Patent ID: 8014616

Claim:
A computer-implemented method of expanding a sparsely sampled digitized image, the method performed by the computer comprising the steps of: acquiring a sparse digital image b, said image comprising a plurality of intensities corresponding to an I-dimensional grid of points; initializing points (x (k) , y (k) ), wherein x (k) is an initial approximation of a first expanded image x recovered from the sparse digital image b and defined by b=RΦ −1 x , wherein R is a Fourier transform matrix, Φ is a wavelet transform matrix, y (k) is a point in the set of subgradients ∂ ( ∑ i = 1 l ⁢ ( ∇ i ⁢ Φ - 1 ⁢ x ( k ) ) 2 ) 1 / 2 , ∇ i is a forward finite difference operator for an i th coordinate, and k is an iteration counter; calculating a first auxiliary variable s (k) from x ( k ) - τ 1 ( α ⁢ ⁢ Φ ⁢ ∑ n ⁢ L n * ⁢ y n ( k ) + Φ ⁢ ⁢ R * ( R ⁢ ⁢ Φ - 1 ⁢ x ( k ) - b ) ) , wherein τ 1 , α are predetermined positive scalar constants, the sum is over all points n in x , and L* is an adjoint of operator L=(∇ 1 , . . . , ∇ I ); calculating a second auxiliary variable t n (k) from y n (k) +τ 2 L n Φ −1 x (k) , wherein τ 2 is a predetermined positive scalar constant; updating x (k+1) from sign(s (k) )max{0,|s (k) |−τ 1 β}, wherein β is a predetermined positive scalar constant, to provide an improved approximation to the first expanded image x ; and updating y n (k+1) from min ⁢ { 1 τ 2 ,  t n ( k )  2 } ⁢ t n ( k )  t n ( k )  2 .