We propose a variational approach to obtain superresolution images from multiple low-resolution frames extracted from video clips. First the displacement between the lowresolution frames and the reference frame are computed by an optical flow algorithm. Then a low-rank model is used to construct the reference frame in high-resolution by incorporating the information of the low-resolution frames. The model has two terms: a 2-norm data fidelity term and a nuclear-norm regularization term. Alternating direction method of multipliers is used to solve the model. Comparison of our methods with other models on synthetic and real video clips show that our resulting images are more accurate with less artifacts. It also provides much finer and discernable details.
We propose to combine cepstrum and nonlinear time–frequency (TF) analysis
to study multiple component oscillatory signals with time-varying frequency and
amplitude and with time-varying non-sinusoidal oscillatory pattern. The concept of
cepstrum is applied to eliminate the wave-shape function influence on the TF analysis,
and we propose a new algorithm, named de-shape synchrosqueezing transform (deshape
SST). The mathematical model, adaptive non-harmonic model, is introduced
and the de-shape SST algorithm is theoretically analyzed. In addition to simulated
signals, several different physiological, musical and biological signals are analyzed to
illustrate the proposed algorithm.
We investigate the energy landscape of the mixed even p-spin model with Ising spin configurations.
We show that for any given energy level between zero and the maximal energy, with
overwhelming probability there exist exponentially many distinct spin configurations such that
their energies stay near this energy level. Furthermore, their magnetizations and overlaps are
concentrated around some fixed constants. In particular, at the level of maximal energy, we
prove that the Hamiltonian exhibits exponentially many orthogonal peaks. This improves the
results of Chatterjee  and Ding-Eldan-Zhai , where the former established a logarithmic
size of the number of the orthogonal peaks, while the latter proved a polynomial size. Our second
main result obtains disorder chaos at zero temperature and at any external field. As a byproduct,
this implies that the fluctuation of the maximal energy is superconcentrated when the external
field vanishes and obeys a Gaussian limit law when the external field is present.