[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

Distance-Regular Graphs

Lecture Notes in Economics and Mathematical Systems

THE theory of Fourier integrals arises out of the elegant pair of reciprocal formulae The Laplace Transform By David Vernon Widder. (Princeton Mathematical Series.) Pp. x + 406. (Princeton: Princeton University Press; London: Oxford University Press, 1941.) 36s. net.

The Laplace Transform

We investigate different notions of linear independence and of ma- trix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in terms of signed tropical determinants or arising from a no- tion of linear independence introduced by Gondran and Minoux. To do this, we revisit the symmetrization of the max-plus algebra, establishing properties of linear spaces, linear systems, and matrices over the symmetrized max-plus algebra. In parallel we develop some general technique to prove combinatorial and polynomial identities for matrices over semirings that we illustrate by a number of examples.

Linear independence over tropical semirings and beyond

We show that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems. In particular, we set up an equivalence between the external representation of tropical convex sets and zero-sum stochastic games, in which tropical polyhedra correspond to deterministic games with finite action spaces. Then, we show that the winning initial positions can be determined from the associated tropical polyhedron. We obtain as a corollary a game theoretical proof of the fact that the tropical rank of a matrix, defined as the maximal size of a submatrix for which the optimal assignment problem has a unique solution, coincides with the maximal number of rows (or columns) of the matrix which are linearly independent in the tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius theory.

Tropical polyhedra are equivalent to mean payoff games

Some general techniques on linear preserver problems

We prove general Cramer type theorems for linear systems over various extensions of the tropical semiring, in which tropical numbers are en- riched with an information of multiplicity, sign, or argument. We obtain exis- tence or uniqueness results, which extend or rene earlier results of Gondran and Minoux (1978), Plus (1990), Gaubert (1992), Richter-Gebert, Sturmfels and Theobald (2005) and Izhakian and Rowen (2009). Computational issues are also discussed; in particular, some of our proofs lead to Jacobi and Gauss- Seidel type algorithms to solve linear systems in suitably extended tropical semirings.

Tropical Cramer Determinants Revisited

Inequalities on the rank of the sum and the product of two matrices over semirings are surveyed. Preferences are given to the factor rank, row and column ranks, term rank, and zero-term rank of matrices over antinegative semirings. During the past century a lot of literature has been devoted to in- vestigations of semirings. Brie∞y, a semiring is essentially a ring where only the zero element is required to have an additive inverse. Therefore, all rings are also semirings. Moreover, among semirings there are such combinatorially interesting systems as the Boolean algebra of subsets of a flnite set(with addition being union and multiplication being intersec- tion), nonnegative integers and reals(with the usual arithmetic), fuzzy scalars(with fuzzy arithmetic), etc. Matrix theory over semirings is an object of much study in the last decades, see for example (9). In particu- lar, many authors have investigated various rank functions for matrices over semirings and their properties, see (1, 3, 6, 7, 8, 12) and references there in. There are classical inequalities for the rank function ‰ of sums and products of matrices over flelds, see, for example (10, 11): The rank-sum inequalities:

/pdf/rank-inequalities-over-semirings-19q6b28sj4.pdf

Alexander Guterman

Papers

Linear independence over tropical semirings and beyond

Tropical polyhedra are equivalent to mean payoff games

Some general techniques on linear preserver problems

Tropical Cramer Determinants Revisited

Rank inequalities over semirings