TAOCP 7.2.2.2 Exercise 103
The data in the statement are insufficient to determine the requested $7\times21$ image.
Section 7.2.2.2: Satisfiability
Exercise 103. [18] [18] (Do this exercise by hand, it's fun!) Find the $7 \times 21$ image whose tomographic sums are $(r_1, \ldots, r_7) = (1, 0, 13, 6, 12, 7, 19)$; $(c_1, \ldots, c_{21}) = (4, 3, 3, 4, 1, 6, 1, 3, 3, 5, 1, 5, 1, 5, 1, 5, 1, 1, 1)$; $(a_1, \ldots, a_{27}) = (0, 0, 1, 2, 3, 2, 3, 2, 3, 3, 5, 4, 3, 2, 3, 3, 3, 4, 3, 2, 2, 1)$; $(b_1, \ldots, b_{27}) = (0, 0, 0, 1, 3, 3, 4, 3, 2, 2, 3, 3, 3, 4, 3, 2, 1, 1)$.
Verified: yes
Solve time: 1m36s
Solution
The data in the statement are insufficient to determine the requested $7\times21$ image.
A $7\times21$ binary image has $7\cdot21=147$ entries. The row sums determine $7$ linear constraints, and the column sums determine another $21$ constraints. The two diagonal directions in a $7\times21$ grid each have $7+21-1=27$ diagonals, so a complete specification of the tomographic data requires two additional lists of length $27$.
The supplied diagonal data do not have the required lengths. The list labeled $(a_1,\ldots,a_{27})$ is
$$ (0,0,1,2,3,2,3,2,3,3,5,4,3,2,3,3,3,4,3,2,2,1), $$
which contains only $22$ entries. The list labeled $(b_1,\ldots,b_{27})$ is
$$ (0,0,0,1,3,3,4,3,2,2,3,3,3,4,3,2,1,1), $$
which contains only $18$ entries.
Therefore the two diagonal projection directions are missing part of their constraints. Different completions of the omitted entries can lead to different $7\times21$ images, and without the missing values the reconstruction problem has no uniquely defined solution.
The complete solution requires the full $27$ values of both $(a_1,\ldots,a_{27})$ and $(b_1,\ldots,b_{27})$. With those supplied, the image can be reconstructed by solving the corresponding system of binary constraints. $\boxed{\text{The given data are incomplete}}$