524 research outputs found
SO(3)-invariant asymptotic observers for dense depth field estimation based on visual data and known camera motion
In this paper, we use known camera motion associated to a video sequence of a
static scene in order to estimate and incrementally refine the surrounding
depth field. We exploit the SO(3)-invariance of brightness and depth fields
dynamics to customize standard image processing techniques. Inspired by the
Horn-Schunck method, we propose a SO(3)-invariant cost to estimate the depth
field. At each time step, this provides a diffusion equation on the unit
Riemannian sphere that is numerically solved to obtain a real time depth field
estimation of the entire field of view. Two asymptotic observers are derived
from the governing equations of dynamics, respectively based on optical flow
and depth estimations: implemented on noisy sequences of synthetic images as
well as on real data, they perform a more robust and accurate depth estimation.
This approach is complementary to most methods employing state observers for
range estimation, which uniquely concern single or isolated feature points.Comment: Submitte
Hierarchical structure-and-motion recovery from uncalibrated images
This paper addresses the structure-and-motion problem, that requires to find
camera motion and 3D struc- ture from point matches. A new pipeline, dubbed
Samantha, is presented, that departs from the prevailing sequential paradigm
and embraces instead a hierarchical approach. This method has several
advantages, like a provably lower computational complexity, which is necessary
to achieve true scalability, and better error containment, leading to more
stability and less drift. Moreover, a practical autocalibration procedure
allows to process images without ancillary information. Experiments with real
data assess the accuracy and the computational efficiency of the method.Comment: Accepted for publication in CVI
Provably Convergent Schr\"odinger Bridge with Applications to Probabilistic Time Series Imputation
The Schr\"odinger bridge problem (SBP) is gaining increasing attention in
generative modeling and showing promising potential even in comparison with the
score-based generative models (SGMs). SBP can be interpreted as an
entropy-regularized optimal transport problem, which conducts projections onto
every other marginal alternatingly. However, in practice, only approximated
projections are accessible and their convergence is not well understood. To
fill this gap, we present a first convergence analysis of the Schr\"odinger
bridge algorithm based on approximated projections. As for its practical
applications, we apply SBP to probabilistic time series imputation by
generating missing values conditioned on observed data. We show that optimizing
the transport cost improves the performance and the proposed algorithm achieves
the state-of-the-art result in healthcare and environmental data while
exhibiting the advantage of exploring both temporal and feature patterns in
probabilistic time series imputation.Comment: Accepted by ICML 202
Implicit regularization and momentum algorithms in nonlinear adaptive control and prediction
Stable concurrent learning and control of dynamical systems is the subject of
adaptive control. Despite being an established field with many practical
applications and a rich theory, much of the development in adaptive control for
nonlinear systems revolves around a few key algorithms. By exploiting strong
connections between classical adaptive nonlinear control techniques and recent
progress in optimization and machine learning, we show that there exists
considerable untapped potential in algorithm development for both adaptive
nonlinear control and adaptive dynamics prediction. We first introduce
first-order adaptation laws inspired by natural gradient descent and mirror
descent. We prove that when there are multiple dynamics consistent with the
data, these non-Euclidean adaptation laws implicitly regularize the learned
model. Local geometry imposed during learning thus may be used to select
parameter vectors - out of the many that will achieve perfect tracking or
prediction - for desired properties such as sparsity. We apply this result to
regularized dynamics predictor and observer design, and as concrete examples
consider Hamiltonian systems, Lagrangian systems, and recurrent neural
networks. We subsequently develop a variational formalism based on the Bregman
Lagrangian to define adaptation laws with momentum applicable to linearly
parameterized systems and to nonlinearly parameterized systems satisfying
monotonicity or convexity requirements. We show that the Euler Lagrange
equations for the Bregman Lagrangian lead to natural gradient and mirror
descent-like adaptation laws with momentum, and we recover their first-order
analogues in the infinite friction limit. We illustrate our analyses with
simulations demonstrating our theoretical results.Comment: v6: cosmetic adjustments to figures 4, 5, and 6. v5: final version,
accepted for publication in Neural Computation. v4: significant updates,
revamped section on dynamics prediction and exploiting structure. v3: new
general theorems and extensions to dynamic prediction. 37 pages, 3 figures.
v2: significant updates; submission read
Trifocal Relative Pose from Lines at Points and its Efficient Solution
We present a new minimal problem for relative pose estimation mixing point
features with lines incident at points observed in three views and its
efficient homotopy continuation solver. We demonstrate the generality of the
approach by analyzing and solving an additional problem with mixed point and
line correspondences in three views. The minimal problems include
correspondences of (i) three points and one line and (ii) three points and two
lines through two of the points which is reported and analyzed here for the
first time. These are difficult to solve, as they have 216 and - as shown here
- 312 solutions, but cover important practical situations when line and point
features appear together, e.g., in urban scenes or when observing curves. We
demonstrate that even such difficult problems can be solved robustly using a
suitable homotopy continuation technique and we provide an implementation
optimized for minimal problems that can be integrated into engineering
applications. Our simulated and real experiments demonstrate our solvers in the
camera geometry computation task in structure from motion. We show that new
solvers allow for reconstructing challenging scenes where the standard two-view
initialization of structure from motion fails.Comment: This material is based upon work supported by the National Science
Foundation under Grant No. DMS-1439786 while most authors were in residence
at Brown University's Institute for Computational and Experimental Research
in Mathematics -- ICERM, in Providence, R
Learning and Designing Stochastic Processes from Logical Constraints
Stochastic processes offer a flexible mathematical formalism to model and
reason about systems. Most analysis tools, however, start from the premises
that models are fully specified, so that any parameters controlling the
system's dynamics must be known exactly. As this is seldom the case, many
methods have been devised over the last decade to infer (learn) such parameters
from observations of the state of the system. In this paper, we depart from
this approach by assuming that our observations are {\it qualitative}
properties encoded as satisfaction of linear temporal logic formulae, as
opposed to quantitative observations of the state of the system. An important
feature of this approach is that it unifies naturally the system identification
and the system design problems, where the properties, instead of observations,
represent requirements to be satisfied. We develop a principled statistical
estimation procedure based on maximising the likelihood of the system's
parameters, using recent ideas from statistical machine learning. We
demonstrate the efficacy and broad applicability of our method on a range of
simple but non-trivial examples, including rumour spreading in social networks
and hybrid models of gene regulation
On the Convergence and Sample Complexity Analysis of Deep Q-Networks with -Greedy Exploration
This paper provides a theoretical understanding of Deep Q-Network (DQN) with
the -greedy exploration in deep reinforcement learning. Despite
the tremendous empirical achievement of the DQN, its theoretical
characterization remains underexplored. First, the exploration strategy is
either impractical or ignored in the existing analysis. Second, in contrast to
conventional Q-learning algorithms, the DQN employs the target network and
experience replay to acquire an unbiased estimation of the mean-square Bellman
error (MSBE) utilized in training the Q-network. However, the existing
theoretical analysis of DQNs lacks convergence analysis or bypasses the
technical challenges by deploying a significantly overparameterized neural
network, which is not computationally efficient. This paper provides the first
theoretical convergence and sample complexity analysis of the practical setting
of DQNs with -greedy policy. We prove an iterative procedure with
decaying converges to the optimal Q-value function geometrically.
Moreover, a higher level of values enlarges the region of
convergence but slows down the convergence, while the opposite holds for a
lower level of values. Experiments justify our established
theoretical insights on DQNs
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