Inversion of weighted Radon transforms via finite Fourier series weight approximations

Figures & data

Fig. 1 (a) Attenuation map $a=a(x)$, (b) emiter activity $f=f(x)$, (c) noiseless emission data $g=P_{W}f(\mathit{\gamma })$, (d) noisy emission data $p=p(\mathit{\gamma })$ (30% noise) for phantom 1. (See Subsections 3.1, 3.3)

Fig. 2 (a) Attenuation map $a=a(x)$, (b) emiter activity $f=f(x)$, (c) noiseless emission data $g=P_{W}f(\mathit{\gamma })$, (d) noisy emission data $p=p(\mathit{\gamma })$ (30% noise) for phantom 2. (See Subsections 3.1, 3.3)

Table 1 Numbers ${\mathit{\sigma }}_{W,D,m}$, ${\mathit{\rho }}_{W,D,m}$ of (2.11), (2.30) for phantom 1, where $D={\mathcal{B}}_R$.

	$m=1$	$m=2$	$m=3$	$m=20$
${\mathit{\sigma }}_{W,D,m}$	0.390	0.584	0.739	1.320
${\mathit{\rho }}_{W,D,m}$	1.399	2.025	2.494	4.532

Table 2 Numbers ${\mathit{\sigma }}_{W,D,m}$, ${\mathit{\rho }}_{W,D,m}$ of (2.11), (2.30) for phantom 2, where $D={\mathcal{B}}_R$.

	$m=1$	$m=2$	$m=3$	$m=20$
${\mathit{\sigma }}_{W,D,m}$	0.489	0.694	0.803	1.296
${\mathit{\rho }}_{W,D,m}$	3.112	4.567	5.436	9.719

Table 3 Relative reconstruction errors $\mathit{\eta }(f_m,f,X),\mathit{\eta }(f_m^{+},f,X)$ for the noiseless case for phantom 1.

	$m=0$	$m=1$	$m=2$	$m=3$	$m=20$
$\mathit{\eta }(f_m,f,X)$	0.331	0.305	0.295	0.293	0.288
$\mathit{\eta }(f_m^{+},f,X)$	0.331	0.305	0.295	0.293	0.287

Table 4 Relative reconstruction errors $\mathit{\eta }(f_m,f,X),\mathit{\eta }(f_m^{+},f,X)$ for the noiseless case for phantom 2.

	$m=0$	$m=1$	$m=2$	$m=3$	$m=20$
$\mathit{\eta }(f_m,f,X)$	0.292	0.179	0.152	0.141	0.137
$\mathit{\eta }(f_m^{+},f,X)$	0.168	0.151	0.138	0.136	0.137

Fig. 3 Phantom 1:(a) true f, (b), (c) approximate reconstructions $f_0$, $f_2$ from the noiseless data g, (d), (e), (f) central horizontal profiles. (See Subsections 3.3, 3.5)

Fig. 4 Phantom 2:(a) true f, (b), (c) approximate reconstructions $f_0$, $f_2$ from the noiseless data g, (d), (e), (f) central horizontal profiles. (See Subsections 3.3, 3.5)

Fig. 5 Phantom 1: (a) true f, (b), (c) approximate reconstructions $f_0$, $f_2$ from the filtered noisy data $\stackrel{\sim }{p}={\mathcal{A}}_{8,8}^{\mathit{sym}}p$ (30% noise), (d), (e), (f) central horizontal profiles. (See Subsections 3.3, 3.5)

Fig. 6 Phantom 2: (a) true f, (b), (c) approximate reconstructions $f_0$, $f_2$ from the filtered noisy data $\stackrel{\sim }{p}={\mathcal{A}}_{8,8}^{\mathit{sym}}p$ (30% noise), (d), (e), (f) central horizontal profiles. (See Subsections 3.3, 3.5)

Table 5 Relative errors $\mathit{\eta }(f_m,f,X),\mathit{\eta }(f_m^{+},f,X)$ for $f_m$ reconstructed from $\stackrel{\sim }{p}={\mathcal{A}}_{8,8}^{\mathit{sym}}p$ for phantom 1.

	$m=0$	$m=1$	$m=2$	$m=3$	$m=20$
$\mathit{\eta }(f_m,f,X)$	0.398	0.380	0.374	0.373	0.368
$\mathit{\eta }(f_m^{+},f,X)$	0.398	0.379	0.373	0.371	0.366

Table 6 Relative errors $\mathit{\eta }(f_m,f,X),\mathit{\eta }(f_m^{+},f,X)$ for $f_m$ reconstructed from $\stackrel{\sim }{p}={\mathcal{A}}_{8,8}^{\mathit{sym}}p$ for phantom 2.

	$m=0$	$m=1$	$m=2$	$m=3$	$m=20$
$\mathit{\eta }(f_m,f,X)$	0.268	0.218	0.218	0.225	0.248
$\mathit{\eta }(f_m^{+},f,X)$	0.214	0.209	0.213	0.221	0.245