Optical security and authentication using nanoscale and thin-film structures

Authentication of encoded information is a popular current trend in optical security. Recent research has proposed the production of secure unclonable ID tags and devices with the use of nanoscale encoding and thin-film deposition fabrication techniques, which are nearly impossible to counterfeit but can be verified using optics and photonics instruments. Present procedures in optical encryption provide secure access to the information, and these techniques are improving daily. Nevertheless, a rightful recipient with access to the decryption key may not be able to validate the authenticity of the message. In other words, there is no simple way to check whether the information has been counterfeited. Metallic nanoparticles may be used in the fabrication process because they provide distinctive polarimetric signatures that can be used for validation. The data is encoded in the optical domain, which can be verified using physical properties with speckle analysis or ellipsometry. Signals obtained from fake and genuine samples are complex and can be difficult to distinguish. For this reason, machine-learning classification algorithms are required in order to determine the authenticity of the encoded data and verify the security of unclonable nanoparticle encoded or thin-film-based ID tags. In this paper, we review recent research on optical validation of messages, ID tags, and codes using nanostructures, thin films, and 3D optical codes. We analyze several case scenarios where optically encoded devices have to be authenticated. Validation requires the combined use of a variety of multi-disciplinary approaches in optical and statistical techniques, and for this reason, the first five sections of this paper are organized as a tutorial. © 2017 Optical Society of America


INTRODUCTION
The publication of the two seminal papers in optical security in 1994 and 1995 initiated the creation of a new sub-discipline within optics [1,2].Twenty years later, papers published in the field of optical security and encryption are numbered and downloaded in the thousands [3][4][5][6].Basic concepts behind optical encryption are relatively straightforward since they are based on the use of a 4f coherent processor: the data to be encrypted (plain-text) is phase-encoded using phase-only random distribution M 1 ; a second random phase mask M 2 is located at the Fourier plane of the first lens.When the system is illuminated by means of a laser source, the propagated field at the recording plane (i.e., the Fourier plane of the second lens) is pseudo-random and noise like.The original plain-text can be accessed from the complex cypher-text (not the intensity), provided the conjugate of phase mask M 2 is known.
In order to improve the encryption capabilities and security, multiple enhancements and variations of the original double random phase encryption setup have been proposed.For instance, encryption systems have been suggested that work in the Fresnel domain [7] or fractional-Fourier domains [8]; wavelength multiplexing [9] and incoherent light [10] are also notable design variables.Those systems based on polarization deserve special mention since they dramatically increase the degrees of freedom of the encryption processor [11][12][13][14].Nevertheless, it was proved that the original double random phase encryption design was flawed due to the linear character of optical transformations involving light propagation [15,16].Multiple alternatives have been suggested in order to make optical security systems more robust [17][18][19][20][21].For instance, the so-called digital optics methods represent a very popular approach in optical-based encryption [22][23][24][25][26][27].These procedures mimic physical processes by means of digital calculations or combine numerical operations with optical procedures.Related to these methods, phase-truncation asymmetric keys [28,29] or optical designs aimed to break symmetries [30] have been proposed.On the other hand, systems that work in low light conditions are being used in security.These arrangements use quantum sources that provide very few photons.Despite operating in photonstarved conditions, which makes the decryption very low light level, validation of the authenticity of the message can be achieved.In those scenarios where it is not required to access the information, photon counting encryption is a very good and secure alternative [31][32][33][34][35][36][37][38][39].Other designs based on computational ghost imaging have been proposed [40,41].
Most of the optical encryption methods rely on the use of holographic encoding of information.A very recent approach on optical security is based on including an extra security layer in order to authenticate messages.Information is encoded using conventional techniques, but the recipient may be able to determine whether the information was sent by a trusted party.The physical support used for the message plays a key role in this problem.The recipient has to be able to read or decrypt the data but also can test the material using optical analytical tools in order to validate the authenticity of the sender.For instance, bank notes printed using inks containing certain amounts of nanoparticles are difficult to be counterfeited [42].Another possible scenario where validation is a critical issue is the electronic industry.A label with the specifications of an integrated circuit is attached to the microchip package.This label can be phaseencoded or produced with metal nanoparticles or thin films.The user can rely on the contents of the information provided if the response to certain optical measurements (speckle statistics, polarimetric signatures, etc.) delivers the correct signal [43][44][45].Gold nanoparticles (AuNP) can be used in the fabrication of pharmaceutical pills and tablets for consumer safety and protection.These samples become polarimetrically tagged, and, thus, authentication can be performed using ellipsometry [46].
As explained above, the message has to be encoded on a device that can be authenticated using optical technologies.Accordingly, production of these codes may require the use of fabrication techniques in the nanoscale such as nanostructures [47][48][49] or thin-film deposition [50].Regarding materials, metallic nanoparticles and, more specifically, AuNP are suggested in the fabrication of optical codes because they provide distinctive polarimetric signatures that can be used for validation [51][52][53][54][55].
This paper aims to review several scenarios of security and authentication of codes tagged with nanoparticles, phase-encoded or produced using thin-film technologies.Since authentication of such structures requires the combined use of several techniques in different disciplines, Sections 2-5 are organized as a tutorial.This includes topics such as Fourier optics, polarimetry, and statistical pattern classification using machine-learning techniques.Accordingly, the paper is organized as follows: Section 2 reviews Fourier optics and wave propagation theory.Within this framework, the behavior of phase-encoded information after propagation is analyzed and related with speckle noise.Double random phase encryption is considered as a special case of phase encoding.Section 3 describes basic concepts on polarized light including Stokes parameters and Mueller matrix formalism.Special emphasis is placed on the measurement of the Mueller matrix using ellipsometers.Section 4 describes the important concept of data analysis.Since multidimensional measurements are required for authentication of the samples, we introduce principal component analysis as a method to reduce dimensionality in complex problems.Section 5 surveys three methods in machine-learning classification used in this review: K-nearest neighbors, multiclass support vector machine, and random forests.
The second part of the paper reviews several experiments on authentication for security.Section 6 analyzes how to distinguish among phase-encoded QR codes made of gold nanoparticles or silver [44]: the samples are illuminated using polarized light, and speckle noise distributions are recorded after propagation.A three-class support vector machine algorithm is used to perform authentication.A similar problem is studied in Section 7. In this case, QR codes are produced using thin films of Cr and Ta 2 O 5 .The dataset is synthetically generated from the measurements of the Mueller matrix.Different scenarios are considered for authentication.For instance, the samples can be imaged using a microscope polarimeter.The information provided by the Mueller components is used to perform authentication by means of correlation.When the samples are phase-encoded, the recorded Mueller components of the speckle signal are used to generate a large dataset of Stokes vectors that feeds a K-nearest neighbors classification system [45].In the next section, we describe how to distinguish among pharmaceutical tablets that have been coated with gold nanoparticles of different sizes.In this case, we analyze the polarization signature of these tablets using a Mueller spectro-polarimeter [46].In Section 9, we train a system based on a random forests classifier to authenticate 3D phase-encoded samples [56].Finally, conclusions and future perspectives on this topic are presented in Section 10.

PROPAGATION OF PHASE-ENCODED INFORMATION
Scalar diffraction theory provides a simple way of describing the propagation of wave-front E [57].Without loss of generality, it is assumed that the electric field propagates along the z axis.Then, the mathematical relationship between the propagated wave-front Ex; y; z and the beam Ex; y; 0 at reference plane z 0 is Ex; y; z FT −1 FTEx; y; 0Hu; v; z; In these formulas, operators FT and FT −1 stand for the Fourier and the inverse Fourier transforms, respectively, Hu; v; z is the transfer function, (u, v) are the spatial frequencies, λ is the wavelength, and i is the imaginary unit.Unit vector s α; β; γ indicates the normal direction to the wave-front at each point.Components α; β; γ are related to the spatial frequencies (u, v) by means of Note that for α; β ≪ 1, the wave is paraxial; i.e., vector s forms small angles with propagation axis z.If this approximation is valid, transfer function Hu; v; z can be simplified and now reads Hu; v; z expikz exp−iπλzu 2 v 2 .In this case, it is said that propagation is described by means of the Fresnel diffraction approximation.
Propagation of light using scalar diffraction theory is calculated by means of the following formula: In Fresnel diffraction conditions, propagation of light is calculated using paraxial approximation: Ex; y; z expikzFT −1 FTEx; y; 0 exp−iπλzu 2 v 2 : Figure 1 shows an example of how Eq. ( 1) is used in practice.Figure 1(a) depicts a QR code encoding the message 0034934021143 (a telephone number).This distribution [Qx; y; 0] is illuminated by a coherent light source.The variables used in the calculation are summarized in Table 1.Random phase-only distributions are commonly used in optical information processing problems (i.e., pattern recognition, optical encryption, etc.).They are described by means of Rx; y; 0 exp2πi randx; y, where randx; y is a function that is uniformly distributed random numbers over [0,1].Phase encoding is performed by attaching a diffusor such as adhesive tape to the sample.Note that thick diffusors are random in 3D by nature, and, thus, thick phase masks cannot be duplicated.
The analysis of the behavior of propagated random phase-only distributions Rx; y; 0 provides interesting information.Figure 2(a) shows the irradiance I R x; y; z jRx; y; zj 2 obtained by calculating the propagated field Rx; y; z using Eq. ( 2) in Fresnel approximation conditions.The set of values in the calculation is the same as that used in the previous simulation and summarized in  [58,59] describes the intensity distribution of speckle patterns using the following formula: where I, hIi, and σ are the intensity data points, its average, and the corresponding standard deviation, respectively, and Γ stands for the Gamma special function.
Parameter n 0 is defined as n 0 hIi∕σ 2 .Equation (3) corresponds to the Gamma probability density function [60].This curve is superimposed on the histograms of Figs.Random phase-encoded images produce a speckle-like noise distribution after propagation.The intensity distribution of a speckle pattern is described using the following formula: Γn 0 : Despite the fact that I C x; y; z and I Q x; y; z can be easily distinguished by visual inspection, due to the statistical nature of the propagated random phase-encoded beam, the log-likelihood test can be used to discriminate among random phase-encoded and non-phase-encoded propagated distributions using a straightforward calculation [43].Note that the log-likelihood statistical test is useful for distinguishing among two distributions of probability.Assuming that I C x; y; z and I Q x; y; z can be considered as classes that represent the null (H 0 ) and alternative (H 1 ) hypothesis, respectively; the log-likelihood condition that validates the null hypothesis (H 0 ) is X x;y logI C x; y; z − logI Q x; y; z > 0: (4)  Calculating the natural logarithm of Eq. ( 4) for the N pixels of the image, Using the data obtained from Figs. 1(b) and 2(b), the following values are obtained: P x;y logI C x; y; z 126629 and P x;y logI Q x; y; z 54165.This means that the null hypothesis is accomplished.As expected, I C x; y; z has a better fit with the Gamma-distribution model [Eq.( 3)] than I Q x; y; z.
As explained in Section 1, the double random phase encryption [2] technique provides a basic theoretical framework for optical encryption.It can be understood as a generalization of random phase encoding [1]. Figure 4 shows a sketch of a 4f optical encryption system: plain-text image Q is random phase-encoded by means of phaseonly mask M 1 , and phase-only key M 2 is placed at the Fourier plane of the first lens.The scalar electric field at the Fourier plane of the second lens (cypher-text) is where sub-indices λf indicate that Fourier transforms have to be scaled accordingly.Ex; y; 4f is a complex function and has to be recorded using holographical or interferometric techniques.Note that the same optical system can be used to decrypt the message: plain-text Q can be accessed provided that phase key M 2 is known: where * is the complex conjugate operator.Note that Eq. ( 7) is valid assuming that Q is a real-valued, positive distribution image.

POLARIZED LIGHT
A quasi-monochromatic transverse electromagnetic field propagating in the z-axis direction is described as E E x r; t; E y r; t; 0. The Stokes vector S S 0 ; S 1 ; S 2 ; S 3 is a convenient way of describing the state of polarization of the wave-front [61].The components of this vector (the Stokes parameters) are defined as Sketch of 4f double random phase encryption system.
where hi stands for the temporal average over time interval T , and Note that S 0 is a measure of the irradiance of the beam, S 1 compares the irradiance of the wave in the x-and y-directions, and S 2 is similar to S 1 but the comparison is performed along two perpendicular directions rotated 45°with respect to the x axis.S 3 provides information of the circular content of the wave.For instance, for a linearly polarized beam with a polarization angle ψ with respect to the x axis, the Stokes vector takes the form S S 0 1; cos 2ψ; sin 2ψ; 0. Elliptically polarized light is described by a Stokes vector: S S 0 1; cos 2ψ cos 2ϵ; sin 2ψ cos 2ϵ; sin 2ϵ; (10) where ϵ is the eccentricity angle.The four Stokes parameters are sometimes combined in a single one, called the degree of polarization (DoP): where P ≤ 1.Note that for fully polarized beams, S 2 0 S 2 1 S 2 2 S 2 3 and P 1, whereas for totally depolarized light, P 0. Sometimes, the Stokes parameters are normalized to S 0 :_ s S 1 ∕S 0 ; S 2 ∕S 0 ; S 3 ∕S 0 and only three components are used.
The Poincaré sphere, defined as the surface j_ sj 1, provides a geometrical interpretation of the states of polarization (see Fig. 5).For instance, linearly polarized light beams are located on the equator of the sphere.Circularly polarized lights are placed on the poles (clockwise, North Pole; counterclockwise, South Pole).Totally polarized light (P 1) appears on the surface of the sphere, whereas unpolarized light (P 0) is set at the very center of the Poincare sphere; partially polarized light states (P < 1) are represented inside the sphere.In summary, an arbitrary state of polarization at the Poincaré sphere is described by S S 0 1; P cos 2ψ cos 2ϵ; P sin 2ψ cos 2ϵ; P sin 2ϵ: (12) When light interacts with an optical element, the state of polarization of the beam can change.The Stokes parameters of the wave-front are transformed according to a linear law represented by a 4 × 4 Mueller matrix, i.e., S 0 M • S. If the beam passes through different polarization devices, the final Stokes vectors are obtained by combining the Mueller matrices of each element, Mueller calculus provides a complete representation of the state of polarization of light because it can be used with incoherent light.The Jones formalism provides a simpler description of polarization, but it can only be used with coherent beams.
The Stokes vector is a convenient way of describing the state of polarization of the wave-front.Samples and optical devices that interact with light are described by the components of the Mueller matrix.When a light beam interacts with matter, the resulting state of polarization is obtained by multiplying the Mueller matrix by the original Stokes vector.If the beam passes through different optical devices, the overall Mueller matrix is obtained by combining the Mueller matrices of each element.
Equation ( 13) below shows the Mueller matrices for a linear polarizer with polarization axes in the 0°, 45°, 90°, and −45°directions.The matrix for a quarter wave plate with fast axis in the horizontal direction is also indicated: Note that the Mueller matrix for a different direction of polarization θ can be obtained using rotation matrix Rθ: Poincaré sphere displaying some states of polarization.
The Stokes parameters can be easily measured with the help of a polarizer and a quarter wave plate in front of the light sensor using the following set of equations: I α;0 stands for the recorded irradiance when the linear polarizer is set at an angle α with respect to the x direction, and I α;π∕2 is the image recorded when a quarter wave plate is used in addition to the polarizer.After recording these six quantities, the Stokes parameters are easily obtained.Mueller calculus provides a simple way of explaining why Eq. ( 16) provides a way of measuring the Stokes parameters.The corresponding Stokes vectors of the state of polarization of the light recorded in each measurement, namely S 0 , S 1 , S 2 , and S 3 , are determined using the Mueller formalism.Let S S 0 ; S 1 ; S 2 ; S 3 ; then Note that the first component of the Stokes vector provides the information of the measured irradiance.For instance, vector S 2 is obtained by subtracting the states of polarization when the beam crosses polarizers M 45°a nd M −45°.In this case, the measured irradiance is S 2 -that is, the third Stokes parameter of the analyzed beam.
Whereas light is characterized by the four parameters of the Stokes vector, samples and optical devices that interact with light are described by the components of the Mueller matrix.In the general case, the Mueller matrix displays 16 different nonvanishing terms.In fact, due to matrix normalization, only 15 components have to be measured.For isotropic samples, the Mueller matrix M I takes a very simplified form because 8 out of the 16 components are zero [62].In this case, M I reads where quantities N , C, and S are related with ellipsometric angles Ψ and Δ by means of N cos2Ψ; Ψ and Δ are related to the p-and s-reflection Fresnel coefficients (r pp and r ss ) by means of ratio ρ: For non-depolarizing anisotropic samples, the differential Mueller matrix L lnM provides a simpler description of the media considered [63].Instead of the 15 independent components of the conventional Mueller matrix, L is fully described by means of six independent terms: In this case, three generalized Fresnel coefficients have to be defined, namely ρ r pp r ss , ρ ps r ps r ss and ρ sp r sp r ss where r ps and r sp stand for the reflected light that is transformed from the p-to the s-polarization and vice versa.These ratios can be calculated using the components of the differential Mueller matrix [62]: where ! and T ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 23 il 01 2 l 13 il 02 2 l 12 − il 03 2 q :

Measurement of the Mueller Matrix
The measurement of the 16 components of the Mueller matrix of a sample is not a simple task.As a general approach, M can be determined by subsequently illuminating the sample with four different beams with independent states of polarization.It assumed that the best results are obtained when the four Stokes parameters of the illuminating beam are on the vertices of a regular tetrahedron inscribed in the Poincaré sphere [64,65].Accordingly, a set of 16 independent linear equations is produced, and thus the problem can be solved.To minimize error, more than four different states of polarization measurements can be made.In this case, the components of the Mueller matrix can be determined by calculating the pseudo-inverse matrix.
Some different methods for measurement of the Mueller matrix have been proposed [66].Among the available techniques, in this paper we used two different approaches adapted to the characteristics of the problem.The first system considered is based on four photoelastic modulators (PEMs) as variable phase retarders.This device presents two main advantages: measurements are made simultaneously, and there are no moving parts [67].A sketch of this setup is presented in Fig. 6(a).PEMs are described as phase plates with a horizontal fast axis with retardation of δ.PEMs are electrically driven, with retardation being a periodic function of time, i.e., δt ∝ coswt ϕ: The Mueller matrix of the sample is obtained by writing the polarization state of light recorded by the detector.The light reflected by the sample with matrix M S is where M PEM0 , M PEM1 , M P1 , θ 0 , θ 1 , and, θ 2 are the matrices and orientation angles of PEM0, PEM1, and P1, respectively.The angles are set in such a way that calculations are simpler.Equivalently, the polarization state at the detector is where the first component of S D is the measured intensity of the detector.Since δt is a harmonic function, the components of the sample matrix M can be obtained using Fourier analysis of the recorded signal.The interested reader can access Ref. [67] for a more comprehensive explanation.
A second instrument for measuring the components of the Mueller matrix has been used.The outline of the apparatus is presented in Fig. 6(b).Again, this device can be described in terms of four main devices: the polarization state generator uses a linear polarizer and a rotating compensator, whereas the reflected light beam is analyzed by means of a rotating compensator and a polarizer.The sample is imaged on a camera, and, thus, the present system is able to perform image polarimetry.
According to Fig. 6(b), the Stokes vector S D of the detected light is determined using the following equation: where M 0 and M 1 are the Mueller matrices of linear polarizers [Eq.(15)] and M RC is the corresponding matrix for a compensator:  Sketch of (a) four photoelastic modulators Mueller matrix polarimeter and (b) Mueller matrix imaging ellipsometer.PEMs are photoelastic modulators.Reprinted with permission from [46].Copyright 2016 Optical Society of America.
Note that theta is a function of time θ wt and δ is the retardation of the compensator.The angular speeds of the two compensators w 0 and w 1 were related by the relationship w 0 p q w 1 , where p and q are integer numbers.The orientation of polarizers M 0 and M 1 was set at 0°and 90°, but other options can be considered.As in the previous case, the components of the Mueller matrix of the sample are determined using Fourier analysis methods [68].

VARIABLES, DATASETS, AND REDUCTION OF DIMENSIONALITY
Classification problems require an accurate selection of the variables used to train and test the system.In some of the practical examples described in this paper, we use different kinds of data: in scenarios where polarization plays a key role, parameters such as the Mueller matrix components, and the Stokes parameters or the DoP, provide additional information.Sometimes these variables are measured for a determined range of wavelengths, providing classification features with a large number of dimensions.In the case of measurements of speckle, recorded data can be adjusted to a Gamma distribution of probability.Here, the histogram of the distribution or statistics such as mean, variance, skewness, and kurtosis can be helpful.These parameters are defined as where x i are the corresponding measurements and N is the total number of pixels.Note that the formulas for the skewness and kurtosis are bias-corrected and differ from the conventional ones.Also, the image entropy [69] is also taken into account: Here, p i is the relative frequency distribution of gray level i.
A training dataset X for N samples is described by means of a N × M matrix where row i represents an M -dimensional vector containing the features that describe sample i. Classes are labeled as integer numbers and arranged as a column vector y y 1 ; y 2 ; …; y n T of dimension  Sometimes, feature vectors can have high dimensions.For instance, a conventional 8-bit image histogram has to be stored in a 256-component vector; a wavelength spectrum can require several hundreds of components depending on the analyzed range and the step size.This huge amount of information may complicate the use of classification techniques, especially when the number of measured samples is small when compared with the dimension of feature vectors.This effect is known as the curse of dimensionality [70].Principal component analysis (PCA) is popular method for reducing dimensionality [71].The idea beyond PCA is to find an alternative base that fits better the characteristics of the measured variables of the problem.For an excellent and intuitive tutorial on PCA, the reader can access Ref. [72].
The PCA approach can be derived using just linear algebra.Let Y be a matrix related to dataset X [Eq.(30)] by means of linear transformation P, i.e., Y PX.In what follows, the mean of the measurements is zero (i.e., the mean has been previously subtracted).Reduction of dimension is based on the idea that two uncorrelated variables can be described as a new variable that is a linear combination of the original ones.Note that the cross variance between two uncorrelated variables is equal to zero.We introduce a covariance matrix for the dataset described in the transformed space S y as Rewriting Eq. ( 31) in terms of P and X, S y becomes where the new matrix A is A XX T .Note that the target of PCA is to find the linear transformation that provides the best representation of dataset X. Best means that the variables have to be uncorrelated and thus S y has to be diagonal.Since A XX T , this matrix can be diagonalizable, i.e., A EDE T ; D is a diagonal matrix and E is the transformation (eigenvectors) matrix.Finally, if we set E P T , then A P T DP and S y becomes diagonal: Note that P T P −1 .In summary, the eigenvectors of A XX T transform S y into a diagonal matrix.It is worthwhile to point out that eigenvectors associated with high eigenvalues (variances) of S Y provide more information than those directions whose eigenvalues are small.Accordingly, these less relevant directions can be avoided, and the dimensionality of the dataset is reduced.
Principal component analysis looks for an alternative base for reducing the dimensionality of dataset X; this base corresponds to the eigenvectors of A XX T .Eigenvectors associated with high eigenvalues provide more information than those directions with small eigenvalues: at the end of the day, less relevant directions are avoided and the dataset dimensionality is reduced.

CLASSIFICATION USING MACHINE-LEARNING TECHNIQUES
In this section, we provide a summary of the machine-learning classification methods [73] used in the present paper.Among different possible techniques, we used K-nearest neighbors (K − NN), support vector machines (SVMs), and random forests (RFs).
The three methods are non-parametric and supervised.Non-parametric means that the number of variables involved in the calculation is determined by the training data.Supervised learning means that the system learns from a training set of data in which each member of the training set is labeled with the class this element belongs to.

K -Nearest Neighbors
K − NN is a machine-learning classification algorithm that is relatively simple to implement and that provides very accurate results with high classification success ratios [74].To determine the class to which a test sample belongs, K − NN searches for the k-closest points of the training dataset.The analyzed test point is assigned to the class to which the majority of the k-closest points belong.Sometimes, this voting procedure is weighted according to the distance; that is, the most distant points are less significant and vice versa.Euclidean distance is the by-default option, but other definitions of distance or weights can be used as well.Note that the proper selection of the number of neighbors k is highly dependent on the problem considered.In general, small values of k define better boundaries, but the system becomes more sensitive to noise.
Hold-out validation of classification is performed by splitting the dataset into training and test subsets.To avoid bias, validation is performed several times.In each trial, the samples used in the training or test sets are randomly selected.Classification accuracy is determined by calculating the ratio of the total number of correctly classified samples, divided by the total number of test samples [75].Several websites provide materials to test and experiment classification methods using real data.For instance, the interested reader can access the scikit-learn website for useful insight on K − NN classification [76,77].
To determine the class of a test sample, k − NN searches for the k-closest points of the training dataset.The analyzed test point is assigned to the class to which the majority of the k-closest points belong.Hold-out validation of classification is performed by splitting the dataset into training and test subsets.To avoid bias, validation is performed several times.In each trial, the samples used in the training or test sets are randomly selected.

Multiclass Support Vector Machine
SVM is a classification method based on determining the best hyper-surface able to distinguish between two classes [78,79].Later on, SVM was generalized in such a way that can be utilized with multiple classes as well [80].The use of SVM is suggested when the number of samples in the datasets is small but they present high dimensionality.
Let x i and y i be the m-dimensional feature vector of sample i and the label that describes the class to which x i belongs, respectively [x i are the rows of Eq. ( 30)].
Possible values for y i are 1 and −1.Let w • x b 0 be the equation of the optimum hyperplane (namely, maximum margin hyperplane) that separates both classes; w is the normal vector to the hyperplane, x is a hyper-point in the feature space, and b is the bias.Optimum hyperplane means here that the distance between the hyperplane and the nearest point of either group is maximized.These points are the so-called support vectors.The hyperplanes that contain the support vectors (margin hyperplanes) are w • x b 1, and the distance between them is 2∕jwj.Note that classes that are not separable in the feature (original) space can be separated in a transformed higher-dimensional space.A 2D sketch of the elements involved in a two-class SVM is presented in Fig. 7.
Linear SVM is enunciated as follows: all the samples of class +1 have to lay in the semi-space defined by w • x b ≥ 1.For the class −1 samples, the condition is w • x b ≤ −1.Accordingly, all the samples of the training set have to satisfy the following conditions: The objective of the algorithm is to find the hyperplane whose distance between margin hyperplanes 2∕jwj is maximized.In other words, the system looks for the hyperplane that satisfies This condition can be refined taking into account that some samples cannot be properly classified.We introduce parameter ξ as the distance between a misclassified sample and the correct margin hyperplane; ξ 0 if the sample is correctly classified.Then, condition ( 35) is rewritten as with conditions y i w • x i b ≥ 1 − ξ i and ξ i ≥ 0 for i 1; …; n; (36) Two-class SVM: margin hyperplanes and support vectors.
where C is a regularization constant that has to be determined in such a way that SVM reports the best possible results.
In general, classes cannot be separated by means of a hyperplane, and a nonlinear approach is required.Feature space is transformed to a new space by means of nonlinear transformation law ϕ.This new space can be of higher dimension, but the decision boundary surface in the transformed space is again a hyperplane.Condition (36) becomes Kernel function K describes a simple way of calculating dot products.Several kernels have been described; in this paper we use the Gaussian radial basis function kernel, defined as Again, C and γ have to be found in order for SVM to produce the best classifications results.
SVM has been generalized to K classes.In order to attack this problem, two approaches are possible: (i) One-versus-one (two classes are taken into account each time): two-class SVM is run KK − 1∕2 times: class A versus B, A versus C, B versus C, etc.In each run, the sample is assigned to a determined class and the class to which the sample belongs is the one that has been assigned a majority of times.This is the approach used in this paper.(ii) One-versus-the-rest: one class is tested against the rest of the samples.

Random Forests
RFs [81] are based on decision trees.Decision trees are commonly used as selection procedures since they provide a visual perspective of the analyzed problem.The main approach behind RFs is that a group of weak learners or classifiers such as trees can be integrated to produce a strong learner or classifier.Averaging trees is also known as bagging.Based on this tool, random forests have been demonstrated as a very powerful classification technique.According to some authors, random forests provide the best possible performance among classification methods [82], but other groups find this statement arguable [83].
As indicated in Eq. ( 30), X is the dataset matrix; each of the n samples of the dataset is described by m features, whereas y provides the information of the class each sample belongs to.X and y are combined in a new matrix _ X (see Fig. 8).
With the information contained in the dataset, a complete decision tree is generated (Fig. 9).Each node is split into leaf nodes using the conditions until all features are used.
RFs are based on selecting at random a limited number of features and samples of the dataset (bagging process).Note that the minimum number of features ν required to create a random tree is v ⌊ ffiffiffi ffi m p ⌋, where the symbol ⌊⌋ stands for the floor operator.
A random decision tree is created using partial data (for instance, yellow and blue ellipses in Fig. 8).The system is trained by generating a large collection of uncorrelated random trees (such as the red box in Fig. 9).RFs classification is based on the assumption that most of the partial random trees will provide a correct prediction of the class.When a test sample has to be classified, its features are used to check the generated random trees.The output of every random tree provides a vote for a specific class for the test sample.The class to which the sample is assigned is obtained by the majority of votes of the set of random trees.
Support vector machine is a classification method based on determining the best hyper-surface able to distinguish among two or more classes.Random forests are based on selecting randomly a limited number of features and samples of the dataset (random tree).The system is trained by generating a large collection of uncorrelated random trees.
Classification is based on the assumption that most of the partial random trees will provide a correct prediction of the class.

AUTHENTICATION OF CODES PRODUCED WITH GOLD NANOPARTICLES
In this section, we demonstrate by experiments that QR codes providing the same information but made of different materials can be distinguished by the combined use of nanoparticle phase encoding, polarized light, statistical analysis of speckle, and machine-learning techniques [44].The problem consists of authenticating a 21 × 21 pixel QR code fabricated with gold nanoparticles (AuNP) from another QR code Example of a complete decision tree.The red box indicates a random selection of nodes and conditions, i.e., a random tree.Optical setup for authentication of codes produced with gold nanoparticles.Reprinted with permission from [44].Copyright 2015 Optical Society of America.
a given sample, all the histograms behave in a very similar way.However, there are slight differences in the histogram when the polarization of the illuminating beam is changed.This fact can be explained in terms of the dichroism present in metal nanoparticles.In other words, optical parameters depend on the direction of polarization of the incident light [51][52][53].
As discussed in Subsection 4.1, PCA provides a way for reducing the dimensionality of the problem.Accordingly, we used this technique to provide an alternative representation of the histograms.Figure 13(d) displays the results of applying this technique to the histograms.Results are shown using a 3D plot, which means that only the first three principal components are taken into account.It is apparent that using the PCA representation, the set of histograms for each sample looks completely disconnected.As in the previous experiment, the measured histograms display small differences depending on the polarization direction of the illuminating beam.Moreover, the profiles of the histograms look very similar for the three cases considered.Taking into account that the histogram of the propagated irradiances can be described as  a speckle-Gamma probability distribution, a Kolmogorov-Smirnov test was used to check the goodness of the fit [84].All the histograms of the three samples passed the test.
Again, we have performed reduction of the histogram dimensions using PCA. Figure 15(d) shows a plot using the three first principal components.Surprisingly, the clouds of points for each sample are not overlapped.
Finally, classification among samples using three-class SVM has been carried out.
Since SVM is appropriate for high-dimensional problems, we fed the SVM algorithm with the information provided by the histograms.Both cases are considered: phaseencoded and non-phase-encoded samples.A hold-out strategy is used to train the classification model (see Subsection 5.1).The dataset is split into training and test subsets, and validation is performed 40 times.As explained above, the samples used in the training or test sets were randomly selected.Table 2 shows accuracy results: no errors were detected during classification.
In summary, we demonstrated that polarized light can be used to distinguish and validate phase-encoded QR codes produced with metallic nanoparticles.The histograms of the speckle distributions obtained after propagation are used to produce a dataset for multiclass SVM.Calculations show that the classification accuracy is 100%.

AUTHENTICATION OF CODES PRODUCED WITH THIN-FILM TECHNIQUES
In the previous section, we demonstrated that codes encoding the same information but produced using different nano-encoded materials can be distinguished because of their different polarimetric signatures.Slight differences in the image histogram are used to train a classification system that is able to perform successful authentication.Now we consider a similar problem: two QR codes providing the same information are made using standard thin-film technology.We demonstrate that they can be distinguished by analyzing the polarimetric properties of the reflected light [45].
In the present study, the measurements were carried out using the Mueller matrix image ellipsometer presented in Subsection 3.1 [Fig.6(b)] [68].
The QR used is shown in Fig. 16(a).It encodes the numerical sequence 0034934021143.The lithographic mask with the QR image was produced using a  Reprinted with permission from [45].Copyright 2015 Optical Society of America.
high-contrast orthochromatic film [Fig.16(b)].The two codes were produced using chromium (Cr) and tantalum pentoxide (Ta 2 O 5 ), respectively.These materials were deposited on a microscope slide (25 mm × 75 mm) using radio-frequency sputtering [Fig.16(c)].The thicknesses of the resulting codes were 120 nm and 20 nm for the Ta 2 O 5 and Cr samples, respectively.Note that the Ta 2 O 5 and Cr codes look very different: whereas the former is very transparent, the latter is quite absorbent [Fig.16(c)].More information about the fabrication procedure of these codes can be found in [45,85,86].
The system is illuminated by means of a LED source in combination with an interferometer filter (λ 535 nm).The state of polarization of the beam is determined by means of a linear polarizer and a rotating achromatic compensator.The beam interacts with the sample with an angle of incidence of 56°and passes through a rotating compensator and a linear polarizer.The resulting distribution of irradiance is imaged using an objective lens and recorded by a CMOS camera.This instrument produces the 16 Muller component images of the sample.
Figures 17(a) and 17(b) display the 16 Mueller images for the two samples considered.
It is quite apparent that eight out of the 16 components of the matrix are zero and, as expected, the two samples are isotropic [Eq.( 18)].The Mueller matrix for isotropic samples is highly symmetrical with only three independent terms (N , C, and S).These quantities are related to ellipsometric angles Ψ and Δ [Eq.(19)] and ratio ρ between the p-and s-reflection Fresnel coefficients [Eq.( 20)].This parameter can be used to distinguish among the two samples.
We calculated coefficient ρ for the two samples, namely ρ Cr and Ta 2 O 5 , and phaseonly filtering is used as a way to distinguish between the two samples [87].This operation is defined as where ⊗ stands for the phase-only filter correlation operator.Figures 18(a) and 18(b) show a 3D representation of correlations jρ Cr ⊗ ρ Cr j and jρ Cr ⊗ ρ Ta 2 O 5 j.Note that the correlation peak for the latter is very small when compared with the autocorrelation.This means that the chromium sample is detected whereas the Ta 2 O 5 is rejected.Therefore, polarimetric coefficient ρ can be used for distinguishing two identical images made of different materials using correlation.
Anisotropic materials such as cellophane [88] can be used for producing more complex structures.Anisotropy has been induced by attaching small strips of adhesive tape In this second experiment, recognition based on phase-only correlation of Fresnel ratios ρ is not as simple as in the scenario presented above.With non-isotropic samples, two new complex-valued ratios, namely ρ sp and ρ ps , can be used to discriminate among samples.The three Fresnel ratios for anisotropic non-depolarizing media are calculated using Eqs.( 21) and ( 22) (see Section 2 and Ref. [62]).
Table 3 shows the value of the cross-correlation maxima for ρ, ρ sp , and ρ ps .Values are normalized to the corresponding autocorrelation maximum.In anisotropic media, the three cross correlations have to be very small in order reject that sample.In this example, the three cross-correlation values are very small and thus the Cr code is detected.
A third scenario has been considered.The objective lens of the ellipsometer was removed, and, thus, the samples cannot be imaged [see Fig. 6(b)].Note that the QRs are phase-encoded, and, therefore, the recorded components of the Mueller matrix are speckle-like noise distributions.In this condition, the code cannot be read, but we demonstrate that the samples can be distinguished using only the polarimetric signature.This is very advantageous from the security point of view because samples can be authenticated as true or counterfeit without accessing the QR code or the information encoded.
Using the Mueller matrix, we can produce a large set of resulting Stokes parameters S 0 MS.We take into account linearly polarized input light S S 0 1; cos 2ψ; sin 2ψ; 0 with ψ ranging from 0°to 179°at intervals of 0.1°.Note that  Components S 0 , S 1 , S 2 , and S 3 and the DoP were used to produce five independent datasets.Using a hold-out strategy, the datasets were divided at random into training and test sets.A two-class K-NN method with k 1 was used to perform classification within the five groups of data.The algorithm was run 50 times.The results are presented in Table 4: in summary, K − NN provides error-free classification using the histograms of S 0 , S 2 , and S 3 or the DoP.

AUTHENTICATION OF PHARMACEUTICAL PILLS COATED WITH GOLD NANOPARTICLES
Pharmaceutical products are produced under strict safety measures following wellestablished protocols.Falsification of specifics could be a major problem for the industry, drugs agencies, and the general public.Criminal activity related with counterfeit medicines produces financial losses for companies and can become a health risk for patients who acquire medicines outside of the regulated commercial networks.Both public administrations and companies are pursuing methods for detecting counterfeiting products in a secure and fast way [89][90][91][92][93].   AuNP are very common in biomedicine: since AuNP are non-toxic and biocompatible, multiple applications have been described [94][95][96][97][98].Among many other applications, AuNP are used as drug carriers and labeling, delivery, heating, or gene-regulating agents.Also, AuNP are used in imaging diagnostics [99,100].
In this section we introduce an approach for tagging pharmaceutical tablets in such a way that falsification becomes virtually impossible.AuNP are used to produce nanoparticle-coated tablets that become polarimetrically labeled.As explained in Section 6, AuNP present dichroic behavior, and, thus, the properties of the scattered light may depend on the state of polarization of the light source.Tablets can be  analyzed by means of polarized light, and according to the polarimetric signature detected, they are classified as true or counterfeit [46].
We produced flat placebo pharmaceutical tablets according to the specifications of Table 5.The size of the tablets is 10 mm.The coating is composed of deionized water, AuNP of different sizes, and hydroxypropyl methylcellulose (HPMC), a very common excipient used in the pharmaceutical and food industries.Three commercial solutions containing AuNP with different sizes and concentrations (provided by Endor Nanotechnologies, Barcelona) were used.The nanoparticles of the solutions used were very homogenous with a dispersity index Đ close to 1.A short description on how the tablets were prepared can be found in [46].In summary, three sets of gold-coated tablets were produced containing, respectively, 4, 12, and 25 nm AuNP.In addition to these sets, we prepared a fourth group of tablets containing no nanoparticles.
The tablets of the four sets look identical and cannot be distinguished by visual inspection (see Fig. 23).A couple of pills of each group were selected at random for characterization.The Mueller matrix components were obtained using a four-PEM Mueller matrix polarimeter; this instrument is depicted in Fig. 6(a) [67].The light source was able to illuminate the sample in the range of 280-700 nm.Results for the eight measured tablets are presented in Fig. 24.
Note that some of the Mueller components in Fig. 24 are very close to zero for any wavelength.Accordingly, the tablets display a behavior that can be identified as Produced tablets used in the experiments.
isotropic because the measured Mueller matrix can be approximately described by means of Eqs. ( 18) and ( 19) (see Section 2).
Figure 25 shows the DoP as a function of the wavelength for some particular states of polarization of the incident light-that is, natural, linear (horizontal, 45°, vertical, and -45°), and circular.For instance, if the sample is illuminated with natural light S 0 1; 0; 0; 0, the DoP of the reflected beam verifies 0.6 < P < 0.7.On the other hand, when S 0 1; −1; 0; 0, P > 0.9.In any case, there is not a strong dependence of the DoP as a function of the wavelength.
Then, the wavelength average of the DoP as a function of polarization angle ψ 0°< ψ < 90° was calculated.Results are presented in Fig. 26.The error bars indicate the standard deviation.Note that the maximum distance between the curves appears for polarization angles 20°< ψ < 35°.Nevertheless, red curves (4 nm) and black curves (no nanoparticles) are very close to each other and almost overlap.
The calculation of the average of the DoP is not enough to perform a trustable classification of the sample.For this reason, we used a classification algorithm such as K − NN.Since the sample is characterized by the Mueller matrix, a dataset with an arbitrary number of states of polarization can be generated.We calculate the state of polarization of the reflected light when the polarization angle ψ ranges from 20°to 35°w ith a step size of 0.057°(0.001rad).Then, the DoP as a function of the wavelength for each angle ψ was obtained.Note that DoP curves can cross each other when λ < 350 nm (see Fig. 25).For this reason, the wavelengths are limited to λ ∈ [350,700].The system was trained using a hold-out strategy: half of the values of DoP are used to train the system, and the other half are for testing purposes.The training and test sets are generated at random.The classifier was trained 1000 times with k 1 (i.e., only the closest neighbor is taken into account).The accuracy of the classification was 1, and no errors were detected.Mueller matrix components as a function of the wavelength.The wavelength and the Mueller components range from 280 to 700 nm and [−1, 1], respectively.For the sake of clarity, the y scale of the graphs is adapted to the dynamic range for each component.Two tablets of each class were used.Black, tablets without nanoparticles; red, tablets with 4 nm AuNP; green, tablets with 12 nm AuNP; blue, tablets with 25 nm AuNP.Reprinted with permission from [46].Copyright 2016 Optical Society of America.
We performed a second classification test using K − NN (k 1).The DoP was calculated in the same way as explained above, but in this case the sets were not generated at random: the training set was produced by selecting one of the pills of each class; the data obtained from the remaining pill was used as the test set.The 16 possible combinations were taken into account.The accuracy of the classification was again 1 for all cases considered.In summary, we demonstrated that pills coated with nanoparticles (AuNP) of different sizes can be successfully classified using polarimetric techniques.This result may open a possible trend in research of anti-counterfeiting of pharmaceutical products.

AUTHENTICATION OF THREE-DIMENSIONAL PHASE-ENCODED SAMPLES
In the previous sections, we analyzed the effect of polarized light on the behavior of codes or pharmaceutical pills produced with AuNP, platinum, chromium, etc.The polarimetric signature is used to distinguish among different classes of samples, and authentication is performed using machine-learning algorithms such as SVM or K − NN.Now, the problem is slightly different because polarization does not play a key role in validation.As we discussed in Section 1, phase encoding using adhesive tape produces speckle-like distributions after propagation.In this section, we extend conventional 2D phase encoding to a more general 3D encoding [101][102][103].We show that 3D phase masks (3DPMs) of different thickness will produce different speckle signals.Consequently, 3D masks attached to codes can be used for authentication purposes [56].DoP as a function of the wavelength for several input states of polarization: S 1; 0; 0; 0, S 1; 1; 0; 0, S 1; −1; 0; 0, S 1; 0; 1; 0, S 1; 0; −1; 0, and S 1; 0; 0; 1.Two tablets of each class were used.Black, tablets without nanoparticles; red, tablets with 4 nm AuNP; green, tablets with 12 nm AuNP; blue, tablets with 25 nm AuNP.Reprinted with permission from [46].Copyright 2016 Optical Society of America.
produced by combining glass and diffuser paper.Note that we use the same QR code for the three cases considered.According to Fig. 27, we refer to the three samples as codes A, B, and C. For illustrative purposes, Figs.28(a) and 28(b) show the QR code printed on transparency film and the two-layer 3D code (code A), respectively.The size of the codes is 4 mm × 4 mm.
The optical setup for measuring the speckle signal is presented in Fig. 28(c).An expanded laser diode with wavelength λ 455 nm illuminates the sample.A linear polarizer is used as an attenuator.The sample is placed in front of a CCD camera with the following specifications (Table 6): The camera records the speckle distributions.Distance d is set at 30, 70, 110, and 150 mm.Because speckle patterns can be influenced by vibrations and other surrounding variables, 20 images were recorded for each distance d [59].Averaged DoP as a function of the polarization angle ψ.Two tablets of each class were used.Black, tablets without nanoparticles; red, tablets with 4 nm AuNP; green, tablets with 12 nm AuNP; blue, tablets with 25 nm AuNP.Reprinted with permission from [46].Copyright 2016 Optical Society of America.
Copyright 2016 Optical Society of America.
increases, the histogram tends to be narrower.Accordingly, distinctive information could be derived to perform classification.As discussed in Section 1, speckle values follow a Gamma-distribution probability [Eq.( 3)].Note that the curve that fits each gray level distribution is superimposed on the histograms.
In the present problem, instead of using the 256-bin histograms to feed the classification system, we calculated the following statistics: mean, variance, skewness,   kurtosis, and entropy [see Eq . 28].These features are used to train a random forest classification system.A total of 12 classes have been considered according to the information presented in Table 7.
In summary, the system has to determine the code and the distance the measurement was carried out.Each class is described by 20 sets of m 5 features: 10 are used to train the classifier, and the remaining 10 are used to test the system.According to the formula that estimates the minimum number of features ν required to create a random tree v ⌊ ffiffiffi ffi m p ⌋, is ν 2 (see Subsection 5.3).The number of random trees is selected by calculating the classification error as a function of the number of trees.The error decays with the number of trees used (see Fig. 30).Note that when the system is trained with 100 random binary trees, the error is less than 1%.Finally, the model is checked with the members of the test set.All test images were successfully classified.
A second experiment was taken into account.50 extra speckle patterns were recorded at arbitrary distances d, ranging from 30 to 150 mm.In particular, distances d 30, 70, 110, and 150 were explicitly excluded.The features obtained from these new images were used to test the classification system.The classifier tried to assign the images to one of the 12 classes of Table 7 using the following criteria: a sample image is assigned to one of the classes if and only if it gets 95% of the votes; otherwise, this speckle image is rejected.As a result, none of these 50 speckle images were assigned to one of the 12 possible classes.
In summary, we demonstrate authentication with 3D codes in a scattering medium: identical QR codes encoded with different 3D phase masks can be distinguished using a random forest classifier.
Classification errors as a function of the number of trees used.Reprinted with permission from [56].Copyright 2016 Optical Society of America.

CONCLUDING REMARKS
In this paper, we have reviewed recent progress on validation and authentication using optics and photonics with nanoparticle and thin-film encoding.Optically encoded data produced using nanoparticles or thin-film technologies can be verified using polarimetric speckle analysis and pattern recognition techniques.Experimental methods include Mueller matrix polarimetry and image analysis.Since measured signals for true and counterfeit samples are very similar and difficult to distinguish, statistical and machine-learning methods are required to perform successful classification.Future trends on this topic may include preparation of more complex samples with nanoparticles and simplified measuring methods that do not require sophisticated laboratory equipment.

Figure 1
Figure 1(b) shows the propagated irradiance I Q x; y; z jQx; y; zj 2 .Note that the information displayed in this figure still resembles the image of the QR [Fig.1(a)].
3(a) and 3(b).Calculated histograms and curves generated using Eq.(3) are not distinguishable.In general, random phase-encoded signals produce specklelike noise distributions after propagation within the Fresnel domain.

Figure 11
Figure11sketches the optical setup.A green laser diode (λ 532 nm) passes through a variable density filter for tuning the total intensity.A linear polarizer and a quarter wave plate are used to generate circularly polarized light.A second polarizer is used to select the direction of polarization.Then, the sample is illuminated and propagated 15 mm where a CCD camera records the light distribution.Note that the light that illuminates the code has to be polarized.Metallic nanoparticles used to produce the code show dichroism, and, thus, absorption is dependent on the direction of polarization (see, for instance,[51][52][53]).The corresponding light distributions after propagation are shown in Figs.12(a)-12(c) (see the caption for details).Note that Figs.12(a) and 12(b) look different despite the initial QRs encoding the same information.The state of polarization of the light that illuminates the sample is modified by rotating the second polarizer from 0°to 179°at intervals of 1°[Fig.11].Accordingly, 180 8-bit images have been recorded and the corresponding histograms are calculated.Note that the variable density filter is set in a way that the camera never saturates.Figures13(a)-13(c) display the histograms of the propagated distributions of samples A, B, and C, respectively, when the polarization angle is set at 0°, 30°, 60°, 90°, 120°, and 150°.For

Figure 18 Phase
Figure 18

Figure 22 Principal
Figure 22

Figures 27 (
Figures 27(a)-27(c) show how the samples were prepared.Three 3D phase masks of different thickness were attached to a printed QR code.The 3D phase masks were

Figures 29 (
Figures 29(a)-29(c) show recorded images and histograms at d 110 mm for samples of classes A, B, and C. Images are presented in false color in order to better visualize the speckle-like character of these distributions.Note that in Fig. 29(b) the illumination was not uniform due to a slight misalignment of the optical setup.The three histograms look slightly different.It seems that when the number of layers

Figure 26
Figure 26 (a) Conventional QR code printed on transparency film, (b) two-layer 3D code (class A), and (c) optical setup.Reprinted with permission from[56].Copyright 2016 Optical Society of America.

Figure 29 Recorded
Figure 29

Table 1 .
Then, Qx; y; 0 is phase-encoded by multiplying this distribution by random phase mask Rx; y; 0, i.e., Cx; y; 0 Rx; y; 0Qx; y; 0. Figure 2(b) shows the irradiance of the propagated joint distribution I C x; y; z jCx; y; zj 2 .Unlike the result presented in Fig. 1(b), I c x; y; z looks noisy and no information from the original image Qx; y; 0 can be inferred.
Moreover, the appearance of Figs.2(a) and 2(b) is very similar; that is, both are noiselike distributions.Figures3(a) and 3(b) show the histograms of I R x; y; z and I C x; y; z, respectively (blue curves).Note that both histograms display an identical exponential decay behavior.At first sight, propagated phase-encoded images can be described as indistinguishable noisy distributions.Statistical optics 11 x 12 x 13 … x 1m x 21 x 22 x 23 … x 2m x 31 x 32 x 33 … x 3m

Table 2 .
SVM Results, Three-Class Problem

Table 5 .
Pharmaceutical Tablets and Coating Components

Table 7 .
Twelve Classes Considered Class 1