Fast 3D image reconstruction method based on SVD decomposition of a block-circulant system matrix

We propose an ultra-fast 3D image reconstruction method based on singular value decomposition (SVD) of a block-circulant system matrix obtained from a cylindrical image representation. PET and SPECT image reconstruction based on SVD of the system matrix has already been demonstrated previously but was limited to the reconstruction of small 2D images due to the difficulty of inverting an ill-conditioned large matrix. In this work, this difficulty is overcome by using a Fourier transformed factorization of the block-circulant matrix to accelerate the SVD decomposition procedure and make it more robust. The Fourier transform is further used to accelerate the matrix-vector operations between the system matrix pseudo-inverse and the projection data resulting in an extremely fast direct image reconstruction method. A maximum acceleration of the method is achieved by taking advantage of all in-plane and axial symmetries between the tubes of response through the use of a 3D cylindrical image representation preserving all symmetries in the system matrix. Using the same projection data, the proposed method delivers images of visual quality comparable to FBP, but 25 times faster. Moreover, for imaging systems with many symmetries, the method is so fast that it can produce higher quality images using all available 3D projection data, taking time comparable to FBP or MLEM, the latter using single plane, i.e. partial 2D projection data only. The new method is therefore ideal for real time 3D image reconstruction allowing for the instant visualization of the image estimate while the patient is being scanned.