Algorithm 1 Sparsity Adaptive Subspace Pursuit Algorithm
Input: Matrix A, Vector Φ, Threshold σ, The maximum number of iterations $N_{\max }$
Initialization:$I^0=\varnothing$, $r_0=\Phi$, $S=2$, $K=S$
	n = 1
	(1) $c^1=A^T{r}^0=A^T\Phi$
	(2) $I^1=\{$ K indices corresponding to the largest magnitude entries in vector $c^1$}.
	(3) $r^1=\Phi -proj(\Phi ,A_{I^1})=\Phi -A_{I^1}{A}_{I^1}^{+}\Phi$
	(4) $J^1=\varnothing$
Iteration($n\ge 2$):
	(1) $c^{n-1}=A^T{r}^{n-1}$
	(2) $J^n=I^{n-1}\cup \{$ K indices corresponding to the largest magnitude entries in vector $c^{n-1}$}
	(3) $x_p=A_{J^n}^{+}\Phi$
	(4) $I=\{$ K indices with the largest magnitude entries of projection coefficients $x^p$}
	(5) $r=resid(\Phi ,A_I)=\Phi -{\Phi }_p=\Phi -A_I{A}_I^{+}\Phi$
	(6) Update the sparsity factor K
		1) if $r>r^{n-1}$ then
			$K=K+S$, $r^n=r^{n-1}$, $I^n=I^{n-1}$
		2) else
			The sparse factor Kstays the same, $r^n=r$, $I^n=I$
		end if
	(7) if halting condition true (i.e., $r{}^n<\sigma$ or $n>N_{\max }$), then quit the iteration
Output:
	The sparse solution to the FMT problem $\widehat{x}$, satisfying ${\widehat{x}}_{\{1,2,3\cdots ,N\}-I^n}=0$ and ${\widehat{x}}_{I^n}=A_{I^n}^{+}\Phi$.