Null functions in three-dimensional imaging of alpha and beta particles

Null functions of an imaging system are functions in the object space that give exactly zero data. Hence, they represent the intrinsic limitations of the imaging system. Null functions exist in all digital imaging systems, because these systems map continuous objects to discrete data. However, the emergence of detectors that measure continuous data, e.g. particle-processing (PP) detectors, has the potential to eliminate null functions. PP detectors process signals produced by each particle and estimate particle attributes, which include two position coordinates and three components of momentum, as continuous variables. We consider Charged-Particle Emission Tomography (CPET), which relies on data collected by a PP detector to reconstruct the 3D distribution of a radioisotope that emits alpha or beta particles, and show empirically that the null functions are significantly reduced for alpha particles if ≥3 attributes are measured or for beta particles with five attributes measured.

. Cross-section view of 3D eigenfunctions in 2D BET (left) and 5D BET (right). The eigenfunctions are with indices (ρ x = 0.01, ρ y = 0) µm −1 , and j = 1, 2, 10 (Row 1, 2, 3, respectively). The scaled eigenvalues (λ, introduced in the main text 1 ) corresponding to each eigenfunction are presented on the lower right corner of each plot. The plots show only the center of the object space.

B Additional SVD analysis results of BET
In BET, structures of interest are labeled with fast-electron-emitting radionuclides; by imaging the fast electrons, information about the structure of interest can be reconstructed. Fast electrons include beta particles, conversion electrons and Auger 15 electrons. Beta particles have broad energy spectra. Conversion electrons and Auger electrons have discrete energy spectra. For beta particles, the energy spectra of electrons and positrons are different due to the effect of the nuclear Coulomb field on the wave function of the beta particles. We are interested in comparing the effects of different energy spectra on the performance of BET. For the comparison, we choose three sources 20 which include 18 F source, 131 I source and a monoenergetic source that emits 400-keV electrons. The isotope 18 F emits positrons (beta plus), 131 I emits electrons (beta minus). The end-point energy of the two isotopes are similar. We choose the energy of the mono-energetic source at 400 keV, which is about 2/3 the end-point energy of 18 F, for convenient 25 comparison of the results obtained with different sources. The spectra of 18 F and 131 I are presented in Figure 2.
The SVD analysis results of BET with an 18 F source are presented in the main text 1 . In this supplement, we present SVD analysis results of BET with a 400-keV-monoenergetic source and BET with an 131 I source. 30 We use the same method, simulation parameters, bin size and test object presented in the paper for BET with 18 F source.
The eigenfunctions that contributed to the measurement components corresponds to eigenvalues above a threshold. We set the threshold as λ min = 10 −6 . The results from BET with a source that emits 400-keV 35 electrons are presented in Figure 3; and the results with the 131 I source are presented in Figure 4.  The coronal cross-section view (xz-plane) of the object (Column 1), measurement components (Column 2) and null components (Column 3) and eigenvalue spectra in BET systems where 400-keV mono-energetic electron sources are used. The results are with particle-processing detectors that measure 2D (Row 1), 3D (Row 2), 4D (Row 3) and 5D (Row 4) information about each detected particle, respectively. The color scale represents the radioactivity of the object. The cutoff for λ n is 10 −6 .
The results with the beta-minus source ( 131 I) and the mono-energetic source (400-keV) are similar to the results with

Figure 4.
The coronal cross-section view (xz-plane) of the object (Column 1), measurement components (Column 2) and null components (Column 3) and eigenvalue spectra in 131 I BET systems with particle-processing detectors that measure 2D (Row 1), 3D (Row 2), 4D (Row 3) and 5D (Row 4) information about each detected particle, respectively. The color scale represents the radioactivity of the object. The cutoff for λ n is 10 −6 . (Results of 18 F are presented in the main text 1 .) beta-plus source ( 18 F) presented in the main text. As the number of attributes measured by the detector increases, the null space reduces. For BET with 131 I source, when the detector measures 5D information, including detection position (x d , y d ), 40 propagation direction (s x , s y ), and residual energy E, the null space is almost completely eliminated. For BET with the 400-keV mono-energetic source, when the detector measures 5D information, part of the object buried deeper in tissue is in the null space. That part of object cannot be recovered even with noise-free data. This might be due to the absence of particles with energy higher than 400-keV. The scaled norm of the null functions ( f null s , introduced in the main text 1 ) vs. the number of attributes measured by the detector is plotted in Figure 5. The norm reduces as the number of measured attributes increases. When q = 3 or 4, the scaled norm of the null functions of BET with 400-keV mono-energetic sources is smaller than that with 131 I sources; but when q = 5, f null s for a 400-keV mono-energetic source is larger than that with a 131 I source. One possible explanation is that: the discreteness of the energy spectra of mono-energetic sources helps in reconstructing the source location when (1) position and energy or (2) position and direction are measured in BET; however, when position, energy and position are all measured for 50 each detected particle, the maximum energy emitted by the source plays a bigger role in determining how well an object can be reconstructed.
In conclusion, the different shape in energy spectra from beta-minus decay and beta-plus decay does not lead to significantly different performance of BET with beta-minus source and beta-plus source. However, the particles emitted with higher energy contributes to the resolution deeper in tissue.

C SVD analysis of αET
We present theoretical calculations for αET with 2D, 3D, 4D, and 5D detectors in this supplemental material.
From Equation (13) in the main text, the kernel function of L † L follows where prf(Â|r, z) is the system response to a point source located at (r, z) andÂ is a vector formed by the attributes measured by the system. The attributes includes detection position represented byr d = (x d ,ŷ d ), propagation direction represented by (ŝ x ,ŝ y ), and residual energyÊ.

60
The symbol A represents a vector of the true underlying attributes; whileÂ represents a vector of the measured attributes, which contains detector estimation uncertainty. In this paper, we focus on the intrinsic limitation of αET systems and temporarily ignore the detector noise. Therefore, we useÂ and A interchangeably in the following discussion. Figure 6. Illustration of the coordinate system of an αET system.

C.1 Point response functions
In a typical αET setup, alpha-particle-emitting sources are in a layer of tissue and a detector is placed at plane z = 0, as illustrated in Figure 6. Alpha particles travel in almost straight lines and lose energy continuously along their tracks 2 . The maximum distance a particle can travel in tissue, which we denote as l 0 , is a function of the emission energy of the particle. With a point source, the alpha particles can be detected only in a circle determined by the source location and 70 the emission energy 3,4 .
To describe the point response functions for αET systems, we assume the coordinate system as illustrated in Figure 6, where a particle is emitted from point (r, z) in tissue and detected at point r d in the detector. We define the displacement between the detection location and the projection of the emission 75 point on the detector plane as ∆r = r d − r. We denote the distance from the emission position to the detection location is denoted as l, where l = |∆r| 2 + z 2 and |∆r| is the magnitude of the vector ∆r.
With this notation, the point response function prf(A|r, z) for αET with a particle-processing detector that measures q-D information for q = {2, 3, 4, 5} are: where cyl(r) is a cylinder function and win(t) is a window function. The cylinder function is unity inside a disc of radius 1 and zero outside: The window function is defined by In the following discussion, we consider depth z < l 0 , so that we can leave out the factor win(z/l 0 ) for now.

C.2 2D detector
For a 2D detector, the 2D Fourier transform of a rotational symmetric function is the zero-th order Hankel transform 5 . The 2D Fourier transform of z/4πl 3 with respect to ∆r is 6 where ρ is the magnitude of ρ. The 2D Fourier transform of a cylinder function is a besinc function 5 , The besinc function is defined as where J 1 (·) is the first-order Bessel function of the first kind 7 . The besinc function is also referred to as a jinc or sombrero function. For αET with a 2D detector, the 2D Fourier transform of prf(A|r, z) with respect to ∆r is where the symbol * represents convolution with respect to the 2D spatial frequency ρ. The kernel function of L † L is k(r − r , z, z ) = τ 2 d 2 r d prf(r d |r, z) prf(r d |r , z ).
Since prf(r d |r, z) = prf(r d −r|z) = prf(r−r d |z), the kernel function is a convolution of the two point response functions. Therefore, the 2D Fourier transform of function k(r − r , z, z ) with respect to (r − r ) is where the superscript * represents complex conjugate. The function K(ρ, z, z ) can be evaluated numerically.
We first evaluate k(r − r , z, z ) numerically, and then calculate K(ρ, z, z ) through Hankel transform.

C.4 4D detector
With a 4D detector that measures position and direction (x d , y d , s x , s y ), the kernel function of L † L operator is