Supplemental Table and Figure
Supplemental Table 1. Image similarity measures used for assessment of spatial
quality of iGRASP images (IG) in comparison with the reference images (IR).
Supplemental Figure 1. Evaluation of spatial quality using image similarity measures
defined in Supplemental Table 1; (a) mean square difference (MSD), (b) intensity
correlation coefficient (ICC), (c) gradient difference (GDF), (d) gradient correlation
coefficient (GCC), (e) mutual information (MI), and (f) mean structural similarity
(MSSIM). Each measure is normalized by its maximum value (minimum for MSD) prior
to calculating the mean and standard deviation.
Supplemental Figure 2. Comparison of images reconstructed with different λ values.
All images were reconstructed using 34 spokes/frame, except the reference image
which was reconstructed using 380 spokes. The images were from a 41-year-old
woman with a 1.6 cm oval mass with irregular margins. Biopsy yielded an invasive
ductal carcinoma, poorly differentiated.
1
Supplemental Table 1. Image similarity measures used for assessment of spatial quality of
iGRASP images (IG) in comparison with the reference images (IR).
Similarity
Measure
Definition
Mean squared
difference
𝑀𝑆𝐷 =
10-12
∑(𝑥,𝑦)(𝐼𝐺 (𝑥, 𝑦) − 𝐼̅𝐺 )(𝐼𝑅 (𝑥, 𝑦) − 𝐼̅𝑅 )
𝐼𝐶𝐶 =
√∑(𝑥,𝑦)(𝐼𝐺 (𝑥, 𝑦) − 𝐼̅𝐺 )2 √∑(𝑥,𝑦)(𝐼𝑅 (𝑥, 𝑦) − 𝐼̅𝑅 )2
Gradient
difference
𝐺𝐷𝐹 = ∑
𝑥,𝑦
𝐺𝐶𝐶 =
𝐴𝑣
𝐴𝑣 + (𝐼𝑑𝑉 (𝑥, 𝑦))
2
̅̅̅̅
̅̅̅̅
∑(𝑥,𝑦)(𝐻𝐺 (𝑥,𝑦)−𝐻
𝐺 )(𝐻𝑅 (𝑥,𝑦)−𝐻
𝑅)
2
2
̅̅̅̅
̅̅̅̅
√∑(𝑥,𝑦)(𝐻(𝑥,𝑦)−𝐻
𝐺 ) √∑(𝑥,𝑦)(𝐻𝑅 (𝑥,𝑦)−𝐻
𝑅)
Mutual
information
Mean structural
similarity
1
2
∑(𝐼𝐺 (𝑥, 𝑦) − 𝐼𝑅 (𝑥, 𝑦))
𝑁
𝑥,𝑦
Intensity crosscorrelation
Gradient crosscorrelation
Ref.
+∑
𝑥,𝑦
+
𝑀𝐼 = ∑ 𝑃𝐺𝑅 (𝑔, 𝑟) 𝑙𝑜𝑔
𝑔,𝑟
𝑀𝑆𝑆𝐼𝑀 =
𝐴ℎ
2
𝐴ℎ + (𝐼𝑑𝐻 (𝑥, 𝑦))
11-14
11,15
̅̅̅̅
̅̅̅̅
∑(𝑥,𝑦)(𝑉𝐺 (𝑥,𝑦)−𝑉
𝐺 )(𝑉𝑅 (𝑥,𝑦)−𝑉
𝑅)
2
2
̅̅̅̅
̅̅̅̅
√∑(𝑥,𝑦)(𝑉(𝑥,𝑦)−𝑉
𝐺 ) √∑(𝑥,𝑦)(𝑉𝑅 (𝑥,𝑦)−𝑉
𝑅)
𝑃𝐺𝑅 (𝑔, 𝑟)
𝑃𝐺 (𝑔)𝑃𝐺𝑅 (𝑟)
(2𝜇𝐺 (𝑥, 𝑦)𝜇𝑅 (𝑥, 𝑦) + 𝐶1 )(2𝜎𝐺𝑅 (𝑥, 𝑦) + 𝐶2 )
1
∑ 2
(𝜇𝐺 (𝑥, 𝑦) + 𝜇𝑅2 (𝑥, 𝑦) + 𝐶1 )(𝜎𝐺2 (𝑥, 𝑦) + 𝜎𝑅2 (𝑥, 𝑦) + 𝐶2 )
𝑁
16-17
18-22
23-25
𝑥,𝑦
Note: N, the number of pixels in each image; x and y, column and row of pixel location; 𝐼̅𝐺 and
𝐼̅𝑅 , the mean values of the images; H(x,y) and V(x,y), the horizontal and vertical gradient images
using Sobel operators; IdH(x,y) and IdV(x,y), difference images of horizontal gradient images
(HG(x,y) - HR(x,y)) or vertical gradient images; Av and Ah, the variances of IdV and IdH; PGR(g,r),
joint probability distribution of intensity values of iGRASP (G) and reference (R) images; PG(g)
and PR(r), marginal probability distributions; (x,y) and (x,y), local mean and standard
deviation weighted by an 1111 circular-symmetric Gaussian weighting function with standard
deviation of 1.5 pixels, normalized to unit sum; C1 and C2, constants to avoid unstability,
(0.01255)2 and (0.03255)2, respectively, in this study.
2
Supplemental Figure 1. Evaluation of spatial quality using image similarity measures
defined in Supplemental Table 1; (a) mean square difference (MSD), (b) intensity
correlation coefficient (ICC), (c) gradient difference (GDF), (d) gradient correlation
coefficient (GCC), (e) mutual information (MI), and (f) mean structural similarity
(MSSIM). Each measure is normalized by its maximum value (minimum for MSD) prior
to calculating the mean and standard deviation.
3
Supplemental Figure 2. Comparison of images reconstructed with different λ values.
All images were reconstructed using 34 spokes/frame, except the reference image
which was reconstructed using 380 spokes. The images were from a 41-year-old
woman with a 1.6 cm oval mass with irregular margins. Biopsy yielded an invasive
ductal carcinoma, poorly differentiated.
4