||𝑦⃗(𝜑)||,𝜑∈−𝜋2,𝜋2,(66) or Γ=𝑆∈Ω∶𝑂𝑆=𝑒₃⃗(𝜗,𝜙),𝜗=tg⁻¹ℎ2𝜋𝜙,𝜙∈−𝜋2,𝜋2.(67) The curve length 𝜃 can be given by 𝑑𝜃=||𝑑𝑒₃⃗(𝜑)||=||||𝑑𝑦⃗(𝜑)𝜃(𝜑)=_{𝜑−𝜋/2}||𝑦⃗(𝜑)||||||,𝑑𝜃𝑑𝜑^𝑑𝜑^=^{𝜑−𝜋/2}|||||𝑑𝑦⃗𝜑_||𝑦⃗(𝜑^)|||||||.(68) For example, we can calculate 𝜃₀=𝜃(𝜋/2)≈3.177 and 𝜀=0.011, when 𝑟=1 and ℎ=3, indicating that the overlapping degree in the Fourier domain is small.

For every 𝜑∈[−𝜋/2,𝜋/2], the three orthogonal unit vectors are𝑒₃⃗=𝑂𝑆=𝑦⃗(𝜑)𝑒₁⃗||𝑦⃗(𝜑)||,=(𝑑/𝑑𝜑)𝑒³⃗𝑒₂⃗||(𝑑/𝑑𝜑)𝑒₃⃗||,=𝑒₃⃗×𝑒₁⃗.(69) The inverse Fourier transform can be expressed with either the Γ coordinates (𝜔₁,𝜔₂,𝜃) or the new coordinates (𝜔₁,𝜔₂,𝜑)Ψ𝑟⃗=^𝜃0₀^∞−∞^∞−∞×||𝜔𝑊𝜔₁,𝜔₂,𝜃exp2𝜋𝑖𝑘⃗⋅𝑟₁⃗Ψ𝑘⃗=||𝑑𝜔₁𝑑𝜔₂𝑑𝜃_{𝜋/2−𝜋/2}^∞−∞^∞−∞×Ψ𝑘⃗||𝜔𝑊𝜔₁,𝜔₂,𝜃(𝜑)exp2𝜋𝑖𝑘⃗⋅𝑟₁⃗||𝑑𝜃𝑑𝜑𝑑𝜔¹𝑑𝜔₂𝑑𝜑,(70) with 𝑘⃗=𝑘⃗𝜔₁,𝜔₂,𝜃(𝜑)=𝜔₁𝑒₁⃗(𝜃(𝜑))+𝜔₂𝑒₂⃗(𝜃(𝜑)).(71) To visualize the motion of the frequency plane ∏(𝑂𝑆), its five positions are marked in Figure 7 at 𝜑₁=−𝜋/2, 𝜑₂=−𝜋/4, 𝜑₃=0,𝜑₄=𝜋/4 and 𝜑₅=𝜋/2, respectively. Note that the initial position Π(𝑂𝑆₁) and the final position Π(𝑂𝑆₅) coincide with each other.

In Figure 7, the two curved sides of the red triangle on the top of the unit sphere is the locus of 𝑒₃⃗, the instantaneous rotation axis of the frequency plane. Similarly, −𝑒₂⃗ and Π(𝑂𝑆₁) form the counterpart triangle on the bottom of the unit sphere. The locus of 𝑒₂⃗ is the division line of the regions with different 𝐽(𝑘⃗) values. Based on (58), the total area of the two triangles is only 0.55% of the surface area of the unit sphere.

Note that a sharp point on the locus of 𝑒₂⃗ appears at 𝜑=𝜑₃=0. The reason is as follows.

Let us define𝜎₀(𝜑)=𝑦⃗(𝜑)⋅𝑑𝑑𝜑𝑦⃗(𝜑)×𝑑𝑑𝜑²²𝑦⃗(𝜑).(72) It can be verified that 𝜎₀(0)=0, 𝜎₀(𝜑)<0 for 𝜑<0, and 𝜎₀(𝜑)>0 for 𝜑>0.

It is not difficult to prove 𝜎==𝑒𝑦₃⃗⃗(𝜑⋅)𝑒⋅₁⃗×𝑑𝑒₁̇⃗𝑑𝜑𝑦⃗(𝜑)×𝑑𝑑𝜑²²𝑦⃗(𝜑)1×1||𝑦⃗(𝜑)||||(𝑑/𝑑𝜑)𝑦⃗(𝜑)||²𝑑𝑑𝜑𝜃³=𝜎₀(𝜑)𝐾(𝜑),(73) where 𝐾(𝜑) is positive. Therefore, 𝜎 and 𝜎₀ have the same zeros and sign.

Based on the motion process of Π(𝑂𝑆), we have 𝐽𝑘⃗=13,,𝑘𝑘⃗⃗oinustisdideeththeetwtwootrtirainanglgelse.s,(74) The overlapped region in the frequency space is due to the motion of the instantaneous rotation axis 𝑒₂⃗ of the frequency plane, or equivalently the discrepancy between the curve Γ and half a great circle.

As shown in Figure 8, the weight factor can be (1−1+1), (1/3+ 1/3+1/3), or (0+1+0), and so forth, for the overlapped region. By dividing the overlapped region into 2 symmetric parts, one can define the weight function as (−1+1+1) on the left side and (1+1−1) on the right side, or (1−1+1) on the left and (1+1−1) on the right, and so forth. Hence, one can design various parallel-beam reconstruction formulas and derive the associated cone-beam formulas using the approach described in [1].

(a)

(b)

(c)

(d)

(e)

(f)

Figure 8 

Examples of possible weight factors. (a) 1−1+1, (b) 1/3+1/3+1/3, (c) 0+1+0, and (d) −1+1+1 for the left half, 1+1−1 for the right half, (e) 1−1+1 for the left half, and 1+1−1 for the right half. The straight line represents a great circle on the unit sphere. The weight factor in (a) for ∏(𝑂𝑆2) is further illustrated in (f) as an example.

Appendices

A. Weighted Λ Reconstruction Formula

𝑓(𝑥,𝑦) is a function in the 2D space with the Fourier transform 𝑓𝑘₁,𝑘₂=^∞−∞^∞−∞𝑓(𝑥,𝑦)exp−𝑖2𝜋𝑘₁𝑥exp−𝑖2𝜋𝑘₂𝑦𝑑𝑥𝑑𝑦.(A.1) Its parallel-beam projection is denoted as 𝑃_𝜗(𝑙)=^∞−∞𝑓(𝑙cos𝜗−𝑣sin𝜗,𝑙sin𝜗+𝑣cos𝜗)𝑑𝑣(A.2) with 𝑙∈(−∞,∞),𝜗∈[−𝜋/2,𝜋/2].

Let 𝑓(𝑥,𝑦) pass through the filter |𝑘₁|, we get a new function Φ=(𝑥,𝑦)_∞−∞^∞−∞||𝑘₁||𝑓𝑘₁,𝑘₂exp𝑖2𝜋𝑘₁𝑥exp𝑖2𝜋𝑘₂𝑦𝑑𝑘₁𝑑𝑘₂=𝑓(𝑥,𝑦)∗1−2𝜋²𝑥².(A.3) Defining the signed polar coordinates (𝑘_𝜌,𝜗) with 𝑘_𝜌∈(−∞,∞),𝜗∈(−𝜋/2,𝜋/2), which satisfy 𝑘₁=𝑘_𝜌cos𝜗,𝑘₂=𝑘_𝜌sin𝜗, we haveΦ(𝑥,𝑦)=^{𝜋/2−𝜋/2}^∞−∞×ex||𝑘p₁||−𝑓𝑖2𝑘𝜋_𝜌𝑘,_𝜌𝜗=(𝑥cos𝜗+𝑦sin𝜗)||𝑘_𝜌||𝑑𝑘_𝜌𝑑𝜗_{𝜋/2−𝜋/2}^∞−∞×𝑓𝑘_2𝜌𝑘_𝜌,𝜗exp−𝑖2𝜋𝑘_𝜌(𝑥cos𝜗+𝑦sin𝜗)=1cos𝜗𝑑𝑘_𝜌𝑑𝜗−4𝜋²^{𝜋/2−𝜋/2}𝜕²𝜕𝑙²||||_{𝑙=𝑥cos𝜗+𝑦sin𝜗}^∞−∞×exp−𝑖2𝜋𝑘_𝜌𝑓𝑘_𝜌,𝜗=1𝑙𝑑𝑘_𝜌cos𝜗𝑑𝜗−4𝜋²^{𝜋/2−𝜋/2}𝜕²𝜕𝑙²||||_{𝑙=𝑥cos𝜗+𝑦sin𝜗}𝑃_𝜗(𝑙)cos𝜗𝑑𝜗.(A.4) Comparing (A.3) and (A.4), we have𝑓(𝑥,𝑦)∗1−2𝜋²𝑥²=1−4𝜋²^{𝜋/2−𝜋/2}𝜕²𝜕𝑙²||||_{𝑙=𝑥cos𝜗+𝑦sin𝜗}𝑃_𝜗(𝑙)cos𝜗𝑑𝜗.(A.5) That is, we have rediscovered the identity (20) in a general way. In contrast to the classical Λ reconstruction formula [1], a weight factor cos𝜗 is involved in (A.5). Hence, we can call (A.5) a weighted Λ reconstruction formula.

B. A Weighted Radon Formula and the Orlov Formula

Let us define 𝑢=𝑟⃗⋅𝑒₃⃗,𝑡=𝑟⃗⋅𝑒₁⃗,𝑠=𝑟⃗⋅𝑒₂⃗,(B.1) and the parallel-beam projection is 𝑃_𝜃(𝑡,𝑠)=^∞−∞Ψ𝑢𝑒₃⃗+𝑡𝑒₁⃗+𝑠𝑒₂⃗𝑑𝑢,(B.2) as shown in Figure 6(b). In the frequency plane Π(𝑂𝑆), we define a signed polar coordinate system (𝜔,𝛼) with 𝜔∈(−∞,∞),𝛼∈[0,𝜋] for a point (𝜔₁,𝜔₂) with 𝜔₁=𝜔cos𝛼,𝜔₂=𝜔cos𝛼. Similarly, let us define the signed polar coordinates (𝑋,𝛽) with 𝑋∈(−∞,∞),𝛽∈[−𝜋/2,𝜋/2] for a point (𝑡,𝑠) with 𝑡=𝑋cos𝛽,𝑠=𝑋sin𝛽.

Let the weight function in (32) be 𝑊(𝜔₁,𝜔₂,𝜃)=𝑊(𝛼,𝜃), we have Ψ=𝑟_𝜃00⃗^∞−∞^∞−∞×||𝜔𝑊𝜔₁,𝜔₂,𝛼exp2𝜋𝑖𝑘⃗⋅𝑟₁⃗Ψ𝑘⃗=||𝑑𝜔₁𝑑𝜔₂𝑑𝜃_𝜃00^𝜋0^∞−∞×Ψ𝑊𝜔(₁𝛼,𝜃)|cos𝛼|exp2𝜋𝑖𝜔₁𝑡+2𝜋𝑖𝜔₂𝑠=,𝜔₂,𝜃𝜔²𝑑𝜔𝑑𝛼𝑑𝜃_𝜃00^𝜋0=−1𝑊(𝛼,𝜃)|cos×𝛼|Ψ(^∞−∞𝜔,𝜔𝛼²,e𝜃x)p𝑑𝜔(2𝑑𝜋𝛼𝑖𝑑𝜔𝜃(𝑡cos𝛼+𝑠sin𝛼))4𝜋²^𝜃0₀^𝜋0𝑊(𝛼,𝜃)𝜕²𝜕𝑙²||||_{𝑙=⃗𝑟⋅⃗𝑛=𝑡cos𝛼+𝑠sin𝛼}=−1×|cos𝛼|𝑑𝛼𝑑𝜃𝑅Ψ𝑙,𝑛⃗4𝜋²∇²^𝜃0₀^𝜋0×|co𝑊s𝛼(|𝛼𝑑,𝛼𝜃𝑑)𝜃𝑅,Ψ𝑡cos𝛼+𝑠sin𝛼,𝑛⃗(B.3) with 𝑛⃗=𝑒₁⃗cos𝛼+𝑒₂⃗sin𝛼.

We call it the weighted Radon formula. When Γ_AB is a great semicircle, we have 𝑊(𝛼,𝜃)=1 and 𝜃₀=𝜋, and the above formula becomes the classic Radon formula [12, 13].

Moreover, it is easy to rederive Orlov's formula [14] from (32) Ψ𝑟⃗=^𝜃0₀^∞−∞^∞−∞×||𝜔𝑊𝜔₁,𝜔₂,𝜃exp2𝜋𝑖𝑘⃗⋅𝑟₁⃗Ψ𝑘⃗=||𝑑𝜔₁𝑑𝜔₂𝑑𝜃_𝜃00^∞−∞^∞−∞×Ψ𝑊𝜔(𝛼₁,𝜃)exp2𝜋𝑖𝜔₁𝑡+2𝜋𝑖𝜔₂𝑠=,𝜔₂,𝜃||𝜔₁||𝑑𝜔₁𝑑𝜔₂𝑑𝜃_𝜃00^∞−∞^∞−∞−14𝜋²𝜕𝜕𝑡²²+𝜕²×exp2𝜋𝑖𝜔₁𝜕𝑠²𝑊(𝛼,𝜃)𝑡+2𝜋𝑖𝜔₂𝑠Ψ𝜔₁,𝜔₂,𝜃||𝜔¹||𝜔²=−1𝑑𝜔₁𝑑𝜔₂𝑑𝜃4𝜋²∇²^𝜃0₀0𝑥0200𝑑^∞−∞^∞−∞|cos𝛼||𝜔|𝑊(𝛼,𝜃)⋅Ψ𝜔¹×exp2𝜋𝑖𝜔₁,𝜔₂,𝜃=−1𝑡+2𝜋𝑖𝜔₂𝑠𝑑𝜔₁𝑑𝜔₂𝑑𝜃4𝜋²∇²^𝜃0₀𝑝_𝜃(𝑡,𝑠)^∗∗𝑔_𝜃(𝑡,𝑠)𝑑𝜃,(B.4) where 𝑔_𝜃(=𝑡,𝑠)_∞−∞^∞−∞|cos𝛼||𝜔|𝑊(𝛼,𝜃)exp2𝜋𝑖𝜔¹𝑡+2𝜋𝑖𝜔₂𝑠𝑑𝜔₁𝑑𝜔₂=|cos𝛼|||𝑋||𝑊(𝛼,𝜃)||^{𝛼=𝜋/2+𝛽}.(B.5) The sign ∗∗ denotes the 2D convolution operation.

We recall that a frequency point has 𝐽(k)⃗ sets of Γ coordinates 𝑘⃗==⋯𝑘⃗¹=𝜔𝑘₁,⃗¹𝜔₂,¹_{𝐽⃗𝑘}𝜃=𝑘⃗²𝜔₁,²𝜔₂,²𝜃𝜔₁,^{𝐽⃗𝑘}𝜔₂,^{𝐽⃗𝑘}𝜃.(B.6) If a given triplet (𝜔₁,𝜔₂,𝜃) is the 𝑗th set of Γ coordinates of 𝑘⃗, 𝜔₁,𝜔₂,𝜃=^𝑗𝜔₁,^𝑗𝜔₂,^𝑗𝜃,(B.7) a new normalized weight function [1] for the triplet can be defined by 𝑊𝜔₁,𝜔₂,𝜃=𝑊(𝛼,𝜃)=1/||𝑘⃗⋅𝑒¹⃗^𝑗𝜃||=1/|cos∑𝛼|^{⃗𝐽(𝑘)𝑗=1}1/||𝑘⃗⋅𝑒₁⃗^𝑗𝜃||∑^{𝑘𝐽(0)𝑗=1}1/||𝑘⁰⋅𝑒₁⃗^𝑗𝜃||,(B.8) and in this case we have 𝑔_𝜃(𝑡,𝑠)=1||𝑋||∑^{𝑘𝐽(0)𝑗=1}1/||𝑘1₀⋅𝑒₁⃗^𝑗𝜃||||||||_{𝑘0=⃗𝑒3×𝑋0}.(B.9) Here, 𝑘⁰=𝑘⃗/|𝑘⃗|, and 𝑋⁰=(𝑡𝑒₁⃗+𝑠𝑒₂⃗)/√𝑡²+𝑠²=𝑒₁⃗cos𝛽+𝑒₂⃗sin𝛽 are unit vectors.

Equation (B.4) with (B.9) is the Orlov formula for a general curve on the unit sphere ((2a) with (2c) in [14]). Note that in [14] different letters were used such as 𝑚↔𝑘⁰,𝜏↔𝑒₃⃗,𝜏̇↔𝑒₁⃗,𝑥^↔(𝑡𝑒₁⃗+𝑠𝑒₂⃗),|𝑥^|↔|𝑋|,∫_𝐺𝑑𝐺_𝜏↔∫^𝜃0₀𝑑𝜃, and so forth, and 𝜏̇^𝑘 should be printed as 𝜏̇_𝑘.

C. Link between the Motion of the Frequency Plane and Differential Geometry

To study the property of a curve Γ on the unit sphere and the associated motion of the frequency plane, one can introduce another curve, Γ, defined as 𝑟⃗(𝜃)=^𝜃0𝑒₃⃗𝜃^𝑑𝜃^+𝑟⃗(0),(C.1) where 𝑟(⃗0) is an arbitrary vector in the 3D space, and 𝜃 (as well as 𝜃^) is an arc-length parameter of the curve Γ. Clearly, 𝑟̇⃗=𝑑𝑑𝜃𝑟⃗(𝜃)=𝑒³⃗,𝑟̈⃗=𝑒₁⃗,𝑟⃛⃗=𝑒₁̇⃗.(C.2) Suppose that the arc-length parameter of the curve Γ is 𝑠. The tangential direction of the curve Γ is T⃗=𝑑𝑑𝑠𝑟⃗=𝑑𝑑𝜃𝑠𝑑𝑑𝜃𝑟⃗(𝜃)=𝑑𝑑𝜃𝑠𝑒³⃗(𝜃).(C.3) Furthermore, we have |||𝑑𝑑𝑠𝑟|||⃗=|||𝑑𝑑𝜃𝑠|||||𝑒³⃗1=|||𝑑𝜃(𝜃)||,𝑑𝑠|||,(C.4) where |(𝑑/𝑑𝑠)𝑟|⃗=|𝑑𝑟/⃗𝑑𝑠|=|𝑑𝑟|⃗/𝑑𝑠=𝑑𝑠/𝑑𝑠=1 has been used.

When the positive directions are the same, we have 𝑑𝜃𝜃𝑑𝑠==𝑠+1,𝑠₀T⃗=𝑒₃⃗,(𝜃).(C.5) Picking up the constant 𝑠₀=0, we have 𝜃=𝑠. Therefore, the symbol 𝜃 represents the arc length parameters of both Γ and Γ. Based on the tangential direction T⃗, the norm and binormal unit vectors [15] can be defined as N⃗=Ṫ⃗|||Ṫ⃗|||=||𝑒𝑒¹₁⃗⃗||=𝑒¹⃗B⃗=T⃗×N⃗=𝑒₃⃗,×𝑒₁⃗=𝑒₂⃗.(C.6) The curvature 𝜅 and the torsion 𝜏 of the curve Γ are 𝜏=𝑟̇⃗⋅𝑟𝜅̈⃗×=𝑟⃛⃗||𝑟=||̈⃗=𝑒₃⃗||𝑒₁⃗||=1,⋅𝑒₁⃗×𝑒₁̇⃗=𝜎,(C.7) and the Darboux vector is Ω⃗=𝜏T⃗+𝜅B⃗=𝑒₂⃗+𝜎𝑒₃⃗.(C.8) The Frenet-Serret Equations of the curve ΓNBṪ̇⃗̇⃗⃗===−−N⃗𝜎T,⃗N+⃗,𝜎B⃗,(C.9) and NBṪ̇⃗̇⃗⃗===ΩΩΩ⃗⃗⃗×××TNB⃗⃗,⃗,,(C.10)

are (39) and (45). In this frame, 𝑟(⃗𝜃) is the original curve (function), and 𝑒₃⃗(𝜃) becomes its derived curve (function). In practice, it is difficult to find the physical meaning of the 3D curve 𝑟(⃗𝜃).

Example C. A circle on the unit sphere and its corresponding curve in the 3D space.

As shown in Figure 4(b), a circle on the unit sphere, marked as 𝐶, can be described as 𝑒₃⃗=sin𝜗₀k⃗+cos𝜗₀cos𝜙i⃗+cos𝜗₀sin𝜙j⃗(C.11) with a fixed parameter 𝜗₀∈(−𝜋/2,+𝜋/2) and a variable 𝜙∈[0,2𝜋].

Using the arc-length parameter 𝜃∈[0,2𝜋cos𝜗₀], we have 𝑒₃⃗(𝜃)=sin𝜗₀k⃗+cos𝜗₀cos𝜃cos𝜗₀i⃗+cos𝜗₀sin𝜃cos𝜗₀j.⃗(C.12) Furthermore, we have 𝑒₁⃗(𝜃)=−sin𝜃cos𝜗₀i⃗+cos𝜃cos𝜗₀𝑒₁̇⃗j,⃗(𝜃)=1cos𝜗₀cos𝜃cos𝜗₀i⃗−1cos𝜗₀sin𝜃cos𝜗₀𝜎=𝑒₃⃗j,⃗Ω⃗=𝑒₂⃗⋅𝑒₁⃗×𝑒₁̇⃗=tg𝜗₀,+𝜎𝑒₃⃗=1cos𝜗₀k.⃗(C.13) Based on the definition (C.1), the corresponding 3D curve 𝐶 is 𝑟⃗(𝜃)=𝜃sin𝜗₀k⃗+cos²𝜗₀sin𝜃cos𝜗₀−cos₂i⃗𝜗₀cos𝜃cos𝜗₀j,⃗(C.14) with the choice 𝑟(⃗0)=−𝑗⃗cos²𝜗₀.

Now, we see that 𝐶 is a helix, whose standard form is 𝑟⃗(𝜙)=𝑏𝜙k⃗+𝑎cos𝜙−𝜋2i⃗+𝑎sin𝜙−𝜋2j,⃗(C.15) with 𝑎=cos²𝜗₀,𝑏=sin𝜗₀cos𝜗₀,𝜙∈[0,2𝜋] (see Figure 4(c)). Its curvature and torsion are 𝜅=1,𝜏=tg𝜗₀.(C.16) The length of the helix 𝐶 for 𝜙∈[0,2𝜋] is 2𝜋√𝑎²+𝑏²=2𝜋cos𝜗₀, which is equal to the circumference of circle 𝐶 on the unit sphere. The radius of the helix 𝐶(𝑎=cos²𝜗₀) is smaller than that of the circle 𝐶(cos𝜗₀). The figure is not drawn to scale. The TNB frame rotates around the 𝑧 direction at a constant velocity Ω⃗=2𝜋2𝜋cos𝜗₀k⃗=1cos𝜗₀k.⃗(C.17)

Acknowledgment

The works of H. Yu and G. Wang are partially supported by the NSF/MRI Program (no. CMMI-0923297) and NIH/NIBIB Grant (no. EB011785). The authors also thank a reviewer for a valuable suggestion.