A complete mathematical treatment is given for describing the topographic primal sketch of the underlying gray tone intensity surface of a digital image. Each picture element is independently classified and assigned a unique descriptive label, invariant under monotonically increasing gray tone transformations from the set ( peak, pit, ridge, ravine, saddle, flat, and hillside), with hillside having subcategories ( inflection point, slope, convex hill, concave hill, and saddle hill). The topographic classification is based on the first and second directional derivatives of the estimated image- intensity surface. A local, facet model, two-dimensional, cubic polynomial fit is done to estimate the image-intensity surface. Zero-crossings of the first directional derivative are identified as locations of interest in the image.
Motivated by (and using tools from) communication complexity, we investigate the relationship between the following two ranks of a 0-1 matrix: its nonnegative rank and its binary rank (the log of the latter being the unambiguous nondeterministic communication complexity). We prove that for partial 0-1 matrices, there can be an exponential separation. For total 0-1 matrices , we show that if the nonnegative rank is at most 3 then the two ranks are equal, and we show a separation by exhibiting a matrix with nonnegative rank 4 and binary rank 5, as well as a family of matrices for which the binary rank is 4/3 times the nonnegative rank.
Abstract. We show that there exist real numbers λ 1 , λ 2 , . . . , λn that occur as the eigenvalues of an entry-wise nonnegative n-by-n matrix but do not occur as the eigenvalues of a symmetric nonnegative n-by-n matrix. This solves a problem posed by Boyle and Handelman, Hershkowitz, and others. In the process, recent work by Boyle and Handelman that solves the nonnegative inverse eigenvalue problem by appending 0's to given spectral data is refined.
1.Let M m,n (M m,n (R)) denote the set of all m-by-n complex (real) matrices, and letThe nonnegative inverse eigenvalue problem (NIEP) asks which sets of n complex numbers λ 1 , λ 2 , . . . , λ n occur as the eigenvalues (spectrum) of some entry-wise nonnegative matrix A ∈ M n . In case the data: λ 1 , . . . , λ n are real, two natural variations suggest themselves:(1) The real nonnegative inverse eigenvalue problem (RNIEP) asks which sets of n real numbers occur as the spectrum of a nonnegative A ∈ M n ; and (2) the symmetric nonnegative inverse eigenvalue problem (SNIEP) asks which sets of n real numbers occur as the spectrum of a symmetric nonnegative matrix A ∈ M n .Each problem remains open in general. Several necessary conditions for NIEP are known. Suppose that λ 1 , . . . , λ n are complex numbers. For every positive integer k define the momentsIf λ 1 , . . . , λ n are the eigenvalues of an n-by-n nonnegative (positive) matrix A, we must have S k (λ) ≥ 0 (S k (λ) > 0) for every positive integer k, because S k (λ) is just the trace of A k . The Perron-Frobenius theorem implies that max 1≤i≤n |λ i | is an eigenvalue of A. (The Perron-Frobenius condition and the nonnegativity of the moments are not independent; see a remark after the statement of Theorem 2.) We also have:
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