Stefano Brocchi

Reconstruction of binary matrices under fixed size neighborhood constraints

Using a dynamic programming approach, we prove that a large variety of matrix reconstruction problems from two projections can be solved in polynomial time whenever the number of rows (or columns) is fixed. We also prove some complexity results for several problems concerning the reconstruction of a binary matrix when a neighborhood constraint occurs.

Solving the two color problem: an heuristic algorithm

The 2-color problem in discrete tomography requires to construct a 2-colored matrix consistent with a given set of projections representing the number of elements of each color in each one of its rows and columns. In this paper, we describe an heuristic algorithm to find a solution of the 2-color problem, that has been recently proved to be NP-complete. The algorithm starts by computing a solution where elements of different colors may overlap, and then it proceeds in searching for switches that leave unaltered the projections but remove the overlaps. Experimental results show that this heuristic approach finds a solution in a short computational time to almost all the randomly generated 2-color instances, and it provides for the remaining ones a high quality approximation.

A tiling system for the class of L-convex polyominoes

A polyomino is said to be L-convex if any two of its cells can be connected by a path entirely contained in the polyomino, and having at most one change of direction. In this paper, answering a problem posed by Castiglione and Vaglica [6], we prove that the class of L-convex polyominoes is tiling recognizable. To reach this goal, first we express the L-convexity constraint in terms of a set of independent properties, then we show that each class of convex polyominoes having one of these properties is tiling recognizable.

Solving Multicolor Discrete Tomography Problems by Using Prior Knowledge

Discrete tomography deals with the reconstruction of discrete sets with given projections relative to a limited number of directions, modeling the situation where a material is studied through x-rays and we desire to reconstruct an image representing the scanned object. In many cases it would be interesting to consider the projections to be related to more than one distinguishable type of cell, called atoms or colors, as in the case of a scan involving materials of different densities, as a bone and a muscle. Unfortunately the general n-color problem with n > 1 is NP-complete, but in this paper we show how several polynomial reconstruction algorithms can be defined by assuming some prior knowledge on the set to be rebuilt. In detail, we study the cases where the union of the colors form a set without switches, a convex polyomino or a convex 8-connected set. We describe some efficient reconstruction algorithms and in a case we give a sufficient condition for uniqueness.

A reconstruction algorithm for a subclass of instances of the 2-color problem

In the field of Discrete Tomography, the 2-color problem consists in reconstructing a matrix whose elements are of two different types, starting from its horizontal and vertical projections. It is known that the 1-color problem admits a polynomial time reconstruction algorithm, while the c-color problem, with c>=2, is NP-hard. Thus, the 2-color problem constitutes an interesting example of a problem just in the frontier between hard and easy problems. In this paper we define a linear time algorithm (in the size of the output matrix) to solve a subclass of its instances, where some values of the horizontal and vertical projections are constant, while the others are upper bounded by a positive number proportional to the dimension of the problem. Our algorithm relies on classical studies for the solution of the 1-color problem.

BCIF: another algorithm for lossless true color image compression

In this paper we present an algorithm for the lossless compression of true color images. Our aim was to develop a practical algorithm with a fast decompression phase. The algorithm executes a block adaptive predictive filtering phase, followed by a color filtering phase that exploits color correlation, and finally compresses the prediction errors through context assignment and Huffman coding. Comparing the proposed algorithm with competing standards as Jpeg2000 and Jpeg-LS, we show how our method yields better compression ratios without having a slower decompression speed.

Smooth sand piles

We define the Smooth Sand Pile Model SmSPM(n), a new granular dynamical system derived from the Sand Pile Model SPM(n). We show a characterization of the reachable states of SmSPM(n), together with some interesting properties of the resulting lattice.

Solving some instances of the 2-color problem

In the field of Discrete Tomography, the 2-color problem consists in determining a matrix whose elements are of two different types, starting from its horizontal and vertical projections. It is known that the one color problem has a polynomial time reconstruction algorithm, while, with k ≥ 2, the k-color problem is NP-complete. Thus, the 2-color problem constitutes an interesting example of a problem just in the frontier between hard and easy problems. In this paper we define a linear time algorithm to solve a set of its instances, where some values of the horizontal and vertical projections are constant, while the others are upper bounded by a positive number proportional to the dimension of the problem. Our algorithm relies on classical studies for the solution of the one color problem.

On the exhaustive generation of k-convex polyominoes

The degree of convexity of a convex polyomino P is the smallest integer k such that any two cells of P can be joined by a monotone path inside P with at most k changes of direction. In this paper we present a simple algorithm for computing the degree of convexity of a convex polyomino and we show how it can be used to design an algorithm that generates, given an integer k, all k-convex polyominoes of area n in constant amortized time, using space O(n). Furthermore, by applying few changes, we are able to generate all convex polyominoes whose degree of convexity is exactly k.

The 1-color problem and the Brylawski model

In discrete tomography, the 1-color problem consists in determining the existence of a binary matrix with row and column sums equal to some given input values arranged in two vectors. These two vectors are said to be compatible if the associated 1-color problem has at least a solution. Here, we start from a vector of projections, and we define an algorithm to compute all the vectors compatible with it, then we show how to arrange them in a partial order structure, and we point out some of its combinatorial properties. Finally, we prove that this poset is a sublattice of the Brylawski lattice too, and we check some common properties.