Perfect radar pulse compression via unimodular fourier multipliers
Lassi Roininen Markku S. Lehtinen Petteri Piiroinen Ilkka I. Virtanen
Inverse Problems & Imaging 2014, 8(3): 831-844 doi: 10.3934/ipi.2014.8.831
We propose a novel framework for studying radar pulse compression with continuous waveforms. Our methodology is based on the recent developments of the mathematical theory of comparison of measurements. First we show that a radar measurement of a time-independent but spatially distributed radar target is rigorously more informative than another one if the modulus of the Fourier transform of the radar code is greater than or equal to the modulus of the Fourier transform of the second radar code. We then motivate the study by spreading a Gaussian pulse into a longer pulse with smaller peak power and re-compressing the spread pulse into its original form. We then review the basic concepts of the theory and pose the conditions for statistically equivalent radar experiments. We show that such experiments can be constructed by spreading the radar pulses via multiplication of their Fourier transforms by unimodular functions. Finally, we show by analytical and numerical methods some examples of the spreading and re-compression of certain simple pulses.
keywords: radar experiments comparison of measurements. Pulse compression
Perfect pulse-compression coding via ARMA algorithms and unimodular transfer functions
Lassi Roininen Markku S. Lehtinen
Inverse Problems & Imaging 2013, 7(2): 649-661 doi: 10.3934/ipi.2013.7.649
We propose a method to construct perfect pulse-compression codes with autoregressive moving average algorithms. We first show the relation between the study of coding and decoding techniques in radar engineering and the study of unimodular polynomials with constrained coefficients. Then we extend the study to unimodular Fourier series and unimodular rational functions. We use the Fourier series and rational functions as transfer functions in the autoregressive moving average algorithms. We show that by a suitable choice of the coefficients, the autoregressive moving average algorithms are realisable, stable and causal. We show examples of some almost perfect codes, i.e. numerically truncated perfect codes. We end by proposing perfect code design principles for practical radar engineering purposes.
keywords: inverse filter. unimodular polynomials Perfect codes autoregressive moving average
Correlation priors
Lassi Roininen Markku S. Lehtinen Sari Lasanen Mikko Orispää Markku Markkanen
Inverse Problems & Imaging 2011, 5(1): 167-184 doi: 10.3934/ipi.2011.5.167
We propose a new class of Gaussian priors, correlation priors. In contrast to some well-known smoothness priors, they have stationary covariances. The correlation priors are given in a parametric form with two parameters: correlation power and correlation length. The first parameter is connected with our prior information on the variance of the unknown. The second parameter is our prior belief on how fast the correlation of the unknown approaches zero. Roughly speaking, the correlation length is the distance beyond which two points of the unknown may be considered independent.
   The prior distribution is constructed to be essentially independent of the discretization so that the a posteriori distribution will be essentially independent of the discretization grid. The covariance of a discrete correlation prior may be formed by combining the Fisher information of a discrete white noise and different-order difference priors. This is interpreted as a combination of virtual measurements of the unknown. Closed-form expressions for the continuous limits are calculated. Also, boundary correction terms for correlation priors on finite intervals are given.
   A numerical example, deconvolution with a Gaussian kernel and a correlation prior, is computed.
keywords: a priori distributions discretization-invariance. Statistical inversion
Hyperpriors for Matérn fields with applications in Bayesian inversion
Lassi Roininen Mark Girolami Sari Lasanen Markku Markkanen
Inverse Problems & Imaging 2019, 13(1): 1-29 doi: 10.3934/ipi.2019001

We introduce non-stationary Matérn field priors with stochastic partial differential equations, and construct correlation length-scaling with hyperpriors. We model both the hyperprior and the Matérn prior as continuous-parameter random fields. As hypermodels, we use Cauchy and Gaussian random fields, which we map suitably to a desired correlation length-scaling range. For computations, we discretise the models with finite difference methods. We consider the convergence of the discretised prior and posterior to the discretisation limit. We apply the developed methodology to certain interpolation, numerical differentiation and deconvolution problems, and show numerically that we can make Bayesian inversion which promotes competing constraints of smoothness and edge-preservation. For computing the conditional mean estimator of the posterior distribution, we use a combination of Gibbs and Metropolis-within-Gibbs sampling algorithms.

keywords: Bayesian statistical estimation inverse problems Matérn fields hypermodels convergence
Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography
Lassi Roininen Janne M. J. Huttunen Sari Lasanen
Inverse Problems & Imaging 2014, 8(2): 561-586 doi: 10.3934/ipi.2014.8.561
We study flexible and proper smoothness priors for Bayesian statistical inverse problems by using Whittle-Matérn Gaussian random fields. We review earlier results on finite-difference approximations of certain Whittle-Matérn random field in $\mathbb{R}^2$. Then we derive finite-element method approximations and show that the discrete approximations can be expressed as solutions of sparse stochastic matrix equations. Such equations are known to be computationally efficient and useful in inverse problems with a large number of unknowns.
    The presented construction of Whittle-Matérn correlation functions allows both isotropic or anisotropic priors with adjustable parameters in correlation length and variance. These parameters can be used, for example, to model spatially varying structural information of unknowns.
    As numerical examples, we apply the developed priors to two-dimensional electrical impedance tomography problems.
keywords: Electrical impedance tomography Bayesian statistical inversion smoothness priors.
Constructing continuous stationary covariances as limits of the second-order stochastic difference equations
Lassi Roininen Petteri Piiroinen Markku Lehtinen
Inverse Problems & Imaging 2013, 7(2): 611-647 doi: 10.3934/ipi.2013.7.611
In Bayesian statistical inverse problems the a priori probability distributions are often given as stochastic difference equations. We derive a certain class of stochastic partial difference equations by starting from second-order stochastic partial differential equations in one and two dimensions. We discuss discretisation schemes on uniform lattices of these stationary continuous-time stochastic processes and convergence of the discrete-time processes to the continuous-time processes. A special emphasis is given to an analytical calculation of the covariance kernels of the processes. We find a representation for the covariance kernels in a simple parametric form with controllable parameters: correlation length and variance. In the discrete-time processes the discretisation step is also given as a parameter. Therefore, the discrete-time covariances can be considered as discretisation-invariant. In the two-dimensional cases we find rotation-invariant and anisotropic representations of the difference equations and the corresponding continuous-time covariance kernels.
keywords: covariance convergence Stochastic difference equation statistical inversion.

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