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Isotopic effects on phonon anharmonicity in layered van der Waals crystals: Isotopically pure hexagonal boron nitride

Ramon Cusco, Luis Artus, James H. Edgar, Song Liu, Guillaume Cassabois,

Bernard Gil

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Narimane Lammari

Mohamad Tarhini

Karim Miladi

Ouahida Louaer

Hassen Abdeslam

Souad Meniai

Hatem Sfar

Abdelhamid Fessi

Elaïssari

Wahida Louaer

Abdeslam Hassen Meniai

Souad Sfar

Hatem Fessi

Abdelhamid Elaissari

Encapsulation methods of active molecules for drug delivery

Keywords: nanocarriers, polymeric nanoparticle, liposome, solid lipid nanoparticle, drug delivery

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Introduction

Over the past few decades, enormous efforts have been devoted to developing nanotechnology as a novel platform to bypass most common hurdles encountered in traditional drug delivery such as poor stability, low bioavailability, poor solubility and the bitter taste of active agents [START_REF] Stella | Nanoparticles: an overview of the preparation methods from preformed polymers[END_REF]. Nanotechnology, a buzzword of this era, has transformed the face of research in numerous sectors, such as textiles [START_REF] Harifi | Application of nanotechnology in sports clothing and flooring for enhanced sport activities, performance, efficiency and comfort: a review[END_REF], electronic engineering [START_REF] Lu | The role of nanotechnology in the development of battery materials for electric vehicles[END_REF], agriculture [START_REF] Prasad | Nanotechnology in Sustainable Agriculture: Recent Developments, Challenges, and Perspectives[END_REF], food [START_REF] He | Nanotechnology in food science: Functionality, applicability, and safety assessment[END_REF] and pharmaceutical/biomedical fields [START_REF] Nikalje | Nanotechnology and its Applications in Medicine[END_REF], [START_REF] Enrico | Chapter 3 -Nanotechnology-Based Drug Delivery of Natural Compounds and Phytochemicals for the Treatment of Cancer and Other Diseases[END_REF]. The application of nanotechnology for diagnosis, treatment, monitoring, and control of biological systems promotes the concept of nanomedicine, an extensively studied field, in which many colloidal drug carriers have been designed, in particular, nanoparticles [START_REF] Nikalje | Nanotechnology and its Applications in Medicine[END_REF], [START_REF] Moghimi | Nanomedicine : current status and future prospects[END_REF], [START_REF] Farjadian | Nanopharmaceuticals and nanomedicines currently on the market : challenges and opportunities[END_REF]. Nanoparticles (NPs) are nano-sized structures dedicated to: i) carry the pharmaceutical agent to a specific tissue, ii) protect the drug from premature degradation, iii) lower the required dose for effective therapy, iv) minimize side effects and thereby v) increase the safety profile and therapeutic index of drugs [START_REF] Lombardo | Smart Nanoparticles for Drug Delivery Application : Development of Versatile Nanocarrier Platforms in Biotechnology and Nanomedicine[END_REF]. Moreover, NPs are employed as an approach to increase the bioavailability of pharmaceutical ingredients by improving the cellular uptake, especially in cancer treatment.

For instance, doxorubicin, a well-known anticancer drug, suffers from its poor pharmacokinetics behavior including poor bioavailability after oral administration. Khdair et al. displayed that encapsulating doxorubicin in chitosan-based-NPs enhances its uptake by MCF-7 and Caco-II tumor cell lines [START_REF] Khdair | Modified-chitosan nanoparticles: Novel drug delivery systems improve oral bioavailability of doxorubicin[END_REF]. Regarding the cutaneous administration, NPs enhances the skin penetration of drugs. A recent study revealed that the encapsulation of psoralen, widely used for psoriasis treatment, in liposomal carriers, increases the skin permeation by 5 fold and reduced the cytokine levels of psoriasis [START_REF] Doppalapudi | Psoralen loaded liposomal nanocarriers for improved skin penetration and efficacy of topical PUVA in psoriasis[END_REF]. Other drugs like docetaxel suffer from their low aqueous solubility which hinders their clinical uses. The nanoencapsulation could also hinder the low aqueous solubility. Qureshi et al. demonstrated that the incorporation of docetaxel in lipid nanocarriers leads to a 686 times increase in water solubility [START_REF] Qureshi | Sustained release docetaxel-incorporated lipid nanoparticles with improved pharmacokinetics for oral and parenteral administration[END_REF].

Depending on the preparation methods, nanocarriers can be broadly divided into two main categories: polymer-based nanocarriers and lipid-based nanocarriers [START_REF] Lombardo | Smart Nanoparticles for Drug Delivery Application : Development of Versatile Nanocarrier Platforms in Biotechnology and Nanomedicine[END_REF]. Amongst the several nanoparticulate systems investigated in drug delivery, polymeric nanoparticles (pNPs), liposomes and solid lipid nanoparticles (SLNs) have received much attention [START_REF] El-Say | Polymeric nanoparticles : Promising platform for drug delivery[END_REF]- [START_REF] Geszke-Moritz | Solid lipid nanoparticles as attractive drug vehicles: Composition, properties and therapeutic strategies[END_REF]. In this context, this chapter highlights the recent development of nanocarrier systems for drug delivery with a focus on the principal properties and preparation methods of the main polymer based-nanocarriers (polymeric nanoparticles) and lipid based-nanocarriers (such as liposomes and solid lipid nanoparticles). Furthermore, the current status of their therapeutic applications is also presented and shortly discussed.

Polymer-based nanoparticles

During the last decades, polymeric nanoparticles have drawn tremendous attention owing to their peculiar characteristics and behaviours arising from their small size [START_REF] Crucho | Polymeric nanoparticles: A study on the preparation variables and characterization methods[END_REF]. As demonstrated by different authors, these nanoparticulate systems show prospects for a wide range of applications due to their substantial benefits including upraised drug stability against photo-or chemical degradation, controlled drug release, increased drug-tissue interaction and consequently decreased adverse effects and improved therapeutic index [START_REF] El-Say | Polymeric nanoparticles : Promising platform for drug delivery[END_REF], [START_REF] Mallakpour | Polymeric nanoparticles : Recent development in synthesis and application[END_REF]- [START_REF] Amoabediny | Overview of preparation methods of polymeric and lipid-based (niosome, solid lipid, liposome) nanoparticles: A comprehensive review[END_REF].

Polymer-based nanoparticles are spherical in shape with a size range ranging from 10 to 1000 nm. They can exist in two different forms, a polymeric layer surrounding a hollow core (nanocapsules), or a polymeric matrix extended from the core to the surface (nanospheres).

The localization of the active molecule within these two forms depends on its chemical structure. In some cases, the drug can also be adsorbed or grafted onto the surface of the particles [START_REF] El-Say | Polymeric nanoparticles : Promising platform for drug delivery[END_REF], [START_REF] Egusquiaguirre | 29 -Nanotherapeutic Platforms for Cancer Treatment: From Preclinical Development to Clinical Application[END_REF], [START_REF] Campos | Chapter 1 -Nanopreparations for skin cancer therapy[END_REF]. Based on the origin of the polymer intended for NPs formulation, two types are used: natural and synthetic polymers. Their main features are their biodegradability and biocompatibility [START_REF] Calzoni | Biocompatible Polymer Nanoparticles for Drug Delivery Applications in Cancer and Neurodegenerative Disorder Therapies[END_REF]. The most widely explored natural polymers are chitosan, alginate and proteins such as albumin [START_REF] Zazo | Current applications of nanoparticles in infectious diseases[END_REF]. Amongst the various synthetic polymers, polylactide-polyglycolide, polylactide, polycaprolactones, and polyacrylates have been extensively used [START_REF] Rao | Polymer nanoparticles : Preparation techniques and size-control parameters[END_REF]. For the preparation of polymer-based particles, various methods have been developed. The choice of the appropriate method depends on many factors such as solubility of the drug in water and organic solvents, physicochemical properties of the polymer and the target carrier specifications (i.e., size, size distribution, surface charge nature, surface porosity, etc…) [START_REF] Crucho | Polymeric nanoparticles: A study on the preparation variables and characterization methods[END_REF], [START_REF] Krishnaswamy | Chapter 2 -Sustainable Delivery Systems Through Green Nanotechnology[END_REF]. The most commonly reported processes used for polymer particles formulations are shown in Figure 1.

Nanoprecipitation method

Nanoprecipitation is a one-step process allowing the instantaneous and reproducible generation of narrow distribution of submicron particles [START_REF] Stella | Nanoparticles: an overview of the preparation methods from preformed polymers[END_REF]. It was first reported by Fessi et al. [START_REF] Fessi | Nanocapsule formation by interfacial polymer deposition following solvent displacement[END_REF]. In this process, an organic solution containing a non-water-soluble polymer (solvent phase) is mixed with water (non-solvent phase). It is interesting to notice that both phases are miscible, and the mixture of the two phases becomes a bad solvent of the selected polymer leading to its precipitation via an instantaneous solvent diffusion mechanism [START_REF] Haghiralsadat | Overview of Preparation Methods of Polymeric and Lipid-Based ( Noisome , Solid Lipid , Liposome ) Nanoparticles : A Comprehensive Review[END_REF]. For nanocapsules preparation, the same process is used except for adding oil to the organic phase in order to form an inner oily core [START_REF] Couvreur | Nanocapsule technology: a review[END_REF]. The benefits of the nanoprecipitation technique include facility, rapidity, reproducibility, and scalability [START_REF]Nanoprecipitation of cellulose acetate using solvent/nonsolvent mixtures as dispersive media -ScienceDirect[END_REF], [START_REF] Albisa | Preparation of Drug-Loaded PLGA-PEG Nanoparticles by Membrane-Assisted Nanoprecipitation[END_REF]. Moreover, in this technique, the use of a stabilizer , in the case of charged polymer, or high production energy is avoided [START_REF] Maaz | Influence of nanoprecipitation method parameters on nanoparticles loaded with gatifloxacin for ocular drug delivery[END_REF]. The main drawback of this method is low loading efficiency for hydrophilic drugs due to the hydrophobicity of a nanoparticle matrix [START_REF] Haghiralsadat | Overview of Preparation Methods of Polymeric and Lipid-Based ( Noisome , Solid Lipid , Liposome ) Nanoparticles : A Comprehensive Review[END_REF]. Nonetheless, several strategies have been asserted to enhance the loading of these molecules [START_REF] Arpicco | Recent studies on the delivery of hydrophilic drugs in nanoparticulate systems[END_REF].

Simple emulsion evaporation method

A simple emulsification evaporation method has been developed by Vanderhoff et al. to design polymeric nanoparticles [START_REF] Vanderhoff | Polymer emulsification process[END_REF]. In this process, the polymer and the drug are first dissolved in a volatile water-immiscible organic solvent. Ethyl acetate, owing to its better toxicological profile, has been widely used as a solvent instead of chloroform and dichloromethane [START_REF] Vauthier | Methods for the preparation and manufacture of polymeric nanoparticles[END_REF]. The organic polymer solution is emulsified with an aqueous phase containing a suitable stabilizer under high shear stress (e.g. by homogenization or ultrasonication) [START_REF] Haghiralsadat | Overview of Preparation Methods of Polymeric and Lipid-Based ( Noisome , Solid Lipid , Liposome ) Nanoparticles : A Comprehensive Review[END_REF]. The organic solvent is then evaporated by continuous stirring at room temperature or under reduced pressure leading to polymer precipitation and nanoparticle formation. The benefits of using this encapsulation method are high encapsulation efficiency of hydrophobic drugs, easy to process and good reproducibility. However, some disadvantages are related to this process; in particular, the poor encapsulation efficiency of hydrophilic drugs, the possible coalescence of droplets during evaporation which may lead to polydisperse polymer particles and the need for a high shear process in order to obtain submicron size with a narrow size distribution [START_REF] Stella | Nanoparticles: an overview of the preparation methods from preformed polymers[END_REF], [START_REF] Crucho | Polymeric nanoparticles: A study on the preparation variables and characterization methods[END_REF].

Double emulsion evaporation method

The double emulsification evaporation method has been proposed as an alternative to nanoprecipitation and simple emulsification evaporation methods in term of encapsulation of hydrophilic drugs [START_REF] Garti | Double emulsions: Progress and applications[END_REF]. For instance, this technique, consisting of two-step emulsification, can encapsulate both hydrophobic and hydrophilic drugs because of the presence of several water and oil layers [START_REF] Iqbal | Preparation of biodegradable PCL particles via double emulsion evaporation method using ultrasound technique[END_REF]. In the first step, the water-in-oil emulsion is formed after the homogenization of an aqueous phase (W1) containing the hydrophilic drug with an oily phase (O) that is generally a polymer solution dissolved in an organic solvent containing the necessary hydrophobic drug. The second step is the emulsification of the first W1/O emulsion in another aqueous phase (W2) containing an appropriate stabilizer, under high shear homogenization or low power sonication to form a double emulsion (W1/O/W2). Although this method can encapsulate both hydrophilic and hydrophobic drugs, it presents several drawbacks such as: low colloidal stability of the resulting particles, high shear stirring that may induce emulsion fragmentation, foam formation, and above-micron size polydisperse particles [START_REF] Stella | Nanoparticles: an overview of the preparation methods from preformed polymers[END_REF].

Ionic gelation

The ionic gelation method has been developed by Calvo et al. as a novel approach that avoids harmful organic solvents or high shear forces [START_REF] Calvo | Novel Hydrophilic Chitosan -Polyethylene Oxide Nanoparticles as Protein Carriers[END_REF]. Among the different natural polysaccharides such as chitosan, some polyelectrolytes and alginates are widely used in the ionic gelation based process. These polyelectrolytes can be crosslinked by electrostatic interaction with an added oppositely charged molecules or macromolecules that can ultimately lead to ionic gelation and therefore to polysaccharide particles formation [START_REF] Yang | Preparation and application of micro/nanoparticles based on natural polysaccharides[END_REF]. The ionic gelation consists of the transition from a liquid to gel state upon complexation between the positively charged amino group of chitosan and the negatively charged tripolyphosphate compound widely used as a polyanion crosslinker leading generally to submicron particles [START_REF] Mohanraj | Nanoparticles -A Review[END_REF]- [START_REF] López-León | Physicochemical characterization of chitosan nanoparticles: electroki-netic and stability behavior[END_REF]. The main benefits of this method are in particular the use of aqueous solvents instead of organic solvents. Moreover, ionic gelation is simple, reproducible and a scalable process [START_REF] Kunjachan | Understanding the mechanism of ionic gelation for synthesis of chitosan nanoparticles using qualitative techniques[END_REF], [START_REF] Usmiati | The Using of Ionic Gelation Method Based on Polysaccharides for Encapsulating the Macromolecules-A Review[END_REF].

Supercritical fluid method

All the above methods often involve the use of toxic solvents and / or surfactants that are dangerous to the environment as well as to physiological systems [START_REF] Miah | Surfactant free synthesis of gold nanoparticles within meso-channels of non-functionalized SBA-15 for its promising catalytic activity[END_REF]. Consequently, current research has been directed to explore safer techniques to develop carriers of …. Supercritical fluids have fascinated the interest of many researchers as they are environmentally friendly solvents [START_REF] Soppimath | Biodegradable polymeric nanoparticles as drug delivery devices[END_REF]. Supercritical CO 2 is the most frequently used supercritical fluid due to its low price, abundancy, non-flammability and non-toxicity. In the supercritical antisolvent process, the drug and the polymer are dissolved in a liquid solvent such as methanol that is totally miscible with the supercritical fluid. A supercritical antisolvent, A supercritical antisolvent is then added to the solution in a closed chamber, at ambient pressure. The insolubility of the solute in the fluid results in the instant precipitation of the solute, leading to particles formation [START_REF] Rani | Polymeric Nanoparticles in Targeting and Delivery of Drugs[END_REF]. Despite the advantage of using environmentally safe materials, some drawbacks are linked to this method since it requires special equipment and is very expensive to be developed as a routine technique [START_REF] Ranjit | Nanoparticle: An overview of preparation, characterization and application[END_REF].

Liposomes

Liposomes were first described by Alec Bangham in 1961 [START_REF] Bangham | Diffusion of univalent ions across the lamellae of swollen phospholipids[END_REF], [START_REF] Li | Composition design and medical application of liposomes[END_REF]. Liposomes are vesicles which are made of one or more lipid bilayers. They can be classified into 3 categories: small unilamellar vesicles (SUV), large unilamellar vesicles (LUV) and multilamellar vesicles (MLV). Liposomes can also be classified according to their surface charge. In fact, they can be non-charged (zwitterionic), cationic or anionic.

The interest of liposomes is related to their ability to contain hydrophilic molecules in their aqueous phase and lipophilic molecules in their lipid bilayers as illustrated in Figure 2 [START_REF] Van Tran | Liposomes for delivery of antioxidants in cosmeceuticals: Challenges and development strategies[END_REF].

Furthermore, liposomes bear a good cell affinity and biodegradability. Lipid bilayers are constructed of phosphatidylcholine and cholesterol. The latter ensures the physical stability of the bilayers.

3.1.Hydration of a Thin Lipid Film

Thin lipid film hydration is the first process used for liposomes preparation. This method consists of the dissolution of lipid ingredients in an organic solvent such as; ethanol, methanol or chloroform in a round-bottom flask [START_REF]Drug carriers in osteoporosis: Preparation, drug encapsulation and applications -ScienceDirect[END_REF]. Subsequent evaporation of the obtained phospholipids solution allows the formation of a lipid film. Its hydration by water or a buffer solution enables the formation of liposomes dispersion (see Figure 3). Sonication, homogenization or extrusion may also be performed to ensure size reduction and particles uniformity [START_REF] Patil | Novel methods for liposome preparation[END_REF].

3.2.Detergent Dialysis

Detergents at their critical micelle concentrations (detergent concentration above which micelles begin to form) can be used to solubilize lipids (see Figure 4). In the detergent dialysis method, phospholipids are solubilized by detergent micelles in an aqueous phase. After detergent removal by dialysis, phospholipid-rich micelles coalescence is obtained, and liposomes are formed. Advantages of this technique include good reproducibility and size uniformity. However, residual detergent retention represents the main drawback [START_REF] Patil | Novel methods for liposome preparation[END_REF]. Dialysis can be performed with a commercial device called LipoPrep (Diachema AG, Switzerland) can also be obtained in dialysis bags engrossed in large detergent free buffers [START_REF] Akbarzadeh | Liposome: classification, preparation, and applications[END_REF]. There are other ways to using dialysis to remove detergent molecules such as chromatographic methods.

3.3.Solvent Injection Technique

In the solvent injection technique, phospholipids are dissolved in an organic solvent such as ethanol or ether, then, this solution is slowly injected into an aqueous buffer solution. SUV dispersion is then obtained. Self-assembly of lipids in the aqueous phase to form liposomes occurs following either ethanol dilution in water or ether evaporation (See Figure 5) [START_REF] Gharib | Hydroxypropyl-ß-cyclodextrin as a membrane protectant during freeze-drying of hydrogenated and non-hydrogenated liposomes and molecule-in-cyclodextrin-in-liposomes: Application to trans-anethole[END_REF].

The ethanol injection method was first described in 1973 [START_REF] Batzri | Single bilayer liposomes prepared without sonication[END_REF]. The main advantage of this method is that narrow size distribution of submicron liposomes (generally under 100 nm) could be obtained without extrusion or sonication [START_REF] Laouini | Preparation, Characterization and Applications of Liposomes: State of the Art[END_REF].

The ether injection method differs from the ethanol injection method since the ether is immiscible with the aqueous phase. The relevance of this method compared to the ethanol injection solvent removal from the product which allows the process to run for an extended period. A more prolonged process was shown to form concentrated liposomes suspension with high loading efficiency [START_REF] Laouini | Preparation, Characterization and Applications of Liposomes: State of the Art[END_REF].

3.4.Reverse-Phase Evaporation Technique

The reverse phase evaporation method enables the preparation of liposomes with high aqueous space-to-lipid ratio. This allows a large percentage of aqueous material to be loaded.

The technique is based on the creation of inverted micelles [START_REF] Akbarzadeh | Liposome: classification, preparation, and applications[END_REF]. The latter are formed when phospholipids solution in an organic phase is hydrated by the addition of water with vigorous mixing. This results in the formation of a water-in-oil emulsion. Subsequent evaporation of the organic phase results in the micelles transformation into a gel-like dispersion (Figure 6). At some point, the gel state collapses and the excess of phospholipids in the environment leads to an aqueous liposomes suspension containing both LUV and MLV [START_REF] Kaddah | Cholesterol modulates the liposome membrane fluidity and permeability for a hydrophilic molecule[END_REF].

Figure 6. Reverse phase evaporation technique

3.5.Membrane contactor technology:

Membrane contactors are used to mix two materials [START_REF] Huang | Progress involving new techniques for liposome preparation[END_REF]. This property is used to enable the mixing of the lipid and the water phase to obtain liposomes (see Figure 7). Jaafar-Maalej et al. used a membrane contactor where phospholipid solution in ethanol was extruded into an aqueous phase [START_REF] Jaafar-Maalej | A new method for liposome preparation using a membrane contactor[END_REF]. The obtained liposomes were prepared by a modified ethanol injection method. In this study, a tubular Shirazu porous glass (SPG) membrane with a mean pore size of 900 nm was used. Laouini et al., 2011, used polypropylene hollow fibers which allowed access to larger membrane areas and uniform flows thus, improving efficiency [START_REF] Laouini | Liposome preparation using a hollow fiber membrane contactor--application to spironolactone encapsulation[END_REF]. Thi Thuy Pham et al. reported the use of membrane contactor technology to produce liposomes on an industrial scale. Their product could be obtained in a very reproducible manner [START_REF] Pham | Liposome and niosome preparation using a membrane contactor for scale-up[END_REF].

Advantages of the membrane contactor technique include greater control of liposome size, high encapsulation efficiency, continuous operation and scale-up of the process [START_REF] Patil | Novel methods for liposome preparation[END_REF].

Solid lipid nanoparticles

Solid lipid nanoparticles (SLNs) are other lipid-based carriers that have emerged as a less toxic, more biocompatible alternative for polymeric based nanoparticles. Moreover, it has been claimed that SLNs actually unite the benefits of lipid emulsion systems, liposomes, and polymeric nanoparticles, while sweeping away temporal and in vivo stability [START_REF] Jaafar-Maalej | Lipid-based carriers: manufacturing and applications for pulmonary route[END_REF]. The general chemical composition of these particles are surfactants and sometimes co-surfactants and a lipid matrix that acquires a solid-state at physiological temperature. The lipid matrix of SLNs should be composed of lipids that have good biocompatibility and a melting point above body temperature. With these characteristics, the commonly used materials to structure the lipid core are fatty acids, fatty alcohols, waxes, and triglycerides. In fact, the crystal structure of the lipid core is critical for the feasibility of SLN systems since it may affect controlled drug release and colloidal stability of the dispersion. The formation of a solid core is determined by components ratio and the choice of the preparation method [START_REF] Bayón-Cordero | Application of Solid Lipid Nanoparticles to Improve the Efficiency of Anticancer Drugs[END_REF].

Different preparation methods were used in the literature to prepare SLNs such as highpressure homogenization, microemulsion, solvent emulsification evaporation, and solvent emulsification diffusion [START_REF]Solid lipid nanocarriers in drug delivery: characterization and design[END_REF]. These techniques are differentiated by the complexity of their execution and the characteristic of their final product [START_REF] Geszke-Moritz | Solid lipid nanoparticles as attractive drug vehicles: Composition, properties and therapeutic strategies[END_REF].

4.1.High-pressure homogenization

High-pressure homogenization (HPH) is widely used in the pharmaceutical industry since it is easy to scale-up and produce high-quality products with small particle size and higher homogeneity. HPH uses a sophisticated device (high-pressure homogenizer) that is able to push liquids through a narrow gap (micron size) with high pressure (from 100 to 2000 bar) creating an acceleration over a very short distance [START_REF] Dumont | Lipid-based nanosuspensions for oral delivery of peptides, a critical review[END_REF].

In brief, the lipid is heated at 5 to 10°C above its melting point, and the drug is dissolved within. After that, two different approaches can be undertaken One approach is hot homogenization, in which the drug-lipid melt is dispersed in a hot aqueous surfactant solution under stirring at the same temperature creating the pre-emulsion. Then, a hot oil-in-water (O/W) nanoemulsion is formed by homogenization using a piston-gap homogenizer. This nanoemulsion is then cooled to room temperature allowing the formation of lipid crystals creating SLNs [START_REF] Müller | Solid lipid nanoparticles (SLN) for controlled drug deliverya review of the state of the art[END_REF].

Another approach is cold homogenization that is started with cooling down the drug-lipid melt allowing the formation of solid lipid microparticles (between 50 and 100 nm) that are then dispersed in a cold surfactant solution shaping a pre-suspension. This pre-suspension is on its turn and is strongly homogenized, breaking down the microparticles and forming SLNs.

The cold HPH approach is used in the case of thermosensitive drugs where at cold conditions, lipid melting is minimized and therefore prevents the diffusion of hydrophilic drugs into the water phase [START_REF] Trotta | Preparation of solid lipid nanoparticles by a solvent emulsification-diffusion technique[END_REF].

4.2.Microemulsion

This method was first developed by Gasco [START_REF] Gasco | Method for producing solid lipid microspheres having a narrow size distribution[END_REF]. It is based on the precipitation of lipid particles by diluting the system with water. Briefly, a microemulsion is formed by adding a mixture of water, surfactant, and co-surfactant to the lipid melt. Taking into account that the water mixtures should be heated to the same temperature as the lipid and added under mild stirring. The formed microemulsions are then diluted in a cold aqueous phase which causes their precipitation into SLNs.

Different types of surfactants and co-surfactants were used in this method such as lecithin, biliary salts, and alcohols. In addition, it is important that during the precipitation step, mixing should be done by mild mechanical agitation to guarantee that the small size of the obtained SLNs is due to precipitation and not agitation. This method has the advantages of being a solvent-free process with gentle preparation conditions. Making it suitable for the encapsulation of various therapeutic peptides and active molecules [START_REF] Dumont | Lipid-based nanosuspensions for oral delivery of peptides, a critical review[END_REF].

4.3.Emulsion methods

Emulsion-diffusion (ED) requires the use of three different liquid phases. The organic phase in which the lipids and the lipophilic drug are dissolved (solvent) and generally miscible with the aqueous phase. An aqueous phase containing a stabilizing agent (non-solvent)and a dilution phase that usually involves a large amount of water. Solvent and non-solvent phases are mutually saturated. The organic phase is emulsified in the water phase under high shear stirring. After that, the water dilution phase is subsequently added allowing the diffusion of the organic solvent to the outer water resulting in the precipitation of lipids and the formation of SLNs. The obtained SLNs by this method are diluted, therefore, a concentration step is followed by ultra-filtration or freeze-drying. The advantage of this technique is the avoidance of temperature use which reflects better stability of the system in general and the drug encapsulated in particular [START_REF] Das | Recent Advances in Lipid Nanoparticle Formulations with Solid Matrix for Oral Drug Delivery[END_REF], [START_REF] Miladi | Particles from preformed polymers as carriers for drug delivery[END_REF].

Emulsion-evaporation (EE) is differentiated from ED by the fact that the non-solvent used should be immiscible with the solvent. In addition, it is critical that the evaporation temperature of the solvent is lower than the non-solvent. After the formation of a simple emulsion, the organic solvent is evaporated leading to the precipitation of the lipids in the non-solvent phase and formation of nanoparticles that are stabilized instantaneously by surfactants. This leads to the formation of O/W nanoemulsion and in some cases, a double emulsion can be done leading to the formation of W/O formulations. A major drawback of EE is the use of a water-immiscible solvent that can be harmful for human administration [START_REF] Dumont | Lipid-based nanosuspensions for oral delivery of peptides, a critical review[END_REF], [START_REF] Fuentes | Design of lipid nanoparticles for the oral delivery of hydrophilic macromolecules[END_REF].

Therapeutic applications:

Numerous active molecules have been encapsulated in nanomaterials to obtain various 363 therapeutic applications. This has led to the emergence of nanomedicine, which is the field 364 that combines nanotechnology with pharmaceutical and biomedical sciences. For instance, 365 polymeric nanoparticles represent the simplest form of soft materials used for nanomedicine 366 applications. This is due to their easy synthesis and wide biomedical applications. In fact, two 367 of the top 10 best-selling drugs in the US in 2013 were polymeric drugs (Copaxone® and 368 Neulasta®), [START_REF] Duncan | Polymer therapeutics: Top 10 selling pharmaceuticals -what next?[END_REF], [START_REF] Bobo | Nanoparticle-Based Medicines: A Review of FDA-Approved Materials and Clinical Trials to Date[END_REF]. Liposomes also represent another feature of nanomedicine. Until properties could be conferred by solid lipid nanoparticles or nanostructured lipid carrier [START_REF] Poovi | Lipid nanoparticles: A challenging approach for oral delivery of BCS Class-II drugs[END_REF].

376

The following Table 1 shows some examples of nanomedicine-based formulations. Fungal infections [START_REF] Alhowyan | Antifungal efficacy of Itraconazole loaded PLGA-nanoparticles stabilized by vitamin-E TPGS: In vitro and ex vivo studies[END_REF] Nanoprecipitation Poly-(D, L-lactic-co-glycolic acid) Cisplatin 126 nm -12 mV

Ovarian cancer [START_REF] Domínguez-Ríos | Cisplatin-loaded PLGA nanoparticles for HER2 targeted ovarian cancer therapy[END_REF] Simple emulsion Poly-(D, L-lactic-co-glycolic acid) and Gly-L-Phe-DThr-Gly-L-Phe-L-Curcumin 200-250 nm -22 mV Alzheimer's disease [START_REF] Barbara | Novel Curcumin loaded nanoparticles engineered for Blood-Brain Barrier crossing and able to disrupt Abeta aggregates[END_REF] Leu-LSer(O-b-D-Glucose)-CONH2 (g7)

Conclusion

Nanomedicine offers an enormous outlook for upgrading current therapies and emerging new treatment tools for illness previously assumed difficult or impossible to treat.

Nanostructured systems have shown exciting benefits due to their size and shape since they can reach areas which are inaccessible using other conventional drug delivery systems protect the drug molecules from premature degradation, sudden clearance, and toxic manifestations.

Nanostructured systems can also deliver the drug to a specific tissue site or cell leading to increased drug amounts at a local site hence enhancing the therapeutic profile. In this chapter, we spotlight recent advances of both lipid and polymer based-nanocarriers such as liposomes, solid lipid nanoparticles and polymeric nanoparticles in the emergence of new concepts for the efficient transport of drug molecules. The first section of this review has provided information about the properties and recent developments in the synthesis of these nanosystems. Several methods have been explored for nanocarrier preparation, nonetheless, the choice of the appropriate approach is a crucial key to gain specific requirements.

Additionally, this paper has reviewed current applications of these carriers. The discussed nanostructured systems provide interesting perspectives for applications in several fields such as drug and gene delivery, biomedical imaging, diagnosis, and theranostics. The specific functionalization of the carrier surface can enhance its biodistribution to target cells and limit its delivery into healthy ones. Consequently, engineering nanocarriers with specific surface [START_REF] Silva | Preparation, characterization and biocompatibility studies on risperidoneloaded solid lipid nanoparticles (SLN): High pressure homogenization versus ultrasound[END_REF] receptors can be considered to be the turning point leading, in the future to a potent use of these systems in large-scale therapeutic applications.

Figure 1 .

 1 Figure 1. Schematic illustration of nanoprecipitation, simple emulsion evaporation, and double emulsion evaporation processes.

Figure 2 .

 2 Figure 2. Liposomes containing hydrophilic and hydrophobic active drug.

Figure 3 .

 3 Figure 3. Hydration of a thin lipid film (example of a hydrophobic drug incorporated in cyclodextrins).

Figure 4 .

 4 Figure 4. Detergent dialysis method.

Figure 5 .

 5 Figure 5. Solvent injection technique.

Figure 7 .

 7 Figure 7. Membrane contactor technique

Figure 8

 8 shows the different preparation techniques of solid lipid nanoparticles.

Figure 8 .

 8 Figure 8. Preparation methods of solid lipid nanoparticles

  369today, around 20 liposomal products have been approved by regulatory agencies, especially 370 for cancer management. Other therapeutic indications of liposome-based products include 371 antifungal and antibacterial drugs, analgesics, vaccines, and photodynamically active agents 372[START_REF] Skupin-Mrugalska | Chapter 6 -Liposome-Based Drug Delivery for Lung Cancer[END_REF]. In addition, lipid-based carriers have been evidenced to be attractive candidates for 373 active encapsulation. In fact, lipid materials could circumvent barriers associated with the oral 374 administration of poorly soluble drugs or drugs with intestinal permeability issues. Such

  375

Table 1 . Examples of therapeutic indications of liposome-based formulations 378 Method Materials Encapsulated molecule Size Zeta Application Reference Polymer-based nanoparticles

 1

	377


Weiliang Jin

Riccardo Messina

Alejandro W Rodriguez

published or not. The documents may come

General formulation of coupled radiative and conductive heat transfer between compact bodies


Yves D Jean
email: yves.jean@lehman.cuny.edu

Orthographic Projection for Optical Signal Processing

Controlled illumination is a powerful tool for solving scene recognition problems. Binary and high frequency illumination primitives are projected into the scene to spatially encode or probe the optical environment. Recently, computer vision researchers have shown that orthogonal functions and computational techniques from the signal processing framework can be mapped directly into the scene using projector-camera systems. These scene-space signal processing algorithms are achieved with illumination-encoded functions as primitives and computations derived from surface reection models. Some examples of this new optical approach include convolution ltering and aliasing-canceling lterbanks. In this paper we present a digital projector calibration method and an orthographic light eld system. The calibrated system produces a light eld structure matching the usual sampling geometry of image processing, well-suited to scene-space algorithms. The result is superior ltering and resolution performance due to higher accuracy optical representation. We evaluate our results by comparing processor-based ltering with a scene-space algorithm.

Introduction

Structured light techniques have proven to be exible and powerful tools for scene analysis problems such as ranging and surface reconstruction (see Salvi et al [START_REF] Salvi | Pattern codication strategies in structured light systems[END_REF] for an overview). They have been used to recover the geometric and optical properties of surfaces and to discern the global and local illumination components [START_REF] Nayar | Fast separation of direct and global components of a scene using high frequency illumination[END_REF]. It is not surprising that controlled illumination has such capabilities because it introduces a known optical parameter into the irradiance that reaches the camera.

A common projector-camera technique is to encode the pixel positions of the projector with binary structured light to resolve point correspondences, via triangulation, with the camera pixels that recover the code [START_REF] Salvi | Pattern codication strategies in structured light systems[END_REF]. Binary primitives, in general, minimize luminance transformations due to surface reection, leaving a code mainly representing the spatial relationships between the projector, camera, and scene objects.

Techniques using high-frequency primitives have been used to probe optical properties such as defocus blur to compute depth [START_REF] Nayar | Real-time focus range sensor[END_REF][START_REF] Zhang | Projection defocus analysis for scene capture and image display[END_REF]. A high frequency pattern inria-00325381, version 1 -29 Sep 2008 is projected into the scene and the reected signal is analyzed for low-pass ltering eects (blur). There are other variations of these techniques highlighting the eectiveness of projector-camera systems in computer vision problems.

Recently, researchers have investigated light patterns based on orthogonal bases from the signal processing domain. The use of analytical functions as structured light primitives allows for new approaches to computer vision problems.

The motivation is to perform computations directly in the scene by modeling signal processing formulations in the optical domain with digital projectors and cameras. The works of Damera-Venkata and Chang [START_REF] Damera-Venkata | Realizing super-resolution with superimposed projection[END_REF], Jean [START_REF] Jean | Scene-space feature detectors[END_REF], and Ghosh et al [START_REF] Ghosh | Brdf acquisition with basis illumination[END_REF] clearly demonstrate that signal processing formulations can be implemented in scene-space by optically encoding analytical functions and leveraging the native reection models in the scene. This is the core computational model and is easily extended with additional optical components and image processing. Mapping signal processing tools into scene-space also requires satisfying the analytical constraints of sampling theory for signal analysis, representation and processing that must be preserved in the optical domain.. In this paper we present an orthographic projection calibration technique designed to produce higher accuracy light elds with only consumer-grade digital projector technology. The calibration technique corrects digital projector o-axis light eld distortion and is applicable to optical systems incorporating digital projectors. The technique is especially eective for producing calibrated orthographic projection systems to bring the regular sampling geometry of signal processing to scene-space techniques. We present our results by implementing a scene-space algorithm and compare the results to processor-based image processing.

The paper is divided into the following sections. In Section 2 we review the related projector-camera literature and scene-space techniques, in particular.

The calibration procedure is outlined in Section 3, and our results and analysis presented in Section 4.

Related work

There are many structured light techniques, using projectors and cameras, where a known illumination pattern is projected into the scene and the spatial or spectral transformation of the reected signal is used to determine the conguration of objects and surfaces. We focus on coded light triangulation and depth from defocus, techniques built on a pattern primitive and related to our work with basis functions. Salvi et al [START_REF] Salvi | Pattern codication strategies in structured light systems[END_REF] provides an overview of optical triangulation coding strategy trade-os depending on the computer vision performance objectives (i.e., segmentation ease, minimal image sequence, etc.). Binary coded light is easier to detect compared to other coding schemes though requiring more projections.

Patterns built from binary primitives, like line stripes are commonly used in triangulation systems [START_REF] Scharstein | High-accuracy stereo depth maps using structured light[END_REF][START_REF] Zhang | Spacetime faces: high resolution capture for modeling and animation[END_REF][START_REF] Zhang | High-resolution, real-time 3-d shape acquisition[END_REF][START_REF] Lanman | Surround structured lighting for full object scanning[END_REF][START_REF] Young | Viewpointcoded structured light[END_REF]. Smoother patterns such as ramps or gray level coding take fewer projections but increase the diculty for code detection.

However, sinusoidal primitives have been used with phase shifting to enhance the resolution of recovered codes [START_REF] Huang | High-speed 3-d shape measurement based on digital fringe projection[END_REF]. Binary coded primitives are exible and have been used in other solution strategies. Young et al [START_REF] Young | Viewpointcoded structured light[END_REF] provide an alternative to a sequence of pattern projections with multiple cameras in a view-point coding method. In Zhang et al [START_REF] Zhang | Spacetime faces: high resolution capture for modeling and animation[END_REF] and Chen et al [START_REF] Chen | Vision processing for realtime 3-d data acquisition based on coded structured light[END_REF] multiple techniques are combined to minimize projection sequences and handle scene motion.

Depth from defocus is another approach to surface reconstruction using projectors and cameras. By measuring the extent of defocus on an imaged surface, the depth from the focal plane can be determined. Again, binary primitives assembled into a high frequency pattern are projected into the scene and reected back to the camera along the same light path [START_REF] Zhang | Projection defocus analysis for scene capture and image display[END_REF][START_REF] Nayar | Real-time focus range sensor[END_REF]. A coaxial conguration between the camera and projector enables the camera to directly image the high frequency pattern after traversing the depth dependent defocus volume in the scene. Another use of high frequency primitives is in an algorithm by Nayar et al [START_REF] Nayar | Fast separation of direct and global components of a scene using high frequency illumination[END_REF] to separate the global and local radiance in a scene.

We present an orthographic projection system and calibration techniques.

Other researchers have produced orthographic systems, for example, Nayar and Anand's [START_REF] Nayar | 3d display using passive optical scatterers[END_REF] system drives a volumetric display with an orthographic light eld.

Lanman et al [START_REF] Lanman | Surround structured lighting for full object scanning[END_REF] use mirrors to transform an orthographic light eld to produce an object surrounding structured light triangulation system. Neither [START_REF] Lanman | Surround structured lighting for full object scanning[END_REF] nor [START_REF] Nayar | 3d display using passive optical scatterers[END_REF] address the projector distortion we examine in this paper, though the lens design optimization in [START_REF] Nayar | 3d display using passive optical scatterers[END_REF] may reduce the eect. The calibration technique we present in this paper corrects the light eld distortion in digital projectors. The technique is general and can be used in any system using digital projectors.

Scene-space techniques

Signal and image processing is a powerful analytical framework with a rich set of tools heavily used by the computer vision community to solve computational problems and for modeling the image formation process [START_REF] Durand | A frequency analysis of light transport[END_REF][START_REF] Nayar | Real-time focus range sensor[END_REF][START_REF] Zhang | Projection defocus analysis for scene capture and image display[END_REF][START_REF] Ramamoorthi | A signal-processing framework for reection[END_REF]. Researchers have developed novel new algorithms by shifting computations from the image to the scene directly. Scene-space techniques [START_REF] Damera-Venkata | Realizing super-resolution with superimposed projection[END_REF][START_REF] Jean | Scene-space feature detectors[END_REF][START_REF] Ghosh | Brdf acquisition with basis illumination[END_REF] implement signal processing computations in the optical domain by encoding scalar functions as illumination images, utilizing optical components, and modeling products and sums from the native surface reection models or bidirectional reectance distribution function (BRDF) in the scene. Equation 1 is the general surface reection equation of the radiance B emitted from location x in direction ω o due to the integration over the hemisphere parametrized by Z of ρ the BRDF and incident illumination L from direction ω i (scaled by the surface orientation term cosθ i ):

B(x, ω o ) = ˆZ ρ(ω i , ω o )L(x, ω i ) cos θ i dω i (1)
Damera-Venkata and Chang [START_REF] Damera-Venkata | Realizing super-resolution with superimposed projection[END_REF] showed that overlapped digital projectors can be used to produce super-resolution images on a Lambertian (B(x) = B(x, •))

screen surface. Their approach takes advantage of the ability to produce aliased optical signals when the low-pass projector lter is insucient [START_REF] Damera-Venkata | On the resolution limits of superimposed projection[END_REF]. They leverage the maximally decimated lterbank framework [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF] to algorithmically determine

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Fig. 1. O-axis lens design of digital projectors [START_REF] Smith | [END_REF]: The optical axis of the lens is above the center of the imaging chip resulting in a raised projection. The normally symmetric pyramidal projection is warped in the process, yet a correct image will appear on a projection surface parallel to the image chip.

the set of aliased images that, combined on a projection screen,

B(x) = i B i (x), (2)
(1 ≤ i ≤ #projectors), cancel the aliasing errors.

In another example, Jean [START_REF] Jean | Scene-space feature detectors[END_REF] presented a technique for optical convolution ltering by leveraging the intrinsic irradiance integration performed by camera pixels. The lter coecients are embedded in the projector illumination and reected from textured Lambertian surfaces. Lambertian reection models a sample-wise multiplication of the surface texture B(x) with the projected lter coecients f (x) which is integrated by a camera pixel, ˆΩ f (u)B(u)du,

(Ω = surface area) producing the lter response. The ltering technique computes the lter response in the scene before the lossy image formation process.

A simple image capture records the camera pixel integration as the convolution integral. This advantage suggests it can be used to recover scene information within the extent of a pixel or super-resolution. Jean tested the technique on planar textured objects with Gaussian derivative lter images. The results show it can perform edge detection with no spatial processing of the images.

Ghosh et al [START_REF] Ghosh | Brdf acquisition with basis illumination[END_REF] designed a BRDF acquisition system that performs a single operation: optical projection of the BRDF of a material sample onto spherical basis functions, with reective optics and a projector-camera pair. It demonstrates the power gained with geometric warping of light elds to solve computer vision problems.

Orthographic projection

Scene-space techniques depend on digital projectors to convert image representations of scalar functions into a projection light eld. The digital projector pixel grid denes the sampling geometry of the function and the distribution of rays in the light eld. However, the projection is problematic for two reasons: it is a pyramidal frustum, and thus, the light rays diverge in depth, and most digital projectors are designed to produce a distorted frustum for practical applications (Figure 1). In the follow subsection, we examine these problems and provide a calibration procedure for frustum distortion correction and an orthographic projector design.

Frustum distortion correction

Consumer digital projectors typically produce an o-axis projection frustum [START_REF] Smith | [END_REF][START_REF] Raskar | A self-correcting projector[END_REF]. (Figure 1 shows a schematic of this optical design [START_REF] Smith | [END_REF]) O-axis optics produces a correct image on a wall or projection screen but distorts the light eld as an artifact.

A simple solution to this problem is to use the lens shift feature found in some digital projectors. Lens shift is a mechanical control of the lens position along the vertical or horizontal direction. Figure 1 shows a vertical lens shift above the center of the imaging chip. Using this control, we can set the lens shift to zero (imaging chip and lens center aligned) creating an on-axis projection which produces a symmetric frustum. In Section 3.3 an optical design to transform a symmetric pyramidal projection to orthographic is provided. However, frustum distortion due to o-axis projection, if not removed with the following lens shift calibration process, will cause distortion in the orthographic light eld.

Projector lens shift calibration

Each lens shift control (horizontal or vertical) independently translates the lens assembly along one axis of a plane parallel to the imaging chip. The independence between the two controls is key to our lens calibration procedure, the lens shift parameters are measured and corrected one at a time. that the base sits at. Position the patterned sheet to intersect the light output from the digital projector producing a (initially distorted) frustum. Figure 2b shows an example. Here is the step-by-step 1. Choose the lens shift control which manipulates the displacement axis parallel to the table surface (when the projector is sitting on the foot pegs this will be the horizontal lens shift controls).

2. Align the center line of the projected grid image with a line on the loose patterned sheet. (see Figure 2b)

3. Simultaneously manipulate the lens shift control and adjust the patterned sheet until the projected grid image produces a symmetric frustum about a line on the surface grid pattern (see Figure 2a). The axis is now calibrated.

Turn the projector onto the side and repeat the procedure beginning with

Step 1 to calibrate the other axis.

Perform the following to verify that the optical axis is parallel to the table surface. Measure the height of the vertical center of the light frustum exiting the digital projector lens. Using a vertical surface as a reference (see Figure 2b, we use a Spectralon target), check that the height of the center horizontal grid image line is equal to the previous measurement and level with the surface. Note that camera calibration [START_REF] Bouget | Camera calibration for matlab[END_REF] of the projector can be performed to measure radial lens distortion. We did not nd it necessary if lens shift calibration using the center grid image lines is performed properly. The orthographic projection systems, [START_REF] Lanman | Surround structured lighting for full object scanning[END_REF][START_REF] Nayar | 3d display using passive optical scatterers[END_REF], both use xed o-axis projectors but do not address frustum distortion correction or compensation. Frustum distortion was not a factor in Damera-Venkata and Chang's work because the scene-space computation occur on the projection screen. Ghosh et al did not address nor correct for frustum distortion with their reective design. However, reective optics is an alternative [START_REF] Smith | [END_REF] to the refractive (Fresnel lens) optics approach presented in the next section. Jean's algorithm is aected by frustum distortion and we will show how calibration improves the resolution and ltering performance.

Projection transformation optics

The symmetric pyramidal light eld from an on-axis digital projector can be transformed into an orthographic light eld with refractive optics (see Figure 3). A Fresnel lens is a collimator of a point source or, equivalently, the rays from a pyramidal light eld. Nayar and Anand [START_REF] Nayar | 3d display using passive optical scatterers[END_REF] and Lanman et al [START_REF] Lanman | Surround structured lighting for full object scanning[END_REF] also used Fresnel lenses in their orthographic systems. Figure 3 shows a schematic for the orthographic system design and a picture of our setup. We now provide a calibration procedure to determine the focal length and lens positioning. The initial step is to establish orthogonal spatial relations amongst the components.

This calibration procedure is a continuation of Section 3.2 and is similar to steps outlined in [START_REF] Lanman | Surround structured lighting for full object scanning[END_REF].

To produce an orthographic light eld with a digital projector we must establish two orthogonal planes to position and place the projector and other components. A third orthogonal plane will dene the optical axis and x the orientation of the components. A at surface, upon which the projector is placed, denes one plane. Using a calibrated frustum from the previous section creates

inria-00325381, version 1 -29 Sep 2008

a parallel relationship between the digital projector's optical axis and the surface. A vertical planar object (a Spectralon target) denes the second plane.

The Spectralon target and the at surface form an orthogonal pair of planes, however, the light eld frustum from the digital projector must be aligned and integrated into this orthogonal relationship.

The lines of the grid image on the at surface between the digital projector and the target (see Figure 2b) are used to align the two objects. With the projector in a xed position, the center horizontal and vertical grid image lines are orthogonal and dene two planes that intersect about the optical axis. The center vertical line is used to position the Spectralon target, relative to the optical axis, and the horizontal lines along the axis are used for perpendicular orientation. This alignment method is used in the following orthographic system design process.

Results

We tested the orthographic projection system by implementing Jean's [START_REF] Jean | Scene-space feature detectors[END_REF] scenespace ltering algorithm to perform edge detection. We used a ViewSonic PJ1158 3LCD digital projector with dual lens shift controls, and an IMPERX VGA-210 monochrome 1CCD camera (12-bit resolution). The test images and lter response images are shown in Figures 4 and5. The test image in Figure 4a, is printed and mounted on a at panel and placed in front of the orthographic projection system (the projector focus plane was set at the print surface). Figure 4b is the camera image of the print from the 1CCD camera, this camera will be used to perform lter integration as dened in Jean's algorithm. Figure 4c is taken with a higher resolution 3CCD camera with a zoom lens to highlight the projected lter pattern composed of a grid of Gaussian derivative primitives (images) [START_REF] Jean | Scene-space feature detectors[END_REF].

First we compare ltering performed on a computer processor to orthographic scene-space ltering. A pair of orthogonal Gaussian derivative edge detection lters are used in both cases to compute edge energy [START_REF] Perona | Detecting and localizing edges composed of steps, peaks and roofs[END_REF]. Using Matlab, the Another test image is shown in Figures 5e and5f, containing a zebra pattern and a natural background. Naturally, the edge image shows the zebra pattern is enhanced but also shows the background features are segmented as well.

The scene-space ltering tests are not meant to represent computational accuracy. Neither the projector illumination nor the camera response were subject to photometric calibration. The BRDF of the prints were not analyzed but approximated as a diuse material embedded with an albedo modeled ink. The goal was to show that even under these sub-optimal experimental conditions strong lter responses were measured.

There are optical performance limitations in our orthographic system design.

For example, Fresnel lenses introduce optical artifacts and are not well-suited for imaging tasks. Digital projectors have large apertures that limit the depth of eld and orthographic projection transformation does not remove this problem.

Thus, there is a practical limit to the eective size of the working volume of the orthographic projector light eld. We expect the emerging solid-state RGB laser projector technology will solve the depth of eld problem. Also, our results are based on a planar test object, generalizing to curved objects with large surface undulations is a natural next step. We will attempt to handle these conditions by the use space-coding techniques to determine surface geometry before projecting lters. This approach can also be used to align camera pixels with lter primitives.

Conclusion

We have presented an orthographic light eld system design and calibration method using consumer digital projector technology. A calibrated orthographic light eld has an optical sampling geometry consistent with signal processing theory. The ecacy of our methods were tested by implementing a scene-space algorithm and clearly show results comparable to computer processor-based computations. The depth of eld limitation must be solved before our method can fully take advantage of signal processing theory and algorithms. Bridging the two domains will enable new computer vision techniques that leverage signal processing algorithms directly in the scene. Our next step is to increase the accuracy of optical computation techniques by compensating for surface orientation.

inria-00325381, version 1 -29 Sep 2008

Figure 2

 2 illustrates and explains the approach we use to calibrate a digital projector with lens shift controls to nd the zero lens shift position producing a symmetric frustum. The procedure requires a physical plane parallel to the lens shift axis to image the displacement. A at table is used as the plane by simply placing the digital projector level on the surface. A regular grid pattern image is projected because it identies groups of pixels to reveal frustum distortion. The regular grid pattern on the plane (dashed lines) intersects the projection image, imaging the frustum outline (solid lines). A solid line extending from the top and bottom of the frustum indicate the optical center. The dotted lines forming an apex are an annotated extrapolation from the truncated frustum of the projected image. The objective of the calibration is to manipulate the lens shift control until the center line is aligned with a plane grid line, splitting the projected frustum into two symmetric halves. The results should mimic the symmetric frustum shown in Figure 2a, indicating that the axis lens shift is calibrated to zero displacement. Here is the step-by-step procedure: Place a large grid patterned sheet on a at table surface (or use a patterned surface or an optical bench). Set the adjustable feet of the digital projector such inria-00325381, version 1 -29 Sep 2008 (a) Overlay of alignment grid pattern and projected grid image (b) Projector example.

Fig. 2 .

 2 Fig.2. Lens shift calibration using a surface grid pattern and a grid image projection: (a) is an illustration of the projection frustum overlayed on the at surface grid pattern (the dashed lines). The projection image, shown as solid lines, is a grid pattern that intersects the surface revealing a truncated frustum used to establish frustum symmetry. Five landmarks in (a) represent the design parameters for an orthographic projection device (see Figure3). Landmark A is the center of projection. B and C represent the front of the digital projector (zoom lens housing) and the Fresnel lens mount position, respectively. D and E delimit the lens aperture. (b) shows our projector calibration setup (projector out of view). The vertical object on the right, is a Spectralon target.

Fig. 3 .

 3 Fig. 3. Orthographic projection device: a) is a schematic of an orthographic projection device, a projector and Fresnel lens form an optical combination to produce an orthographic eld. Figure (b) shows a completed assembly. A grid image is projected onto the Fresnel lens and appears unscaled a distance away on a reective surface, verify an orthographic light eld.

Figure

  Figure 2a shows landmarks A, B, C, and D, relative to the frustum of a digital projector. The gure illustrates and denes the ve landmarks which are the parameters necessary to construct an orthographic projection device as shown in Figure 3. Given A, B, C, and D we can calculate the focal length (f ocal length = |A -C|), aperture (aperture = |D -E|), and mounting position (lens mount displacement = |B -C|) of the Fresnel lens. Replacing the Spectralon target with a Fresnel lens (see Figure 3b) converts the projector's light eld to orthographic. The vertical and horizontal center lines of the grid image must align about the center of the Fresnel lens. The choice of Fresnel lens parameters and mount position are determined by varying the digital projector zoom control to nd the largest image size and position matching a commercially available Fresnel lens aperture and focal length. Once the design parameters are determined, a system like Figure 3b can be assembled. We used an 8.5 X 10.5 aperture Fresnel lens with an 18 focal length (Edmund Optics #43-015) in the gure.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Test of the orthographic system: (a) is the original test image. (b) is the image mounted on a poster board and viewed through a monochrome camera. (c) is a close up of Gaussian derivative lters (7 × 7, σx = 2, σy = 2) projected on the poster and viewed through a 3CCD camera. (d) is a monochrome version of the test image. (e), computed using Matlab, is a threshold image after ltering (d) with the Gaussian derivative lters. After printing and mounting (d) on a board, it is illuminated with uniform illumination (box functions) and viewed through a monochrome camera (e).(f) is a scene-space lter image with our orthographic projection system and using Jean's algorithm and the same Gaussian derivative lters.


Rahul Swaminathan
email: rahul.swaminathan@telekom.de

Ao Wu

Haoyuan Dong

Depth From Distortions

Traditionally, images distortions have been regarded as a flaw that needed to be corrected. Much effort has been focused on "undistortion" methods to remedy this shortcoming. However, we see distortions as encoding crucial information about scene structure. In this paper we describe how scene depth is encoded in the distortions of images acquired with non-single viewpoint cameras or by mosaic construction. We present our framework to exploit these distortions for 3D euclidean reconstruction of scene features from a single image or view. We present methods specifically designed for features such as straight lines, circles or conics and show how it applies to general planar curves as well. Rigorous experimentation using simulations and synthetic images from catadioptric sensors and mosaics are presented.

Introduction

Historically, distortions have been considered as an unwanted artifact. As a result, various attempts have been made to calibrate for and remove these distortions. In contrast, this paper presents a framework to exploit images distortions to estimate 3D depth of the scene. Furthermore, we also use distortion for partial camera calibration.

Distortions in images stem from various factors related to the underlying imaging geometry. We specifically refer to imaging geometry, as this includes not only the optical components such as lenses and mirrors of the camera but also the method of image capture such as mosaicing. A complete taxonomy of distortions, based on imaging geometries, has already been studied [START_REF] Swaminathan | A Perspective on Distortions[END_REF]. This work specifically studied the class of distortions called caustic distortions, present in multi-viewpoint images. While distortions in single viewpoint images depend on the imaging geometry alone and can be removed, caustic distortions depend also on scene depth making it impossible to remove unless scene depth is known [START_REF] Swaminathan | A Perspective on Distortions[END_REF].

In this paper we focus on depth dependant caustic distortions and methods to exploit them for 3D reconstruction.

Multi-Viewpoint Imagery and Depth.

As the name suggests, non-single viewpoint cameras do not possess a single viewpoint but rather a locus of viewpoints. These viewpoint loci are usually modeled using caustics [START_REF] Swaminathan | Non-Single Viewpoint Catadioptric Cameras: Geometry and Analysis[END_REF][START_REF] Grossberg | The Raxel Imaging Model and Ray-Based Calibration[END_REF]. The simplest way to acquire an MVI is by mosaicing [START_REF] Peleg | Mosaicing on adaptive manifolds[END_REF]. In this case we can model the viewpoint locus simply as the camera trajectory. Another common source of MVIs are catadioptric systems built using lens based cameras and curved mirrors. In general, any curved mirror and lens based camera yields a non-single viewpoint imaging system (see [START_REF] Yagi | Real-Time Generation of Environmental Map and Obstacle Avoidance Using Omnidirectional Image Sensor with Conic Mirror[END_REF][START_REF] Bogner | Introduction to Panoramic Imaging[END_REF][START_REF] Chahl | Reflective Surfaces for Panoramic Imaging[END_REF][START_REF] Bruckstein | Omniview Cameras with Curved Surface Mirrors[END_REF][START_REF] Svoboda | Central Panoramic Cameras: Design and Geometry[END_REF][START_REF] Gächter | Mirror Design for an Omnidirectinal Camera with a Uniform Cylindrical Projection when using the SVAVISCA Sensor[END_REF][START_REF] Hicks | Geometric Distributions for Catadioptric Sensor Design[END_REF]for examples). Only specific combinations of mirror shapes, lenses and positioning lead to a single-viewpoint systems [START_REF] Baker | A Theory of Catadioptric Image Formation[END_REF].

A consequence of viewpoint loci, is that every pixel observes the scene from a different viewpoint. As a result one cannot compute perspective views from MVIs without also knowing scene depth at every image point in the MVI. Any attempt to compute a "near perspective" view leads to distortions called caustic distortions [START_REF] Swaminathan | A Perspective on Distortions[END_REF]. These distortions therefore encode scene depth that we exploit for 3D reconstruction.

In our framework, the user identifies contours in the image as corresponding to scene lines, circles, or a general curve of known functional form. This is sufficient to compute the Euclidean structure of the selected contours as well as perform partial camera calibration. Note that, unlike in the case of single-viewmetrology [START_REF] Criminisi | Single View Metrology[END_REF], we do not require metric information nor the existence of parallel lines nor planes for reconstruction.

While the reconstruction of lines using non-single viewpoint catadioptric systems has been demonstrated in the past [START_REF] Caglioti | On the Localization of Straight Lines in 3D Space from Single 2D Images[END_REF][START_REF] Lanman | Reconstructing a 3D Line from a Single Catadioptric Image[END_REF], the full extent of the problem and its scope to general curves and surfaces has not been fully explored beyond its initial proposal in [START_REF] Swaminathan | Non-Perspective Imaging Systems[END_REF]. In fact, the problem applies to more than just catadioptric cameras. Image mosaics, general multi-viewpoint imaging and the study of monocular moving observer and target based trajectory triangulation and tracking [START_REF] Naeve | Structure from Translational Observer Motion[END_REF][START_REF] Avidan | Trajectory triangulation: 3d reconstruction of moving points from a monocular image sequen[END_REF] all share the same geometric property that we exploit.

We begin by introducing our framework to reconstruct scene structures form one multi-viewpoint image. We present the general framework for planar curves including conics and then present special cases for circles and lines. In general, any scene curve with known functional form can be reconstructed using our approach. We then provide extensive experimental analysis of our reconstruction approach using catadioptric systems. We study the various factors influencing robust reconstruction including imaging system parameters. Finally we provide ground truth verification of our approach on rendered images.

Caustic Distortions and Reconstruction

We now present the general theory of 3D reconstruction from one multi-viewpoint image. As mentioned before, the lack of a single viewpoint in MVIs causes image distortions called caustic distortions [START_REF] Swaminathan | A Perspective on Distortions[END_REF] that depend not only on the imaging geometry but also on scene depth. It is impossible to estimate depth at every pixel from one MVI for arbitrary scenes. However, most urban scenes are rather structured [START_REF] Criminisi | Single View Metrology[END_REF] and contain features including lines and curves providing added constraints that we exploit. We begin with the general case of planar curves including conics and then focus on special formulations for circles and lines. However, the method applies directly to non-planar curves and surfaces as well.

Reconstructing Planar Curves.

Considering the multi-viewpoint image in Fig. 1(a), a curve C in the scene on plane Π is imaged as the distorted curve c. We assume the imaging geometry is known for example by calibration [START_REF] Grossberg | The Raxel Imaging Model and Ray-Based Calibration[END_REF][START_REF] Swaminathan | Non-Single Viewpoint Catadioptric Cameras: Geometry and Analysis[END_REF][START_REF] Sturm | A generic concept for camera calibration[END_REF]. Thus, for any pixel p on c, the viewpoint S c (p), and view direction V i (p) are known. Conversely, we can define the curve C as the intersection of the viewing rays V i with the scene plane Π. Furthermore, any other plane Π ′ intersecting the ray bundle V i results in a curve other than C. Thus, determining C is equivalent to determining Π. 3We assume that we know the functional form F of the curve C, such that for any point P on C, F (P ) = 0. This implies we only know the family to which the curve belongs such as a circle, hyperbola, etc and not its exact curve parameters. Reconstruction of C is then posed as a search for the plane parameters Π = [A, B, C, D] for which the points of intersection with viewing rays V i satisfies the functional form F . The objective function is then:

arg min {A,B,C,D} N F (P ) , (1)
where P denotes the point of intersection of viewing ray V i (p) with plane Π. We apply this approach to conic curves in the experiments to follow. Furthermore, the method directly applies to curves and surfaces in 3D as well.

Assume

Reconstructing Circles.

Robust reconstruction of planar curves, requires a good starting estimate for the four plane parameters Π = [A, B, C, D]. Circles are special within the space planar curves as they allow for an analytic estimate of three (the plane normal) out of the four parameters, making search faster and more robust.

We first observe that under perspective projection the image of a circle is an ellipse. Also, one can determine the orientation of the circle plane given the elliptic image. We use this property to analytically derive the orientation of the scene plane (3 parameters) to start the non-linear search. Referring to Fig. 2(a) we ignore the viewpoint locus and project the rays V i (p) from a single point onto a virtual plane to hallucinate a near perspective image of the circle (see Fig. 2(b)). 4 We then use the near elliptic image to estimate the orientation of the plane containing the circle (see [START_REF] Wu | 3D Reconstruction from a Single Image[END_REF] for details). Using the orientation estimate as a starting point for non-linear search, we optimize for the plane Π that results in a circular intersection with the viewing rays (see Fig. 2(c)). This two stage approach performs more robustly and faster than a complete non-linear search.

Reconstructing Lines.

Lines represent a very special and simple case for reconstruction. It has been shown that: given four distinct lines in 3D, there exist zero, one, two, or various infinities of lines intersecting the given lines [START_REF] Teller | Determining the lines through four lines[END_REF]. In the case of multi-viewpoint images, the skewed viewing rays pass through two lines in theory. One line with positive depth is the scene line and the other, it's geometric complement. In [START_REF] Teller | Determining the lines through four lines[END_REF], the authors derived a linear solution for the 3D line using a six parameter plücker coordinates. This was later used for catadioptric systems [START_REF] Caglioti | On the Localization of Straight Lines in 3D Space from Single 2D Images[END_REF][START_REF] Lanman | Reconstructing a 3D Line from a Single Catadioptric Image[END_REF].

We now present a four parameter linear solution to uniquely estimate the scene line for the case of axially non-central imaging systems. By axially noncentral we mean that the viewing rays of the multi-viewpoint image pass through a central axis. Most catadioptric systems are axially non-central because they utilize rotationally symmetric mirrors whose axis is aligned with the optical axis of the lens. Fig. 1(b), shows such an imaging system where the axis is conveniently chosen to be the Z-Axis. Any line L (other than the axis) can be defined as the intersection between plane Π 0 = [A 0 B 0 1 0], passing through the origin but not parallel to the Z-Axis, and plane Π z = [A z B z 0 1], parallel to the Z-Axis but not passing through the origin.

For a point p on the distorted image l, the viewpoint S c (p) = [0 0 S z c ] on the Z-axis, and viewing directionV i (p) = [V x i , V y i , V z i ] are computed. Now, scene points along these viewing rays can be parameterized as:

P = S c (p) + d • V i (p).
The point P lies on the scene line iff it lies on both planes Π 0 and Π z giving:

d • V x i , d • V y i , S z c + d • V z i , 1 • A 0 , B 0 , 1, 0 ′ = 0 (2) d • V x i , d • V y i , S z c + d • V z i , 1 • A z , B z , 0, 1 ′ = 0 (3)
Now eliminating d from Eqs. [START_REF] Swaminathan | Non-Single Viewpoint Catadioptric Cameras: Geometry and Analysis[END_REF][START_REF] Grossberg | The Raxel Imaging Model and Ray-Based Calibration[END_REF] gives the constraint:

V x i , V y i , -V x i S z c , -V y i S z c • A 0 , B 0 , A z , B z ′ = -V z i (4)
Thus, solving for the 3D position of the scene line reduces to a linear problem of estimating the four parameters of the intersecting planes.

Reconstruction Using Catadioptric Cameras

We now present exhaustive experimental evaluation and analysis of reconstruction using non-single viewpoint catadioptric cameras. We focus on catadioptric cameras using commonly used quadric (spherical, parabolic or hyperbolic) mirrors. We begin with numerous simulations with noise and study the effect of various imaging and scene parameters on reconstruction. Next, we present a real working experiment with rendered images. A rendered scene is used to compare the quality of reconstruction, using the entire pipeline including the user interaction, against known ground truth.

A complete understanding of how noise affects reconstruction also involves understanding the effects of mirror shape, size as well as the position of the reconstructed structure in the scene. We present these dimensions of error analysis in the subsequent sections.

Reconstruction of Conics.

We begin with the reconstruction of general conic shaped objects in the scene. These include elliptic, parabolic and hyperbolic curves in the scene. We consider circles and lines subsequently as special cases of conics. Fig. 3(a) shows a typical catadioptric image of a scene containing an ellipse (in blue), parabola(green) and a hyperbola(black), acquired using a perspective lens and a spherical reflector. The images with other quadric mirrors are similar. Under zero noise conditions all catadioptric systems reconstruct the conics perfectly as seen in Fig. 3(b). With increase in noise in pixels along the image contour, reconstruction errors predictably increase. We present two sample noise level of 0.5 and 1.0 pixels in Figs. 3(c,d).

Reconstruction of Circles.

Circles as mentioned earlier represent a special case of conics due to their symmetry. We use the two stage method described in Section 2.2 to reconstruct scene circles. Figs. 4(a-h) show results of reconstructing multiple circles in the scene using various types of catadioptric imaging systems. In Fig. 4(a) the typically distorted image of the circles acquired with a spherical mirror based system is shown. The images due to the other mirror shapes are similar in nature and hence not presented. Under zero noise conditions, every catadioptric systems obviously perform perfect reconstruction as seen in Fig. 4(b). Subsequent reconstructions using a spherical (c,d), parabolic (e,f) and hyperbolic (g,h) mirrors with uniform distributions of 0.5 and 1.0 pixels respectively are also presented.

A careful observation shows that the spherical mirror based catadioptric camera outperforms both, the parabolic and hyperbolic mirror based systems. A more complete comparison of these mirror shapes is provided after studying reconstructions of lines.

Line reconstruction with noise.

We now present results of the simplest form case of reconstructing scene structures that of reconstructing scene lines from a single non-single viewpoint image.

Figs. 5(a-h), show the reconstruction of various lines in the scene at different positions using various types of catadioptric cameras. Similar to the results with circle reconstruction, in Fig. 5(a) we present a typical distorted image of the scene line in a spherical mirror based system. In Fig. 5(b), we present the reconstruction of these lines from the image points under no noise conditions. Again, all mirror based systems reconstruct the lines precisely. Figs. 5(c,d),(e,f),(g,h) now present reconstructions of the same scene lines using a spherical, parabolic, and hyperbolic mirror and under two noise conditions (0.5, 1.0 pixels) respectively.

Again, it can be observed that the spherical mirror based systems Figs. 5(c,d) outperform both the parabolic and hyperbolic mirror based systems.

Reconstruction and Mirror Shapes.

In the last two sections we noticed that the spherical mirror seemed to outperform the other mirror shapes in reconstruction. This observation behoved us to verify this hypothesis experimentally.

We chose lines for these experiments as their reconstruction is linear and thus faster to compute. As seen in Fig. 6(a), after tens of thousands of simulations of line reconstructions under various noise conditions, it is clear that the spherical mirror based system consistently has lower reconstruction error (black curve), than any other mirror based catadioptric system. The tests were conducted under comparable conditions such as scene structure depth and field of view of the cameras. Moreover, the hyperbolic mirror is actually far worse than any other mirror based system. Another intriguing phenomenon, was the linear relationship between reconstruction error and noise.

Reconstruction and Mirror Size.

Robustness of reconstruction from non-single viewpoint images depends heavily on the size of the underlying viewpoint locus or caustic. The bigger the mirror, the larger the caustic is likely to be. This means, for a given scene structure, its image would undergo greater caustic distortion. A consequence of this is higher robustness to noise during reconstruction. We studied this hypothesis for the case of spherical mirror as they were experimentally shown to be the best of the three mirror shapes studied. Fig. 6(b) shows a comparison of reconstruction error against mirror radius. Note that the mirror was appropriately located below the entrance pupil at increasing distances so as to ensure the system had the same effective field of view. As can be seen from the plots, the larger the radius, the better the reconstruction.

Reconstruction and Target Position.

The above analysis of mirror shape and size assumed arbitrary scene structure (line) positions. The error presented was the average across multiple line positions and orientations. However, the position of the scene structure is also critical to robust reconstruction. This again has close relationship to the underlying caustic or viewpoint locus surface's shape and size.

Referring to Fig. 7(a), we consider a spherical mirror and its associated caustic or viewpoint locus. Considering the vertical cross sectional plane, we consider scene lines equidistant from the spherical mirror centre orthogonal to this plane. We know from theory that any structure corresponding to the plane passing through the axis or the horizon (shown in red) cannot be reconstructed because the viewing rays are coplanar. Thus close to these ray direction, reconstruction errors should be large. For other viewing direction along the mirror the errors should be minimal. Results from our experiment (see Fig. 7(b)) in fact verify our prediction. Errors starting at θ = 0 along the axis are high and then decrease initially before rising again towards the horizon. Beyond the horizon, reconstruction errors decrease once again only to increase towards end of the field of view. Moreover, reconstruction error below the horizon (where the caustic is large and spread out) seems to be lower than in the section between the apex and the horizon where the caustic is compact).

The effect of mirror shape, size and location in fact are all closely tied to the local caustic size and extent. Analogous to stereo baseline, the larger the caustic the better the reconstruction (see [START_REF] Wu | 3D Reconstruction from a Single Image[END_REF][START_REF] Swaminathan | Depth cues in multi-viewpoint imaging[END_REF] for details).

Experiment on Rendered Image.

We finally present experimental results on rendered images of typical urban interiors. We use rendered images because while we still have ground truth, we can still test the entire pipeline used for reconstruction including the user interaction and automatic gradient based contour detection. The details of the user interface and tools for semi-automatic detection of edge contours in the distorted image is beyond the scope of this paper (see [START_REF] Swaminathan | Depth cues in multi-viewpoint imaging[END_REF] for details). We present results under two scenarios: (1) a non-central catadioptric imaging system and (2) a centre strip mosaic.

Fig. 8 (a) shows a catadioptric image acquired with a spherical mirror of radius 7.5cm and a perspective lens based camera rendered using PovRay. The room is roughly 5 × 5m in size. We assume the catadioptric system is calibrated. Using our semi-automatic tool, edge contours in the image are identified as being lines, or circles respectively as in Fig. 8 (b). We then reconstruct the various scene structures using the proposed methods. The reconstructed features (shown in red) are then rendered in the original scene for visualization from two perspectives as seen in Fig. 8 (c,d). Fig. 9 (a) shows a centre-strip mosaic computed by rotating a camera off axis along a known circular trajectory. Again the user identifies edges in the mosaic using the semi-automatic tool as belonging to scene features such as lines or circles respectively. Figs. 9 (b,c) show the reconstructed features (solid red and blue lines) in the room using the proposed methods. We show two views to avoid perspective ambiguities in visualizing the reconstruction.

As seen in both cases of the catadioptric imaging system as well as the mosaic, accurate reconstruction is possible from a single multi-viewpoint image given identifiable structures in the scene.

Conclusion

In this paper we presented a general framework for 3D reconstruction from a single image acquired either as mosaics or using non-single viewpoint cameras. In general any curve in the scene of known functional form can be reconstructed in this manner. We presented a detailed noise analysis by reconstructing numerous scene curves and lines using various catadioptric systems. We also analyzed the factors affecting reconstruction quality such as imaging system design and the scene itself. We find that reconstruction robustness depends on various aspects that effectively measure the size of the underlying caustic or viewpoint locus. In some sense this is similar to the baseline in stereo.

Reconstruction of rendered scenes simulating a catadioptric sensor and mosaic image was performed for comparison with ground truth. While distortions have been historically regarded as an unwanted artifact, we showed how they can be exploited to our advantage for reconstruction.

Fig. 1 .

 1 Fig. 1. Distortions in non-single viewpoint images encode scene depth. (a) The skewed viewing rays constrain the shape of the curve formed on the intersection with any plane Π ′ . Only the real scene plane Π would intersect to form the known scene curve C. (b) An axially non-single viewpoint system imaging a scene line. Proper parameterization of the scene line enables its estimation by solving linear equations in four unknowns.

Fig. 2 .

 2 Fig. 2. Reconstructing circles using two steps. (a) Assume the system to be single viewpoint and hallucinate a perspective image. (b) Fit an ellipse to this hallucinated image and thereof determine orientation of the scene circle. (c) Perform full non-linear search using orientation estimate to determine circle in scene.

Fig. 3 .

 3 Fig. 3. Reconstruction results of conics in the scene using quadric mirror based nonsingle viewpoint catadioptric cameras. (a) Catadioptric image using a perspective lens and a spherical mirror shows the distorted image of these scene conics. (b) We harness exploit the depth encoded due to caustic distortions to reconstruct the conics in 3D accurately (noiseless case). Reconstruction degrades gracefully with increasing noise as shown under simulations of uniform noise distributions of (c) 0.5 and (d) 1.0 pixels.

Fig. 4 .

 4 Fig.[START_REF] Peleg | Mosaicing on adaptive manifolds[END_REF]. Reconstruction of circles in the scene. (a) The distorted image of circles in the scene acquired using a perspective lens and a spherical mirror based catadioptric camera. (b) Under noiseless conditions all catadioptric systems produce accurate 3D reconstructions. Reconstructions using a spherical (c,d), parabolic (e,f), and hyperbolic (g,h) mirror degrade gracefully with increasing noise in the uniform interval (0.5,1.0) pixels.

Fig. 5 .Fig. 6 .

 56 Fig.5. Reconstruction of scene lines using non-single viewpoint catadioptric cameras. (a) An example of distortions in the images of scene lines acquired using a spherical mirror based system. (b) Under noiseless conditions all catadioptric systems produce accurate 3D reconstructions. Reconstructions using a spherical (c,d), parabolic (e,f), and hyperbolic (g,h) mirror degrade gracefully with increasing noise in the uniform interval (0.5,1.0) pixels. Spherical mirror based system seems to outperform other catadioptric systems.

Fig. 7 .

 7 Fig. 7. Experimental study of line position and reconstruction errors. (a) Setup showing the axis and the horizon rays where the viewing rays are coplanar making reconstruction impossible. Scene lines were synthesized and tested for reconstruction errors across the vertical field of view. (b) Errors are as expected high at these singular points and become lower elsewhere.

Fig. 8 .

 8 Fig. 8. Reconstruction of a rendered scene using the depth from distortions framework. (a) The original catadioptric image of a room 5 × 5m in size. (b) Points along contours in the image are semi-automatically extracted and reconstructed. (c,d) With the knowledge of which contour is a line or circle in the scene, we perform Euclidean reconstruction of these structures using the proposed methods. The reconstructed structures (in red) are superimposed on the ground truth views of the scene.

Fig. 9 .

 9 Fig. 9. Reconstruction of a room from a single centre-strip mosaic. (a) The original mosaic image computed by collating the centre column of pixels of an off-axis rotating camera. Marked edge contours semi-automatically detected are then reconstructed. (b,c) Two views from within and outside the room of the reconstructed features overlaid (solid lines) on the ground truth scene structure.

It should be noted that this approach works only when the viewing rays V i (p) form a set of skewed rays in 3D space. If the rays are coplanar, reconstruction is impossible.

Note that since the imaging system is non-central, the image produced will not be elliptical. However, this provides a good starting estimate for the non-linear search.