Indian Institute of Science Bangalore
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Understanding Catchment Scale Processes: Hydrological Modelling and Information-Theoretic Approaches
No full textCatchments are complex environmental systems, and they serve as the fundamental units
for hydrological classification. They are self-organizing systems whose form, drainage
network, ground and channel slopes, channel hydraulic geometries, soils and vegetation,
are all a result of adaptive ecological, geomorphic and land-forming processes. The hydrological
responses of a catchment are predominantly governed by complex interactions
among processes occurring at various spatial and temporal scales. As hydrological processes
exhibit non-linear behaviour at all scales, it is important to explore their intricate
relationships and have a detailed understanding of the catchment behaviour. Quantification
of morphometric indices and hydrological signatures provide vital information about
the complex system properties and the functional behaviour of catchments. Evaluation
of catchment characteristics can significantly improve the scientific understanding of the
variability of hydrological processes at various scales and provide useful insights for the
development of scaling relationships.
Hydrological modelling serves as a powerful tool in assimilating the complex behaviour
of hydrological systems. The performance and applicability of each hydrological model
can differ between catchments due to several catchment characteristics and dominant hydrological
processes. With a wide variety of model structures, it is important to evaluate
how different hydrological models capture the process dynamics in various catchments.
Many a time, the use of a single model can lead to simulation uncertainties, especially in
catchments of poor input data availability and in large-scale modelling exercises. Hence,
effective modelling strategies should be designed in such a way that the inclusion of more
than one hydrological model is ensured, and an ensemble approach should be adopted,
especially in highly heterogeneous catchments.
The application of information-theoretic measures has been found to be extremely useful
in tackling various problems related to hydrological modelling and understanding process
relationships. Information theory serves as a powerful tool in computing the information
content in a variable as well as the amount of information one variable provides about
another. Also, such measures do not require any prior assumptions on the characteristics
of the underlying distributions. Hence, they can be widely applied to address a variety
of problems in the hydrological domain.
The key focus of the research presented in this thesis is to evaluate catchment scale
hydrological process relationships by adopting a model-oriented approach in a regionally
complex catchment. A holistic study of the catchment scale processes is carried out
by combining a model-based analysis and applying statistical evaluation methods and
information-theoretic measures. The study area chosen for the analyses is the Cauvery
River Basin, a major river basin in peninsular India.
The thesis contributes towards providing an understanding of hydrological processes
at the catchment scale by combining the knowledge gained through hydrological modelling
with information-theoretic measures. Catchment characteristics are quantified by
evaluating various geomorphologic indices which serve as a baseline for building better
modelling strategies. Three hydrological models, namely, GWAVA (Global Water
AVailability Assessment) model, SWAT (Soil Water Assessment Tool) and VIC (Variable
Infiltration Capacity) model, are set up for the study region, and their individual
performances along with an ensemble mean simulation are investigated. Additionally,
to develop deeper insights into the long-term hydro-climatology and distribution of water
resources within the study region, a synthesis of hydrological model evaluations and
statistical methods is adopted. To further explore the relationships between various hydrological
fluxes simulated using a physically-based hydrological model, a methodology
is suggested through the application of information-theoretic measures such as Shannon
Entropy and Mutual Information
Shock tube experimental and advanced computational investigations on pyrolysis of cyclohexane derivatives and C2 + C2 reaction
No full textMy thesis involves studying the pyrolysis/thermal decomposition of cyclohexane derivatives (important constituents of conventional transportation fuel). From the literature as well as the composition analysis of two fuels (JP-7(eq) and RP-1) performed in our lab using GC-MS, we found that cyclohexane derivatives can be categorized into different types: single side-chain alkylcyclohexane; multiple side-chain alkylcyclohexane, and decalin (two fused cyclohexane ring).
We have taken iso-propylcyclohexane as a member of single side-chain alkylcyclohexane and 1,3,5-trimethylcyclohexane as a member of multiple side-chain alkylcyclohexane. We have carried out a detailed theoretical study on the various possible reactions of both molecules and associated intermediates/radicals involved in pyrolysis. State-of-the-art quantum chemical calculations have allowed us to probe the transition states for all the reactions and calculate the high-pressure rate constants. We have also carried out experimental investigations using a single pulse shock tube for both molecules. Concentrations of all the major and minor products have been measured using GC-FID and GC-MS. Experimental measurements have been used to validate the kinetic mechanism proposed in the theoretical study. We have used Chemkin for kinetic simulation. We have proposed detailed kinetic models for the pyrolysis of iso-propylcyclohexane as well as 1,3,5-trimethylcyclohexane.
For decalin, we have done only a detailed theoretical investigation of its thermal decomposition and proposed a detailed kinetics mechanism of its decomposition based on our calculations. We have also calculated high-pressure limit rate parameters for all the elementary reactions.
Other important part of my thesis involves the computational investigation of C2+C2 reaction. The addition of these pure carbon molecules is very important in combustion. Our calculation shows that the C2+C2 reaction leads to C3+C, which is not a barrier-less reaction, occurring via C4. We have calculated pressure-dependent rate parameters to determine the temperature and pressure condition in which C2+C2 leads to C3+C. We also found that spin states of C2 play a key role in this reaction
Studies on modeling the mechanics of slender elastic ribbons
No full textRibbons are slender structures characterized by three disparate geometric dimensions: length >> width >> thickness. Such a dimensional disparity enables ribbons to bend, buckle, twist and crease response to simple loading conditions. Their nonlinear deformation behavior, once considered a hindrance, is now routinely exploited in engineering applications related to stretchable electronics and flexible robotics. Such applications demand a systematic understanding of the mechanics of elastic ribbons using experiments, modeling, and simulations. This thesis is a step in this direction.
Experiments using annulus-shaped ribbons and Moebius strips serve as our point of departure. The critical challenge in these experiments lies in measuring complex three-dimensional deformations observed. Routinely used techniques turn out to be inadequate, either due to the compliance of ribbon structures (e.g., contact probes, strain gauges) or due to the large displacements and rotations involved (DIC). We leverage novel computer vision techniques developed in the lab to faithfully digitize shapes and sample deformation maps of ribbons in the experiments. These measurements lead us to the main contributions of this thesis--- a detailed examination of the predictive capabilities of commonly used modeling approaches and the formulation of a dedicated one-dimensional ribbon model.
The physical appearance of ribbons motivates modeling them either as thin elastic plates or as elastic rods having a slender cross-section. Widespread adoption of the von Karman plate theory and the Kirchhoff rod model exemplifies this dichotomy. Somewhat surprisingly, comparing finite element simulations of these models with experimental measurements reveals both approaches to be deficient, even in simpler scenarios than ones where they are routinely used. These studies show that it is essential to permit large displacements and rotations in ribbon models and that compliance in the direction of the width, though small, plays an important role. Indeed, the experiments with annular ribbons and Moebius strips are designed to highlight these deformation features.
We propose adopting the small-strain Cosserat plate theory instead. The model's generality, along with a robust finite element implementation that addresses issues of numerical locking by adopting high order elements and approximating large rotations using exponential maps, translates to excellent agreement with experimental measurements. The model faithfully reproduces measured shapes, displacement fields and curvature distributions, as well as bifurcations and energy localization phenomena observed in experiments.
We then propose a dedicated reduced-order one-dimensional ribbon model by systematically incorporating kinematic assumptions in the plate theory. The model is Sadowsky-type theory that requires one additional field to describe lateral curvatures along the width of the ribbon. We examine the model's predictions through challenging examples, including one involving twist-induced snap-through. The model promises to be a valuable tool to develop insights into the mechanics of ribbons, besides being a compelling alternative to the Sadowsky and Wunderlich ribbon models routinely used in the literature
Towards Electronic Scale Dynamics of Chemical Bonding in Isolated Molecules and Solvated Species
No full textThe chemical bonding in molecules and molecular clusters is determined by the nature of the electron density distribution between the atoms. Over the past century, different theories such as Lewis’s bond, valence-bond theory, molecular orbital theory, and frontier molecular orbital theory, have explained static chemical bond in terms of electron density and their symmetry. Over the past century, “dynamic” nature of the chemical bond has been explored by monitoring the nuclear dynamics in femtosecond and picosecond time regimes. In my thesis work, mostly a computational attempt has been made to explore electronic-scale dynamic nature of chemical bonding in isolated molecules and molecules in solvent environment. As electron density can move over a few angström distance within attosecond time scale, dynamic nature of chemical bonding can be named as attochemistry. To explore attochemistry, vertical ionization scheme has been adopted along with the time-dependent NBO analysis.1
Our discussion of the “dynamic” nature of the chemical bonding must begin with an intriguing comparative study of the potential-energy curves for neutral H2 and cationic H2+, and analogous neutral Li2 and cationic Li2+ species. These curves are well-documented in the literature at different levels of theory. Note that neutral H2 exhibits stronger bonding than neutral Li2 (the potential well of neutral H2 is quite deeper than that of neutral Li2). On the other hand, respective cationic potential energy curves evidence that while the H-H bond becomes weaker following ionization, the same leads to strengthening of the Li-Li bond, revealing a different fate for the adiabatic ions of the respective species. Here, one may ask an important question very relevant to the subject of the present thesis, “Does the vertical ion incorporate the electronic-scale change in the chemical bonding which ultimately dictates the fate of the adiabatic ion?” Characters of the NBOs are determined from the natural hybrid orbitals (NHOs) shows that neutral σH-H is formed by the overlap of two pure 1s orbitals (NBO analysis predicts more than 99% 1s contribution). Similarly, σLi-Li bond also originates from overlap between (almost) pure 2s orbitals (NBO analysis predicts more than 95% 2s contribution). The vertical ionization of the H2 molecule removes an electron from bonding orbital . NBO analysis again predicts more than 99% contribution of the 1s orbital to singly occupied orbital (with α spin) in the vertical H2+ ion. This makes the H-H bond strength weaker upon vertical ionization and as a result, the H-H bond distance of the adiabatic H2+ ion becomes considerably longer than that of the neutral H2 species. Compared with the neutral Li2, the bonding hybrids of the vertical Li2+ ion develops significant sp-hybrid character, resulting in much more directional nature. The wB97xD/6-311+G(d,p) level of theory predicts that bonding NHOs of the vertical Li2+ ion to be of sp0.25 form (with more than 19% p character). In contrast, the corresponding hybridization in neutral Li2 is sp0.05 (with less than 5% p character). This enhanced p-character in NHOs of the vertical Li2+ ion also leads to greater inter-nuclear separation than that of neutral Li2 at the equilibrium geometry. Thus, above comparison of the chemical bonding in Li2 and Li2+ shows that strength of the chemical bond does not merely depend on the number of electrons in the bonding NBO (bond order); rather, it depends on the difference in the bonding hybrids, particularly with regard to the directional p-character. Similarly using NBO fate of a noncovalent bond can be predicted from NBO analysis. For the present discussion, CO:H2O hydrogen bonded complex is taken as an example. The highest E(2) value for the CO:H2O noncovalently bonded complex can be seen (increased after ionization), which also shows the NBOs responsible for the relevant charge transfer. The nC to σOH* charge transfer dominantly contributes to the O-C…..H noncovalent bonding interaction.
NBO analysis suggests that the vertical ionization-induced change of the chemical bonding in the Li2 species, in comparison with the H2 species, occurs at the electronic level without involving any nuclear movement. This electronic change of chemical bonding in the vertical ion dictates the fate (structure and energetics) of the adiabatic ion of the respective species. Using the scheme of the quantum dynamics simulation, one can simulate the temporal evolution of the change of chemical bonding in Li2 species following the vertical ionization scheme. Clearly, time dependent NBO analysis predicts extremely fast periodic expansion and contraction of the electron density in the inter-atomic space following the vertical ionization. This periodic movement represents the initial step of the ionization-induced re-hybridization dynamics of the Li-Li σ-chemical bond. No experiment with attosecond metrology has been performed thus far to unravel this re-hybridization dynamics.
Similarly, a systematic theoretical investigation of the attochemistry of solvated molecules would help one design attosecond experiments under ambient conditions to explore the attochemistry in a liquid environment. With this goal in mind, for the first time, we have explored the attochemistry of molecules surrounded by different non-polar solvent environments. To model solvation effects on the attochemistry of molecules containing gold–chalcogen linkages, we have used an implicit solvent model (Polarizable Continuum Model) under the density functional theory (DFT) formalism for non-polar solvents. We have found that the charge migration time scale in molecules becomes faster in the presence of the solvent environment as compared to that under vacuum. Charge oscillation does not damp quickly in molecules surrounded by the solvent environment as compared to that under vacuum. Furthermore, the direction of the charge migration may change in molecules when they are surrounded by the solvent environment as compared to that under vacuum. Thus, the present work has laid the foundation, for the first time, for thinking of the attochemistry into the realm of liquids.2
In addition to the above-mentioned computational efforts, certain preliminary experimental efforts have also been undertaken. Two important steps towards exploring non-equilibrium solvation effect on attosecond chemical bonding are to prepare a suitable model sample and activate liquid beam inside a vacuum. In this regard, an experimental attempt has been made to carry out laser ablation in liquid to synthesize metal-carbyne3 system which can be a good model system for exploring the attosecond non-equilibrium solvation. Finally liquid beam nozzle has been constructed to activate liquid jet inside the vacuum.
High harmonic generation spectroscopy is a self-probing spectroscopy which can probe the evolving attosecond electronic structure following ionization of a species. Hence, dependency of the HHG spectrum on the direction of recombination of the ejected electron has been explored.4 In the present contribution, taking the ion, atoms, and molecule as examples, we show that V(x) can be constructed from one-dimensional molecular electrostatic potential (MEP) of the respective cation to access theoretical HHG spectra not only of simple atoms but also of multielectron complex molecules. We have shown that the MEP-based formalism not only successfully reproduces the ionization energy of the molecular system but also gives explanation for the orientation-dependent HHG spectral intensity change in terms of the nodal plane of the molecular orbital from where the electron is removed during the HHG process. Further extension of the present one-dimensional model to the two- or three-dimensional potentials is expected to address many interesting and practical issues such as quantum interference and effects of different shapes of orbitals. We intend to explore those effects in the future.
In brief this thesis contributes to our fundamental understanding of electronic scale dynamics of chemical bonding. Our hitherto-taken attempts and the progresses in exploring electronic-scale dynamics of chemical bonding are discussed in the following chapters
Ion insertion and Ion-host Interactions in Diketopyrrolopyrrole-based π-Conjugated Polymers
No full textπ-conjugated organic semiconductors are a class of technologically important materials with broad applicability in areas as diverse as energy conversion and storage, bioelectronics, and consumer electronics. This is because their properties can be tailored to suit a host of applications by, for example, subtle changes to their structure, conjugation length, side chains, crystallinity, and doping. Diketopyrrolopyrrole (DPP)-based π-conjugated polymers exhibit high ambipolar carrier mobilities and high performance in organic field effect transistors and organic photovoltaics.1 Owing to their low band gap, redox activity and favorable redox potentials, DPP-based π-conjugated polymers could also be suitable as channel materials for organic electrochemical transistors, electrodes for ion-insertion batteries, neuromorphic devices and sensing elements in bioelectronic applications. Electrochemical potential-dependent control of polymer redox coupled with ion-insertion, and the subsequent electrochemical doping-induced changes to the electronic conductivity of the semiconducting host, are central to the realization of these functional devices.
In this work, we used organic electrochemical transistors (OECTs) as a platform to simultaneously probe the mechanistic aspects of ion-insertion and electrochemical doping in DPP-based π-conjugated polymers.2 Based on OECT measurements, we discuss the role of ion-size, polymer side chains and electrolyte composition in dictating ion-insertion and transconductance of OECT devices. We corroborate these findings with spectroelectrochemical and electrochemical impedance measurements. Strikingly, our results show a polymer side-chain dependent asymmetry for the insertion of cations and anions. These cannot be fully understood when the electrochemical dopant is assumed to be in a polymer host matrix with a uniform dielectric constant. We discuss whether electrochemical dopant and host polymer interactions should be treated like ion-solvent interactions in classical electrolyte solutions.
Based on these mechanistic insights, we developed DPP-based OECTs that have balanced n- and p-type OECT device performance. We then configured arrays of single-component DPP-based OECTs into complementary like logic circuits to demonstrate NOT, 2-input NAND and 2-input NOR logic gates.3 Since the NAND and NOR circuits are configured from identical single-component OECTs, the NAND and NOR operations are fully voltage reconfigurable. Our work is a significant step towards building OECT-based bioelectronics and polymorphic circuits. Further, owing to their fast redox kinetics, we show that DPP-based polymers could be suitable as cathode materials in Li-ion batteries. We demonstrate fast and stable cycling of DPP-based cathodes in a Li-metal based half-cell with an average voltage of 2.2 V, 70% capacity retention at 500 C and high cycling stability of up to 1000 cycles.4
Our work is an important step towards a mechanistic understanding of ion-insertion and electrochemical doping in π-conjugated organic semiconductors. This understanding of mixed ionic and electronic conduction of these materials will be critical for a host of applications such as for example, i) interfacing circuits that mediate communication between an ionic biological circuitry and a fully electronic circuitry; ii) electrodes for energy storage devices; iii) electrochromic windows, iv) neuromorphic devices and v) polymorphic circuits with concealed logic operations.
References:
1. Catherine Kanimozhi, , Nir Yaacobi-Gross, Kang Wei Chou, Aram Amassian, Thomas D. Anthopoulos, and Satish Patil. “Diketopyrrolopyrrole–Diketopyrrolopyrrole-Based Conjugated Copolymer for High-Mobility Organic Field-Effect Transistors.” Journal of the American Chemical Society 134, no. 40 (2012): 16532–35.
2. Jonathan Rivnay, Sahika Inal, Alberto Salleo, Róisín M. Owens, Magnus Berggren, and George G. Malliaras. “Organic Electrochemical Transistors.” Nature Reviews Materials 3, no. 2 (2018): 17086.
3. Jibin J. Samuel, Ashutosh Garudapalli, Aiswarya Abhisek Mohapatra, Chandrasekhar Gangadharappa, Satish Patil, and Naga Phani B. Aetukuri. “Single-Component CMOS-Like Logic Using Diketopyrrolopyrrole-Based Ambipolar Organic Electrochemical Transistors.” Advanced Functional Materials 31, no. 45 (2021): 2102903.
4. Jibin J. Samuel, Varun Kumar Karrothu, Ram Kumar Canjeevaram Balasubramanyam, Aiswarya Abhisek Mohapatra, Chandrasekhar Gangadharappa, Varun Ravi Kankanallu, Satish Patil, and Naga Phani B. Aetukuri. “Ionic Charge Storage in Diketopyrrolopyrrole-Based Redox-Active Conjugated Polymers.” The Journal of Physical Chemistry C 125, no. 8 (2021): 4449–57
Data Efficient Domain Generalization
No full textDeep neural networks has brought tremendous success in many areas of computer vision, such
as image classification, retrieval, segmentation , etc. However, this success is
mostly measured under two conditions namely (1) the underlying distribution of the test data
is the same as the distribution of the data used for training the network and (2) The classes
available for testing is the same as the one in training. These assumptions are very restrictive
in nature and may not hold in real-life. Since new data categories are continuously being
discovered, so it is important for the trained models to generalize to classes which has not
been seen during training. Also, since the conditions under which the data is captured keeps
on changing, so it is important for the trained model to generalize across unseen domains,
which it has not encountered during training. Also, in general, the information about the class
(whether it belongs to a seen or unseen class) or domain will not be known a-priori.
Recently, researchers have started to address the challenging scenarios associated with
a deep network, when the testing conditions in terms of classes and domains are relaxed.
Towards that end, domain generalization (DG) for tasks like image classification, object detection,
etc. have gained significant attention. In this work, we focus on the image classification
task. In the first work, we address the scenario where the test data domain can be different
and in the second work, we address the even more general scenario (ZSDG), where both the
class and domain of the test data can be different from that of the training data. In DG, a
deep model is trained to generalize well on an unknown target domain, leveraging data from
multiple source domains during training for the task of image classification. In ZSDG, the aim
is to train the model using multiple source domains and attributes of the classes such that it
can generalize well to novel classes from out-of-distribution target data
Panchromatic study of star clusters: binaries, blue lurkers, blue stragglers and membership
No full textBinary systems can evolve into immensely different exotic systems such as blue straggle stars (BSSs), yellow straggler stars, cataclysmic variables, type Ia supernovae depending on their initial mass, the orbital parameters and evolution. The aim of this thesis is to understand the demographics of post-mass-transfer systems (BSSs, white dwarfs and blue lurkers) present in the open clusters and how they are formed. First, we identified the cluster members using Gaia EDR3 data in six open clusters. Two of the clusters, M67 and King2, were studied in detail using UVIT, Gaia, GALEX, 2MASS and other archival photometric data. The comprehensive panchromatic study showed that (i) there is a robust mass-transfer pathway for BSSs, and blue lurkers in M67, (ii) at least 15% of BSSs in King 2 were formed via binary mass transfer. We also created a homogeneous catalogue of open cluster BSSs using Gaia DR2 data. The analysis of 868 BSSs across 208 clusters showed that (i) BSS frequency increases with age, (ii) there is a power-law relation between cluster mass and maximum number of BSSs, (iii) the formation mechanism of BSSs is dominated by binary mass transfer (54-67%) though there exists a 10-16% chance of BSSs forming through more than 2 stellar interactions. This study demonstrates that there exists an extensive variety in the demographics of binary products, and the UV observations are vital for their detection and characterisation
Standoff Target Tracking Guidance using Line-of-Sight Distance Bifurcation
No full textUnmanned aerial vehicle (UAV) applications that continue to receive significant attention are target acquisition and tracking. There exist many paths to track a stationary target from a prescribed altitude. The most widely used ones are straight lines and circular orbits. Circling a target at a constant radial distance is known as standoff tracking or circumnavigation. In this regard, an evolved requirement by the UAV is to reach a specified radial distance from the target within the stipulated time and continue circumnavigating the target with that radial distance. In doing so, it is desired that the UAV uses an easily computable guidance command with deterministic performance characteristics.
The thesis addresses the standoff target tracking problem by considering a modified two-parameter transcritical bifurcation in UAV-target line-of-sight distance dynamics. Appropriate choice of the bifurcation parameters results in the existence of a stable equilibrium point of the proposed line-of-sight distance dynamics which corresponds to the desired standoff radius. Further analysis relates the control parameters to the desired settling time, that is, the time taken by the UAV to settle on the desired standoff circle. A closed-form analytical expression is derived for the set of achievable settling times as a function of the two bifurcation parameters, the UAV speed, and initial separation. Simulation studies are carried out by considering a second-order heading-hold autopilot, a first-order speed control, and a limited turn rate for the UAV. Additional simulation studies are performed for realistic scenarios considering the presence of wind, noisy sensor measurements, and variable initial conditions. Simulation results demonstrate the robustness of the proposed guidance algorithm in achieving standoff target tracking with a constraint on the settling time. Overall, the proposed method offers a simple and easy-to-implement guidance solution
Edible Battery Design for Bio-Medical applications
No full textThis thesis is focused on the development of digestible electronics for healthcare.
The vision is the development of an edible pill that performs diagnosis inside the
body. Any such edible device needs an energy source made of non-toxic materials.
This thesis discusses the development of such an energy source.
The main idea is based on using a galvanic cell. This thesis constraints itself in
exploring Zn/Cu based cells using lemon extract-pectin blend extracts. Cells based
on a solid electrolyte, gel based electrolyte and a liquid electrolyte contained in a
hardened shell of isomalt are discussed. The load characterisitcs of the cells show
promise to deliver energy to power the device for 100 s. Furthermore, the thesis
details how the cells could be used as a sensor to sense the pH.
To summarize, the thesis develops the means and methods to develop a compact
edible power source
On Plug-and-Play Regularization using Linear Denoisers
No full textThe problem of inverting a given measurement model comes up in several computational imaging applications. For example, in CT and MRI, we are required to reconstruct a high-resolution image from incomplete noisy measurements, whereas in superresolution and deblurring, we try to infer the ground-truth from low-resolution or blurred images. Traditionally, this is done by minimizing , where is a data-fidelity (or loss) function that is determined by the acquisition process, and is a regularization (or penalty) function that is based on a subjective prior on the target image. The solution is obtained numerically using iterative algorithms such as ISTA or ADMM.
While several forms of regularization and associated optimization methods have been proposed in the imaging literature of the last few decades, the use of denoisers (aka denoising priors) for image regularization is a relatively recent phenomenon. This has partly been triggered by advances in image denoising in the last 20 years, leading to the development of powerful image denoisers such as BM3D and DnCNN. In this thesis, we look at a recent protocol called Plug-and-Play (PnP) regularization, where image denoisers are deployed within iterative algorithms for image regularization. PnP consists of replacing the proximal map --- an analytical operator at the core of ISTA and ADMM --- associated with the regularizer with an image denoiser. This is motivated by the intuition that off-the-shelf denoisers such as BM3D and DnCNN offer better image priors than traditional hand-crafted regularizers such as total variation. While PnP does not use an explicit regularizer, it still makes use of the data-fidelity function . However, since the replacement of the proximal map with a denoiser is ad-hoc, the optimization perspective is lost --- it is not clear if the PnP iterations can be interpreted as optimizing some objective function . Remarkably, PnP reconstructions are of high quality and competitive with state-of-the-art methods. Following this, researchers have tried explaining why plugging a denoiser within an inversion algorithm should work in the first place, why it produces high-quality images, and whether the final reconstruction is optimal in some sense.
In this thesis, we try answering such questions, some of which have been the topic of active research in the imaging community in recent years. Specifically, we consider the following questions.
--> Fixed-point convergence: Under what conditions does the sequence of iterates generated by a PnP algorithm converge? Moreover, are these conditions met by existing real-world denoisers?
--> Optimality and objective convergence: Can we interpret PnP as an algorithm that minimizes for some appropriate ?
Moreover, does the algorithm converge to a solution of this objective function?
--> Exact and robust recovery: Under what conditions can we recover the ground-truth exactly via PnP? And is the reconstruction robust to noise in the measurements?
While early work on PnP has attempted to answer some of these questions, many of the underlying assumptions are either strong or unverifiable. This is essentially because denoisers such as BM3D and DnCNN are mathematically complex, nonlinear and difficult to characterize. A first step in understanding complex nonlinear phenomena is often to develop an understanding of some linear approximation. In this spirit, we focus our attention on denoisers that are linear. In fact, there exists a broad class of real-world denoisers that are linear and whose performance is quite decent; examples include kernel filters (e.g. NLM, bilateral filter) and their symmetrized counterparts. This class has a simple characterization that helps to keep the analysis tractable and the assumptions verifiable. Our main contributions lie in resolving the aforementioned questions for PnP algorithms where the plugged denoiser belongs to this class. We summarize them below.
--> We prove fixed-point convergence of the PnP version of ISTA under mild assumptions on the measurement model.
--> Based on the theory of proximal maps, we prove that a PnP algorithm in fact minimizes a convex objective function , subject to some algorithmic modifications that arise from the algebraic properties of the denoiser. Notably, unlike previous results, our analysis applies to non-symmetric linear filters.
--> Under certain verifiable assumptions, we prove that a signal can be recovered exactly (resp. robustly) from clean (resp. noisy) measurements using PnP regularization. As a more profound application, in the spirit of classical compressed sensing, we are able to derive probabilistic guarantees on exact and robust recovery for the compressed sensing problem where the sensing matrix is random. An implication of our analysis is that the range of the linear denoiser plays the role of a signal prior and its dimension essentially controls the size of the set of recoverable signals. In particular, we are able to derive the sample complexity of compressed sensing as a function of distortion error and success rate.
We validate our theoretical findings numerically, discuss their implications and mention possible future research directions