A method for reconstructing a streaming signal x n from streaming measurements y m by maintaining, using sliding windows, a working set of measurements y i-1, a snapshot of a measurement system A i-1, an internal working signal estimate x̂ i -1 of the streaming signal x n, and an external working signal estimate x i -1 of the streaming signal x n, comprising the steps of: refining the internal working signal estimate x̂ i -1, by calculating a residual r i, wherein r i =y i-1 -A i-1 x̂ i -1, to obtain a refined internal working signal estimate x̃ i; refreshing the external working signal estimate by combining the external working signal estimate x i -1 with the refined internal working estimate x̃ i, to obtain refreshed external working signal estimate x i; committing first R coefficients x̂ R i + j of the refreshed external working signal estimate to an output, wherein x̂ R i + j = x i j, j= 1,..., R, and R is an integer downsampling rate; receiving a next streaming measurement y i+M and a corresponding next measurement vector a {i+M},{R(i-1)+1,Ri}, wherein y i + M = ∑ n = R i − 1 + 1 n = Ri x n a i + M, R i − 1 + 1, Ri; updating the working set of measurements, the snapshot of the measurement system, and the internal working signal estimate to incorporate the next streaming measurement and the corresponding measurement vector; and removing an oldest measurement and an effect of the committed subset of coefficients from the working set of measurements and removing a corresponding oldest measurement vector from the working snapshot of the measurement system; wherein the refining further comprises: identifying locations of signal components that account for said residual; combining locations of the current internal signal estimate components with the locations of the signal components that account for said residual; estimating the signal components over the combined locations produced by the combining using the working set of measurements and the snapshot of the measurement system; and keeping only most significant components and the associated locations from the estimating, and producing the most significant components for the internal working signal estimate. refining the internal working signal estimate x̂ i -1, by calculating a residual r i, wherein r i =y i-1 -A i-1 x̂ i -1, to obtain a refined internal working signal estimate x̃ i; refreshing the external working signal estimate by combining the external working signal estimate x i -1 with the refined internal working estimate x̃ i, to obtain refreshed external working signal estimate x i; committing first R coefficients x̂ R i + j of the refreshed external working signal estimate to an output, wherein x̂ R i + j = x i j, j= 1,..., R, and R is an integer downsampling rate; receiving a next streaming measurement y i+M and a corresponding next measurement vector a {i+M},{R(i-1)+1,Ri}, wherein y i + M = ∑ n = R i − 1 + 1 n = Ri x n a i + M, R i − 1 + 1, Ri; updating the working set of measurements, the snapshot of the measurement system, and the internal working signal estimate to incorporate the next streaming measurement and the corresponding measurement vector; and removing an oldest measurement and an effect of the committed subset of coefficients from the working set of measurements and removing a corresponding oldest measurement vector from the working snapshot of the measurement system; wherein the refining further comprises: identifying locations of signal components that account for said residual; combining locations of the current internal signal estimate components with the locations of the signal components that account for said residual; estimating the signal components over the combined locations produced by the combining using the working set of measurements and the snapshot of the measurement system; and keeping only most significant components and the associated locations from the estimating, and producing the most significant components for the internal working signal estimate. identifying locations of signal components that account for said residual; combining locations of the current internal signal estimate components with the locations of the signal components that account for said residual; estimating the signal components over the combined locations produced by the combining using the working set of measurements and the snapshot of the measurement system; and keeping only most significant components and the associated locations from the estimating, and producing the most significant components for the internal working signal estimate.