Complex Exponential Signal Recovery And Its Application In MRS

Signals are generally modeled as a superposition of exponential functions in spec-troscopy of biology,chemistry and medical imaging.Due to the broad bandwidth,costly experiments,hardware limitation or other inevitable reasons,data sampling sometimes cannot satisfy the Nyquist principle,resulting in data missing and signal distortion.How to recover the full signal from partial measurements becomes an active research topic in the signal processing community.Take magnetic resonance spectroscopy as an example.The time domain signal in magnetic resonance spectroscopy can be modelled by a superposi-tion of limited exponential functions,and the corresponding spectrum in Fourier transform domain can be used to determine the molecular structure,for example,human tissues de-tection,compound group analyses and protein structure interpretation.However,the long acquisition of magnetic resonance spectroscopy seriously pretends its practical applica-tions.To accelerate data acquisition,how to reconstruct high-quality magnetic resonance spectroscopy from sparse sampling is a very challenging and advancing problem.According to the compressed sensing theory,an exponential signals may be exactly reconstructed from its partial observations if it enj oys a sparse representation in the discrete Fourier transform.However,the basis mismatch due to frequencies taking values on a continuous domain,and the broad peak due to damping factor lead to the loss of sparsity and hence worsens the performance of compressed sensing.Besides sparsity,another general hypothesis of signals is low rankness.It is proved that the rank of the structured Toeplitz or Hankel matrix arranged by the exponential signal equals to the number of component exponential functions.When the number of exponentials is far less than the signal dimension,the signal can be exactly recovered by enforcing low rankness on the structured matrix.The low rank structured matrix recovery does not rely the sparsity of the signal in the discrete Fourier transform domain and thus overcome the basis mismatch and broad peak reconstruction problems.However,this type of methods are still challenged in two ways:the frequency separation is required for signal reconstruction,and thus leads to failure in concentrated peak reconstruction;the high computational complexity prevents to reconstruct high-dimensional(? 3 dimensions)exponential signals.This thesis focus on how to sufficiently exploit the property of exponential signals in reconstruction,expecting to reconstruct high-quality magnetic resonance spectroscopy.The main work includes:(1)We propose an one-dimensional exponential signal recovery via Vandermonde factorization.A numerical algorithm is developed and its convergence is theoretically analyzed.Experiments demonstrate that the required measurements for the proposed method to reconstruct the signal is less than the state-of-are low rank struc-tured matrix recovery,implying that the data acquisition in the indirect dimension of 2-dimensional magnetic resonance spectroscopy can be further accelerated.Moreover,it is observed that there is no required frequency separation condition for the proposed method,and thus can be applied in concentrated peak reconstruction,extending applications of sparse sampling in bio-medical magnetic resonance spectroscopy.(2)We propose a high-dimensional(>3 dimensions)exponential signal recovery based on tensor factorization and low rank Hankel matrix regularization.We develop an alternating direction itera-tive algorithm and further analyze the convergence and complexity.Experimental results on simulated and real magnetic resonance spectroscopy data show that the proposed ap-proach can successfully recover the signal from very limited measurements.Experiments show that the proposed method requires much less measurements to recover signals than the compared typical tensor completion methods.For the tested 3-dimensional magnetic resonance spectrum,just 10%measurements are required to reconstruct the high-quality spectrum by the proposed method,implying that data acquisition speed can be acceler-ated by 10 times.Therefore,the proposed method may serve as a versatility method for studying complex samples,such as protein,using high-dimensional magnetic resonance spectroscopy.Note that the proposed approaches in thesis are not limited to reconstruct simulated exponential signals and real magnetic resonance spectroscopy,but general exponential signals arising from,for example,radar array,harmonic analysis and analog-to-digital conversion.In addition,some questions for the proposed approaches remain to be further explored,including how to derive the theoretical condition for exact recovery and develop efficient algorithms to accommodate larger datasets.