Linear equations over finite abelian groups
One of the oldest problems in algebra is the solution of a system of linear equations over certain domains. The Gaussian elimination algorithm provides an effective solution over fields. The Smith normal form algorithm yields a method over the integers. Here we consider another variation of this theme. Let A be a finite (additive) abelian group, let αi,j ∈ End(A) for 1 ≤ i ≤ n and 1 ≤ j ≤ m and let b1, . . . , bm ∈ A. Then we want to determine the set S of all (x1, . . . , xn) ∈ A n with
Solving linear equations over finitely generated abelian groups
We discuss various methods and their effectiveness for solving linear equations over finitely generated abelian groups. More precisely, if $\varphi\colon G\to H$ is a homomorphism of finitely generated abelian groups and $b\in H$, we discuss various algorithms for checking whether $b\in \im\varphi$ holds and if so, for computing a pre-image of $b$ in $G$ together with the kernel of $\varphi$.
An operational method for estimating signal to noise ratios from data acquired with imaging spectrometers
A method, using the concept of local means and local standard deviations of small imaging blocks and using a box counting procedure, has been developed for unsupervised estimation of the average signal to noise ratios of images in which the noise is additive. The method has been applied to simulated images with Gaussian noise, to images acquired with the Airborne Visible Infrared Imaging Spectrometer and with the Geophysical and Environmental Research Imaging Spectrometer. The method is compared with other techniques for estimating signal to noise ratios from imaging data. It is believed that the method is generally applicable to images having many small homogeneous blocks. Because the method does not take into account interband effects, it is not appropriate to use the method to estimate signal to noise ratios from images having nonnegligible interband radiometric calibration errors at spatial scales less than or equal to the size of the small imaging blocks used in the noise estimation.