5,250 research outputs found
Scalable Frames and Convex Geometry
The recently introduced and characterized scalable frames can be considered
as those frames which allow for perfect preconditioning in the sense that the
frame vectors can be rescaled to yield a tight frame. In this paper we define
-scalability, a refinement of scalability based on the number of non-zero
weights used in the rescaling process, and study the connection between this
notion and elements from convex geometry. Finally, we provide results on the
topology of scalable frames. In particular, we prove that the set of scalable
frames with "small" redundancy is nowhere dense in the set of frames.Comment: 14 pages, to appear in Contemporary Mat
Measures of scalability
Scalable frames are frames with the property that the frame vectors can be
rescaled resulting in tight frames. However, if a frame is not scalable, one
has to aim for an approximate procedure. For this, in this paper we introduce
three novel quantitative measures of the closeness to scalability for frames in
finite dimensional real Euclidean spaces. Besides the natural measure of
scalability given by the distance of a frame to the set of scalable frames,
another measure is obtained by optimizing a quadratic functional, while the
third is given by the volume of the ellipsoid of minimal volume containing the
symmetrized frame. After proving that these measures are equivalent in a
certain sense, we establish bounds on the probability of a randomly selected
frame to be scalable. In the process, we also derive new necessary and
sufficient conditions for a frame to be scalable.Comment: 27 pages, 5 figure
On Optimal Frame Conditioners
A (unit norm) frame is scalable if its vectors can be rescaled so as to
result into a tight frame. Tight frames can be considered optimally conditioned
because the condition number of their frame operators is unity. In this paper
we reformulate the scalability problem as a convex optimization question. In
particular, we present examples of various formulations of the problem along
with numerical results obtained by using our methods on randomly generated
frames.Comment: 11 page
Scalable Dense Non-rigid Structure-from-Motion: A Grassmannian Perspective
This paper addresses the task of dense non-rigid structure-from-motion
(NRSfM) using multiple images. State-of-the-art methods to this problem are
often hurdled by scalability, expensive computations, and noisy measurements.
Further, recent methods to NRSfM usually either assume a small number of sparse
feature points or ignore local non-linearities of shape deformations, and thus
cannot reliably model complex non-rigid deformations. To address these issues,
in this paper, we propose a new approach for dense NRSfM by modeling the
problem on a Grassmann manifold. Specifically, we assume the complex non-rigid
deformations lie on a union of local linear subspaces both spatially and
temporally. This naturally allows for a compact representation of the complex
non-rigid deformation over frames. We provide experimental results on several
synthetic and real benchmark datasets. The procured results clearly demonstrate
that our method, apart from being scalable and more accurate than
state-of-the-art methods, is also more robust to noise and generalizes to
highly non-linear deformations.Comment: 10 pages, 7 figure, 4 tables. Accepted for publication in Conference
on Computer Vision and Pattern Recognition (CVPR), 2018, typos fixed and
acknowledgement adde
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