In multidimensional signal processing, Multidimensional signal restoration refers to the problem of estimating the original input signal from observations of the distorted or noise contaminated version of the original signal using some prior information about the input signal and /or the distortion process. Multidimensional signal processing systems such as audio, image and video processing systems often receive as input, signals that undergo distortions like blurring, band-limiting etc. during signal acquisition or transmission and it may be vital to recover the original signal for further filtering. Multidimensional signal restoration is an inverse problem, where only the distorted signal is observed and some information about the distortion process and/or input signal properties is known. A general class of iterative methods have been developed for the multidimensional restoration problem with successful applications to multidimensional deconvolution, signal extrapolation and denoising.
Definition In general, the multidimensional signal restoration problem can be represented by an equation of the form,
y ( n 1 , n 2 , . . . . , n m ) = D [ x ( n 1 , n 2 , . . . . . , n m ) ] {\displaystyle y(n_{1},n_{2},....,n_{m})=D[x(n_{1},n_{2},.....,n_{m})]}
where y ( n 1 , n 2 , . . . . , n m ) {\displaystyle y(n_{1},n_{2},....,n_{m})} represents the observed m-dimensional distorted output signal, x ( n 1 , n 2 , . . . . , n m ) {\displaystyle x(n_{1},n_{2},....,n_{m})} represents the m-dimensional undistorted input signal and D [ ⋅ ] {\displaystyle D[\cdot ]} represents the distortion operator acting upon the input signal. D [ ⋅ ] {\displaystyle D[\cdot ]} can be used to model a wide range of transformations such as blurring, additive noise, time limiting, band limiting etc. of multidimensional signals. A simple straightforward solution to above equation is of the form,
x ( n 1 , n 2 , . . . . , n m ) = D − 1 [ y ( n 1 , n 2 , . . . . . , n m ) ] {\displaystyle x(n_{1},n_{2},....,n_{m})=D^{-1}[y(n_{1},n_{2},.....,n_{m})]}
where D − 1 [ ⋅ ] {\displaystyle D^{-1}[\cdot ]} is the inverse distortion operator. However, in most cases of practical use, it may be extremely difficult to implement the inverse distortion operator D − 1 [ ⋅ ] {\displaystyle D^{-1}[\cdot ]} or the such an inverse distortion operator may not exist and even in situations where the distortion operator D [ ⋅ ] {\displaystyle D[\cdot ]} is known and its inverse can be approximately implemented, the resultant reconstructed signal x ~ ( n 1 , n 2 , . . . . , n m ) {\displaystyle {\tilde {x}}(n_{1},n_{2},....,n_{m})} can have very large reconstruction errors due to the inaccuracies present in the estimation of the inverse operator D − 1 [ ⋅ ] {\displaystyle D^{-1}[\cdot ]} . A general class of iterative methods based on the idea of successive approximation is used to estimate the unknown input signal x ( n 1 , n 2 , . . . . , n m ) {\displaystyle x(n_{1},n_{2},....,n_{m})} .
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