Image super-resolution (SR) restoration amounts to reconstruct a high-quality image from its degraded measurement. Such resolution-enhancing technology prove to be essential in medical MR imaging where high-quality image are not prone to acquire or unavailable in the presence of uncontrolled motion such as moving subjects and organs or fetal and neonatal magnetic resonance imaging.

The SR image techniques can be classified into frequency domain, interpolation, regularization [1] and learning based approaches [2, 3]. The first three categories get a high-resolution (HR) image from a set of low-resolution (LR) input images, while in the last approach the high frequency information of the single LR image is enhanced by retrieving the most likely high-frequency information from the HR training samples. Therefore how to reveal the underlying relations between the HR and the LR patch spaces is the key issue. For the contents of local patches across an image vary significantly, a set of subdictionaries are designed by clustering the training HR data. In this work, we propose to cluster the pre-collected HR example patches to generate subdictionary and select the proper subdictionary for any image patch according to the frequency spectrum feature in Fourier domain, because the characteristic of the Fourier spectrogram can reflect the feature complexity, local directionality and the texture periodicity of the image patch simultaneously.

As shown in Figure 1, we present typical image patches vs their Fourier frequency spectrograms. The first patch is uniform in gray intensity, and its frequency spectrum almost concentrate at zero. The patches 2 to 6 just have edge of single direction in image domain, corresponding to a spectrum line in frequency spectrogram perpendicular to the image edges, while the patches 7 to 12 containing slightly more complex edges presents two cross spectrum lines. The image patches of the last group are composed of more complicated structures and fine features, leading to spectra lines including a cross and a slash. However, when the patch structure are too mass, their frequency spectrograms will not exhibit definite spectrum feature as patches 16 to 18. On this basis, we apply the frequency spectra to do the patch clustering instead of K-means based on the Euclidian distance similarity [4] or structured sparsity with mixture Gaussian distributions [5], which are not quite reliable. With pixel set in direction θ denoted as S_θ={i(1),i(2)…i(n)}, the number of directions and the major direction of the spectrum lines in Fourier spectrograms can be determined as shown in Figure 2, and then all the image patches are clustered into 8 classes with the major spectum direction θ_max using the scheme illustrated in Figure 3.

In the dictionary training process, once the cluster are completed, they are stored as a trained dictionary composed of multiple subdictionaries, which can be directly used in the reconstruction. Combined with sparsity constraint and weighted regularization terms, the SR reconstruction model can be formulated as follow

$$min‖DHX-Y‖_2^2 +λ‖ΦX‖_1+β‖M∇X‖_1$$

The first item is reconstruction fidelity energy where D means to down sampling and H denotes as deblurring operator, and the last item is a prior regularity to enhance the image edges, where weighting matrix M is defined that large gradient is zeros while uniform region is one. The second item is a sparse constrain in the transform domain Φ, herein is the PCA bases of the image patches of same θ_max in each subdictionary. A brain MR image and its patch Fourier spectrograms are demonstrated in Figure 4 (patch size 15×15), where first 100 patches are selected for each clusters. It is observed that all the patches seperated into different groups have similar structure complexity and feature direction. Thus, the subdictionary construted from this kind of clustering is more adaptive and also behaves better sparsity. This optimization model can be solved in two steps by iterative soft threshold algorithm and conjugate gradient algorithm. Figure 5 demonstrated the SR reconstructed image of our proposed method (PSNR=30.06, MSSIM=0.95) compared to the original HR image and bicubic interplation method (PSNR=23.22, MSSIM=0.87), with image high resolution of 256×256 and low resolution of 128×128. We can see that our method can recover more fine details as shown in zoomed views with better evaluation parameters.