Since each grid square is crossed by exactly one line from each family, we can organize the calculation, in our example, by tabulating the numbers family by family, and then adding up the tables. These tables are precisely the arrays given on the previous page. Here each of the tables is labelled by the slope of the corresponding familty of lines; as before, one of those lines is picked put in red.
horizontal X stands for 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0_0_0_0 0 0 0_0_0_0 0 0 0 8 8 8|8 8 8 8|8 8|8 8 8 8|8 8 8 8 8 8|8 8_8_8|8 8|8_8_8 8|8 8 8 4 4 4|4 4|4 4 4 4 4 4|4 4|4 4 4 4 4 4|4 4|4 4 4 4 4 4|4 4|4 4 4 4 4 4|4 4|4_4_4_4_4_4|4 4|4 4 4 X X X|X X X X X X X X X X|X X X X X X|X_X_X_X_X_X_X_X_X_X|X X X 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
slope = 1/2 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 3 0 0 0 0_0_1_1 2 2 2_2_2_2 3 3 2 0 0 0|1 1 2 2|2 2|2 2 3 3|2 2 3 0 1 1|2 2_2_2|2 2|3_3_2 2|3 3 3 1 2 2|2 2|2 2 3 3 2 2|3 3|3 3 3 2 2 2|2 2|3 3 2 2 3 3|3 3|3 3 2 2 2 2|3 3|2_2_3_3_3_3|3 3|2 2 1 2 3 3|2 2 3 3 3 3 3 3|2 2|1 1 0 3 2 2|3_3_3_3_3_3_2_2_1_1|0 0 0 2 3 3 3 3 3 3 2 2 1 1 0 0 0 0 0 3 3 3 3 3 2 2 1 1 0 0 0 0 0 0 0 3 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 |
slope = 1 0 0 0 0 0 0 1 2 3 4 3 2 3 4 4 4 0 0 0 0 0 1 2 3 4 3 2 3 4 4 4 4 0 0 0 0_1_2_3 4 3 2_3_4_4 4 4 4 0 0 0|1 2 3 4|3 2|3 4 4 4|4 4 4 0 0 1|2 3_4_3|2 3|4_4_4 4|4 4 3 0 1 2|3 4|3 2 3 4 4 4|4 4|4 3 2 1 2 3|4 3|2 3 4 4 4 4|4 4|3 2 1 2 3 4|3 2|3_4_4_4_4_4|4 3|2 1 0 3 4 3|2 3 4 4 4 4 4 4 3 2|1 0 0 4 3 2|3_4_4_4_4_4_4_3_2_1|0 0 0 3 2 3 4 4 4 4 4 4 3 2 1 0 0 0 0 2 3 4 4 4 4 4 4 3 2 1 0 0 0 0 0 3 4 4 4 4 4 4 3 2 1 0 0 0 0 0 0 |
slope = 2 0 0 0 0 0 1 2 3 2 1 1 2 2 2 3 2 0 0 0 0 1 2 3 3 1 1 2 2 2 3 3 2 0 0 0 0_1_2_3 2 1 1_2_2_2 3 2 1 0 0 0|1 2 3 3|1 1|2 2 2 3|3 2 1 0 0 0|1 2_3_2|1 1|2_2_2 3|2 1 0 0 0 1|2 3|3 1 1 2 2 2|3 2|1 0 0 0 0 1|2 3|2 1 1 2 2 2|3 2|1 0 0 0 1 2|3 3|1_1_2_2_2_3|3 2|1 0 0 0 1 2|3 2 1 1 2 2 2 3 2 1|0 0 0 1 2 3|3_1_1_2_2_2_3_3_2_1|0 0 0 1 2 3 2 1 1 2 2 2 3 2 1 0 0 0 0 2 3 3 1 1 2 2 2 3 3 2 1 0 0 0 0 2 3 2 1 1 2 2 2 3 2 1 0 0 0 0 0 |
vertical 0 0 0 7 7 4 4 2 2 4 4 7 7 0 0 0 0 0 0 7 7 4 4 2 2 4 4 7 7 0 0 0 0 0 0 7_7_4_4 2 2 4_4_7_7 0 0 0 0 0 0|7 7 4 4|2 2|4 4 7 7|0 0 0 0 0 0|7 7_4_4|2 2|4_4_7 7|0 0 0 0 0 0|7 7|4 4 2 2 4 4|7 7|0 0 0 0 0 0|7 7|4 4 2 2 4 4|7 7|0 0 0 0 0 0|7 7|4_4_2_2_4_4|7 7|0 0 0 0 0 0|7 7 4 4 2 2 4 4 7 7|0 0 0 0 0 0|7_7_4_4_2_2_4_4_7_7|0 0 0 0 0 0 7 7 4 4 2 2 4 4 7 7 0 0 0 0 0 0 7 7 4 4 2 2 4 4 7 7 0 0 0 0 0 0 7 7 4 4 2 2 4 4 7 7 0 0 0 |
slope = -2 2 3 2 2 2 1 1 2 3 2 1 0 0 0 0 0 2 3 3 2 2 2 1 1 3 3 2 1 0 0 0 0 1 2 3 2_2_2_1 1 2 3_2_1_0 0 0 0 1 2 3|3 2 2 2|1 1|3 3 2 1|0 0 0 0 1 2|3 2_2_2|1 1|2_3_2 1|0 0 0 0 1 2|3 3|2 2 2 1 1 3|3 2|1 0 0 0 0 1|2 3|2 2 2 1 1 2|3 2|1 0 0 0 0 1|2 3|3_2_2_2_1_1|3 3|2 1 0 0 0 0|1 2 3 2 2 2 1 1 2 3|2 1 0 0 0 0|1_2_3_3_2_2_2_1_1_3|3 2 1 0 0 0 0 1 2 3 2 2 2 1 1 2 3 2 1 0 0 0 0 1 2 3 3 2 2 2 1 1 3 3 2 0 0 0 0 0 1 2 3 2 2 2 1 1 2 3 2 |
slope = -1 4 4 4 3 2 3 4 3 2 1 0 0 0 0 0 0 4 4 4 4 3 2 3 4 3 2 1 0 0 0 0 0 4 4 4 4_4_3_2 3 4 3_2_1_0 0 0 0 4 4 4|4 4 4 3|2 3|4 3 2 1|0 0 0 3 4 4|4 4_4_4|3 2|3_4_3 2|1 0 0 2 3 4|4 4|4 4 4 3 2 3|4 3|2 1 0 1 2 3|4 4|4 4 4 4 3 2|3 4|3 2 1 0 1 2|3 4|4_4_4_4_4_3|2 3|4 3 2 0 0 1|2 3 4 4 4 4 4 4 3 2|3 4 3 0 0 0|1_2_3_4_4_4_4_4_4_3|2 3 4 0 0 0 0 1 2 3 4 4 4 4 4 4 3 2 3 0 0 0 0 0 1 2 3 4 4 4 4 4 4 3 2 0 0 0 0 0 0 1 2 3 4 4 4 4 4 4 3 |
slope = -1/2 2 2 2 2 2 1 1 0 0 0 0 0 0 0 0 0 3 2 2 2 2 2 2 1 1 0 0 0 0 0 0 0 2 3 3 2_2_2_2 2 2 1_1_0_0 0 0 0 3 2 2|3 3 2 2|2 2|2 2 1 1|0 0 0 3 3 3|2 2_3_3|2 2|2_2_2 2|1 1 0 3 3 3|3 3|2 2 3 3 2 2|2 2|2 2 1 2 3 3|3 3|3 3 2 2 3 3|2 2|2 2 2 1 2 2|3 3|3_3_3_3_2_2|3 3|2 2 2 0 1 1|2 2 3 3 3 3 3 3 2 2|3 3 2 0 0 0|1_1_2_2_3_3_3_3_3_3|2 2 3 0 0 0 0 0 1 1 2 2 3 3 3 3 3 3 2 0 0 0 0 0 0 0 1 1 2 2 3 3 3 3 3 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 3 |
8 9 8 14 13 10 13 12 12 13 10 13 14 8 9 8 9 9 9 15 15 13 15 15 15 15 13 15 15 9 9 9 9 9 9 15 15 13 15 15 15 15 13 15 15 9 9 9 16 16 17 28 29 28 28 21 21 28 28 29 28 17 16 16 14 17 19 29 30 30 28 21 21 28 30 30 29 19 17 14 10 14 18 28 30 24 21 22 22 21 24 30 27 17 13 10 10 13 17 28 29 24 24 21 21 24 24 29 28 17 13 10 9 13 17 28 29 24 24 24 24 24 24 29 28 17 13 9 15 19 20 29 31 32 31 30 30 31 32 31 29 20 19 15 18 17 17 29 30 30 32 30 30 32 30 30 29 17 17 18 6 7 9 16 17 17 20 18 18 20 17 17 16 9 7 6 7 9 10 15 16 15 17 16 16 17 15 16 15 10 9 7 8 10 9 14 14 12 14 12 12 14 12 14 14 9 10 8 |
There are mathematical algorithms that allow the exact reconstruction of a 2-dimensional object (or, more generally, a function) from its Radon transform. These are called ``inversion formulas'' and can be found, for example, in Sigurdur Helgason's The Radon Transform, Second edition, Birkhäuser, Boston 1999 or Gerald Folland's Introduction to Partial Differential Equations, Princeton University Press, Princeton NJ 1976. Both these books are based on graduate courses.
3. A numerical simulation: the reconstruction
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