Image reconstruction using redundant Haar wavelets

An image reconstruction method using a weighted four dimension (4D) redundant Haar wavelet comprising:

• receiving under-sampled k-space data including 4D images;
• determining a reconstruction of an image of the 4D images by solving a minimization of a function including a data fidelity term and a penalty term controlling an incorporation of piecewise constant structures in spatial and temporal directions;
• averaging the under-sampled k-space data in a temporal direction to obtain an averaged k-space data and
• determining a coil-profile based on the averaged k-space data,
• wherein determining the reconstruction of the image further comprises:
• updating a solution based on the k-space data, the averaged k-space data, and the coil-profile in view of the data fidelity term;
• converting the solution into an equivalent three-dimensional tensor;
• updating the equivalent three-dimensional tensor using the spatial penalization;
• and
• outputting the reconstruction of the image corresponding to the equivalent three-dimensional tensor updated using the spatial penalization.

An image reconstruction method comprising:

• inputting k-space data;
• averaging the k-space data in a temporal direction to obtain an averaged k-space data;
• determining a coil-profile based on the averaged k-space data;
• updating a solution based on the k-space data, the averaged k-space data, and the coil-profile in view of a data fidelity term;
• converting the solution into an equivalent three-dimensional tensor;
• updating the equivalent three-dimensional tensor using a wavelet transformation;
• and
• outputting a reconstructed image corresponding to the equivalent three-dimensional tensor updated using the wavelet transformation.