Recently, negative refraction of electromagnetic waves in photonic crystals was demonstrated experimentally and subwavelength images were observed. However, these investigations all focused on the periodic structure. Here, we report a new theoretical and experimental finding that negative refraction can appear in some transparent quasicrystalline photonic structures. The photonic quasicrystals (PQCs) exhibit an effective refractive index close to ÿ1 in a certain frequency window. The index shows small spatial dispersion, consistent with the nearly homogeneous geometry of the quasicrystal. More interestingly, a superlens based on the 2D PQCs can form a non-near-field subwavelength image whose position varies with the source distance. These properties make PQCs promising for application in a range of optical devices.

l1-SPIRiT is a fast magnetic resonance imaging (MRI) method which combines parallel imaging (PI)with compressed sensing (CS) by performing a joint l1-norm and l2-norm optimization procedure.The original l1-SPIRiT method uses two-dimensional (2D) Wavelet transform to exploit the intra-coil data redundancies and a joint sparsity model to exploit the inter-coil data redundancies. In this work, we propose to stack all the coil images into a three-dimensional (3D) matrix, and then a novel 3D Walsh transform-based sparsity basis is applied to simultaneously reduce the intra-coil and inter-coil data redundancies. Both the 2D Wavelet transform-based and the proposed 3D Walsh transform-based sparsity bases were investigated in the l1-SPIRiT method. The experimental results show that the proposed 3D Walsh transform-based l1-SPIRiT method outperformed the original l1-SPIRiT in terms of image quality and computational efficiency.

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