21-09-2012, 03:12 PM
Sparse Super-Resolution with Space Matching Pursuits
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Abstract
Super-resolution image zooming is possible when
the image has some geometric regularity. Directional interpolation
algorithms are used in industry, with ad-hoc regularity
measurements. Sparse signal decompositions in dictionaries of
curvelets or bandlets find indirectly the directions of regularity by
optimizing the sparsity. However, super-resolution interpolations
in such dictionaries do not outperform cubic spline interpolations.
It is necessary to further constraint the sparse representation,
which is done through projections over structured vector spaces.
A space matching pursuit algorithm is introduced to compute
image decompositions over spaces of bandlets, from which a
super-resolution image zooming is derived. Numerical experiments
illustrate the efficiency of this super-resolution procedure
compared to cubic spline interpolations.
INTRODUCTION
Zooming operators that increase the size of images are often
needed for digital display of images or videos. When images
are aliased, linear interpolations [11] introduce artifacts such as
Gibbs oscillations or zigzag along edges, and restore a blurred
image. Better images can be estimated with super-resolution
procedures which take advantage of this aliasing together with
some geometric image properties.
A super-resolution algorithm computes a signal estimation
in a space of dimension larger than the input signal size.
Super-resolution algorithms are necessarily non-linear and can
recover high frequency information by taking advantage of
prior signal information. Super-resolution zooming algorithms
are possible by interpolating the image along directions for
which it is geometrically regular. Directional interpolations,
usually known as edge-directed or content-adaptive interpolation,
interpolate along directions that are computed with adhoc
directional regularity estimations [8], [13]. These algorithms
are used in industry with good numerical results.
From Directional Interpolation to Sparsity
If a signal f has some directional geometric regularity then
it has a sparse representation in a dictionary D = fgpgp2¡
of curvelets [2], [1] or bandlets [7], [6], [10]. Finding an
appropriate direction of interpolation can be connected to
sparse super-resolution estimation, although we shall see that
this sparse super-resolution estimation may not perform well
for image interpolation. In the following we shall concentrate
on a dictionary of curvelets but the same conclusions apply
to a bandlet dictionary. A curvelet is an elongated oscillatory
waveform whose Fourier transform is concentrated along a
particular direction of the Fourier plane, as illustrated in Fig. 2.
Bandlet Sparsity and Interpolations
A band space Wb is selected so that wavelet coefficients
have a high energy variation within the corresponding band
given that its angle is selected among local directions of
regularity. To verify that wavelet coefficients are indeed regular
in the band, we check that they have a sparse representation
in a bandlet basis obtained by computing a directional wavelet
transform of these wavelet coefficients [7], [6]. If it is indeed
the case, then wavelet coefficients are interpolated along the
band direction which is a direction of regularity. Otherwise,
coefficients in the bands are interpolated with a cubic spline
interpolation.
CONCLUSION
Super-resolution with directional interpolations requires accurate
estimation of the image directional regularity. We introduced
an algorithm which computes directions of regularity
by optimizing a structured sparse signal representation. This
structured representation is computed by projecting the signal
over vector spaces chosen from a dictionary, as opposed to
a projection over individual vectors. It is computed with a
cascade of space matching pursuits. This algorithm reduces the
amount of computations compared to an optimization using individual
dictionary vectors and provides more accurate superresolution
image estimations. When the image has regular
geometric structures, the algorithm provides a better SNR and
a better visual image quality than a cubic spline interpolation.