A tissue characterization with susceptibility e.g., susceptibility weighted imaging (SWI), quantitative susceptibility mapping (QSM), susceptibility tensor imaging (STI), gets more attention in high field MRI. We propose a method to remove background phase using a discrete wavelet transform (DWT) by which slowly varying low frequency distortions are substantially reduced in the resultant susceptibility map compared to existing techniques.In general three processing steps are needed in the QSM analysis: phase unwrapping, background phase removal, and susceptibility mapping or reconstruction by solving an inverse problem. The former two steps are preprocessing needed to obtain accurate susceptibility maps. Various phase unwrapping methods have been proposed ^1,2. Background phase results from global geometry, air-tissue interfaces, and any field inhomogeneity, which needs to be removed, thereby leaving the phase only due to tissue susceptibility. To remove the background phase, Laplace equation (sum of the second order partial derivatives of the background phase along x, y, and z directions makes zero) is used. There are several ways to implement the Laplace equation in the forms of convolution and deconvolution at the spatial and spatial frequency domains ³. Some (wrinkle like) low frequency distortions are observed in the reconstructed susceptibility maps similar to Gibbs phenomena, which are partially due to the division of the Laplacian kernel in the spatial frequency domain to implement the deconvolution process.
We propose a method to remove the background phase by DWT. Three dimensional DWT is applied iteratively for decomposing the phase images into high and low frequency bands. Three stage decompositions are made by using a 4th order symlet wavelets, a modified version of Daubechies wavelets. The background removed phase images are obtained by reconstructing the decomposed images without the lowest frequency band. Since the background phase consists of local dc and linear terms mostly, they can be removed by a high pass filtering ⁴. Since DWT is performed entirely in the spatial domain, no conversion or division of the Laplacian kernel in the spatial frequency domain is needed, which reduces the low frequency distortions in the susceptibility map.Various background phase removal methods are tested with a series of parameters for a numerical phantom which consists of 12 cylinders with different susceptibilities from 0.02 to 0.24 ppm. Figure 1 shows the susceptibility maps obtained by removing the background phase with SHARP 3, and the proposed DWT method. Other processing procedures including phase unwrapping and susceptibility reconstruction were the same for both cases. The susceptibility values reconstructed in the 12 cylinders of the phantom are also shown. As seen in Fig.1, more accurate susceptibility map is obtained by using the DWT method compared to SHARP. Figure 2 shows in-vivo susceptibility maps of head for volunteers obtained at 7T and 3T MRI. Three dimension gradient echo sequence was used to obtain the phase images. Two echoes were obtained with TR=30ms, TE1=8.1ms, TE2=20.3ms, flip angle=10 degrees, FOV=224x224x40mm, matrix size=448x448x40. The phase images were obtained by subtracting the phase of the first echo from that of the second echo. As seen in Fig.2, low frequency distortions are dominant in the susceptibility map using SHARP (a), while such artifact is not seen in the map using DWT (b) (see the ellipses marked). The distortions are more serious at higher field (7T).From computer simulation using a numerical phantom and in-vivo experiments at 7T and 3T MRI, more accurate susceptibility maps are obtained when the background phase is removed by DWT compared to other existing techniques including SHARP. Since DWT applies entirely in the spatial domain, no conversion or division of the kernel in the spatial frequency domain is needed, which reduces low frequency distortion in the resultant susceptibility maps. More theoretical background of the DWT on relationship with the Laplace equation may need to be further investigated.Using a numerical phantom, various processing elements for QSM are analyzed. The computer simulation shows that the DWT method efficiently removes the background phase without introducing low frequency distortions at the resultant susceptibility maps. The proposed DWT method also shows superb performances over exiting techniques in the QSM experiments with volunteer subjects at 7.0T and 3.0T MRI, without slowly varying distortions at the susceptibility maps.