Inverse problems for unilevel block α‐circulants
We consider the following inverse problems for the class Cα of unilevel block α‐circulants C=[Cs−αr]r,s=0k−1 , where k > 1, C0, C1, …, Ck−1∈Cd1×d2 , α ∈ {1,2, … ,k −1}, gcd(α,k) = 1, and ∥ ⋅ ∥ is the Frobenius norm.
Characterization and properties of matrices with k-involutory symmetries
We say that a matrix R∈Cn×n is k-involutory if its minimal polynomial is xk-1 for some k⩾2, so Rk-1=R-1 and the eigenvalues of R are 1, ζ, ζ2,…,ζk-1, where ζ=e2πi/k. Let α,μ∈{0,1,…,k-1}. If R∈Cm×m,A∈Cm×n,S∈Cn×n and R and S are k-involutory, we say that A is (R,S,α,μ)-symmetric if RAS-α=ζμA. We show that an (R,S,α,μ)-symmetric matrix A can be usefully represented in terms of matrices Fl∈Ccαl+μ×dl,0⩽l⩽k-1, where cl and dl are respectively the dimensions of the ζl-eigenspaces of R and S. This continues a theme initiated in an earlier paper with the same title, in which we assumed that α=1. We say that a k-involution is equidimensional with width d if all of its eigenspaces have dimension d. We show that if R and S are equidimensional k-involutions with widths d1 and d2 respectively, then (R,S,α,μ)-symmetric matrices are closely related to generalized α-circulants [ζμrAs-αr]r,s=0k-1, where A0,A1,…,Ak-1∈Cd1×d2. For this case our results are new even if α=1. We also give an explicit formula for the Moore-Penrose inverse of a unilevel block circulant [As-αr]r,s=0k-1 for any α∈{0,1,…,k-1}, generalizing a result previously obtained for the case where gcd(α,k)=1.
DEM generation from new VIR sensors: IKONOS, ASTER and Landsat-7
DEM generation is a living R&D topic since new-launched satellite images can be used to generate stereoscopic images. Using a coplanarity equation model developed at CCRS and an automatic matching method developed at PCI, DEMs are generated with new VIR stereo data. Accuracy of 4.9 m with high-resolution IKONOS images, and 33 m and 101 m with medium-resolution ASTER and Landsat-7 ETM/sup +/ images respectively are obtained. The same matching accuracy of one-pixel is translated into worse results with Landsat due to a small B/H ratio of 0.15.