Many pharmaceuticals undergo a drying stage, during which the wet solid, or “cake”, is converted into a powder consisting of particles with the requisite size distribution. These powders are subsequently encapsulated into solid dosage forms/pills.
Agitation filter dryer (AFD) is one of the most common dryers used in the industry since it integrates both filtration and drying unit operations are integrated into one equipment. However, the formation of aggregates during the drying process has become a challenging problem because of unbreakable hard lumps and product degradation due to the trapped moisture in the powder. During the drying process, the temperature and pressure (vacuum) are controlled (depending on the solvent used and the temperature stability of the solid) and the cake is agitated or intermittently agitated to prevent unwanted agglomeration. Even though the parameters of the drying processes and the overall industrial environments where these processes take place are generally well controlled, the evolution of the particle sizes during agitation is not fully predictable. Particle size distribution (PSD) in cakes may range from a few 1 micron to 1000 microns with the possibility of even larger agglomerates forming. The goal of the drying process is to maintain the initial particle size and to prevent ‘hard agglomerates’ (crystals bonded together) from forming. The presence of large hard agglomerates is undesired as they can cause content uniformity issues in drug formulation and can cause a batch to be rejected for not meeting PSD specifications. Even though the breaking of hard aggregates is possible in some cases, the delumping of the powder requires additional equipment and time, and it adds cost to the process. In the end, the delumping process also can result in loss of products. Thus, it is crucial to monitor the evolution of particle sizes quantitatively and in real time, to detect the beginnings of agglomeration as early as possible and correct through feedback control on process parameters (e.g., temperature, agitation speed).
No real-time online monitoring methods exist presently that can detect and prevent such abnormally large agglomerants early on. Imaging by a standard camera from a distance compatible with the manufacturing setting (˜0.5-1m away from the powder) does not provide sufficient spatial resolution to extract the PSD. Probe-based imaging with the fiber bundle is invasive and it only captures a very small field of view; thus, it may miss large agglomerates forming elsewhere; moreover, there is a risk of the powder obscuring the viewing field, rendering the imaging operation impossible. Instead, manufacturers commonly rely on trained personnel to visually observe the mixing-but this can be subjective. Machine vision to analyze the appearance of the cake surfaces and detect agglomerates are generally limited. Alternatively, it is possible at fixed time intervals to extract a sample from the cake and pass it through a particle size analyzer instrument. However, this method is invasive and slow and, thus, not suitable for industrial use.
Extracting quantitative information about highly scattering surfaces from an imaging system is a challenging problem because the phase of the scattered light undergoes multiple folds upon propagation, resulting in complex speckle patterns.
The speckle is an encoding of spatially variant patterns on the cake surface, which include particle locations and sizes. However, no explicit relationships exist to allow easy inverse mapping from the speckle to the statistical properties of the spatially variant particle features.
BRIEF SUMMARY
In an example embodiment, the present invention is a method of monitoring a particle size distribution (PSD). The method comprises illuminating a plurality of particles having a particle size distribution (PSD) by an at least partially coherent beam to produce scattered light; capturing the scattered light by a pixelated photoelectric detector, thereby creating an ensemble of raw speckled images; based on the ensemble of the raw speckled images, computing an ensemble-averaged intensity correlation; providing the ensemble-averaged intensity correlation to an inverse module, the inverse module being configured to determine the PSD based on the ensemble-averaged intensity correlation; and obtaining from the inverse module the PSD.
In another example embodiment, the present invention is a device for monitoring a particle size distribution (PSD). The device comprises an illumination module adapted to illuminate a plurality of particles having a particle size distribution (PSD) by an at least partially coherent beam to produce scattered light; a pixelated photoelectric detector configured to capture the scattered light and create an ensemble of raw speckled images; a correlation-computing module configured to compute an ensemble-averaged intensity correlation based on the ensemble of the raw speckled images; and an inverse module configured to determine the PSD based on the ensemble-averaged intensity correlation.
In yet another example embodiment, the present invention is a computer program product for monitoring a particle size distribution (PSD). The computer program product comprises a computer readable storage medium having program instructions embodied therewith, the program instructions executable by a processor to cause the processor to perform a method comprising: illuminating a plurality of particles having a particle size distribution (PSD) by an at least partially coherent beam to produce scattered light; capturing the scattered light by a pixelated photoelectric detector, thereby creating an ensemble of raw speckled images; based on the ensemble of the raw speckled images, computing an ensemble-averaged intensity correlation; providing the ensemble-averaged intensity correlation to an inverse module, the inverse module being configured to determine the PSD based on the ensemble-averaged intensity correlation; and obtaining from the inverse module the PSD.
The following Detailed Description references the accompanying drawings which form a part this application, and which show, by way of illustration, specific example implementations. Other implementations may be made without departing from the scope of the disclosure.
The present disclosure provides methods, systems, and computer program products for monitoring a particle size distribution (PSD). Various embodiments of the present disclosure are discussed in detail below. While specific implementations are discussed, it should be understood that this is done for illustration purposes only. A person skilled in the relevant art will recognize that other components and configurations may be used without parting from the spirit and scope of the disclosure.
In a first example embodiment, a method of monitoring a particle size distribution (PSD) is provided. The first example embodiment is illustrated in
As used herein, “a partially coherent beam” refers to a beam that includes multiple coherent modes in space and/or time, wherein the modes can be mutually incoherent.
As used herein, a “pixelated” detector includes a class of digital or analog detector where the information is returned in the form of samples called “pixels.” The samples may be based on a sampling scheme, which may include uniform sampling, non-uniform sampling, masked sampling, and/or random sampling. For uniform sampling, the spacing between pixel locations is substantially constant. For non-uniform sampling, the spacing between pixel locations varies according to a predetermined formula, e.g. quadratic (also known as “linear chirp”) and/or other schemes. Masked sampling is a technique where samples are skipped (or blocked) according to a predetermined scheme, e.g. every second or every third pixel. Masked sampling can be physically implemented by superimposing an absorbing mask over the detector or at an image-conjugate plane. Random sampling is a technique where the pixel locations are determined according to a random number generator obeying a pre-chosen probability distribution. Although the examples and figures described herein are based on a uniform sampling scheme, what is presented herein may be implemented according to any the other sampling schemes, such as the sampling schemes described herein, without departing from the scope and spirit of what is described herein.
In various aspects of the first example embodiment, the ensemble-averaged intensity correlation may be an ensemble-averaged intensity autocorrelation. In further aspects, the inverse module may include a machine learning module, a gradient descent module, a non-linear solver, a curve-fitting module, or a differential evolution algorithm module. In an example embodiment, the inverse module may be configured to generate the PSD. In another example embodiment, the inverse module may be configured to generate a cumulative distribution function (CDF) and to differentiate the CDF to generate the PSD.
In various aspects of the first example embodiment, the inverse module may include the machine learning module. In further aspects, the machine module may include a neural network. Providing the ensemble-averaged intensity correlation to the inverse module may include providing the ensemble-averaged intensity correlation to the neural network configured to determine the PSD. In some aspects, the neural network may be a convolutional neural network. The convolutional neural network may include, for example, at least one skip connection. In other example embodiments, the convolutional neural network may include a plurality of stages. Each stage of the plurality of stages may include at least one skip connection and at least one batch-normalization and activation layer. In some aspects, the convolutional neural network may include a linear layer.
In various aspects of the first example embodiment, the plurality of particles may be a dry powder. The dry powder may be ground. Grinding the plurality of particles may be discontinued when the PSD shows agglomeration. The plurality of particles may be a wet powder. The wet powder may be agitated. Agitating the plurality of particles may be discontinued when the PSD shows agglomeration.
As used herein, “differentiating the CDF” to produce a PSD involves taking the first derivative of the CDF to produce the PSD.
In a second example embodiment, a device is provided for monitoring a particle size distribution (PSD). The second example embodiment is illustrated in
In various aspects of the second example embodiment, the correlation-computing module may be configured to compute an ensemble-averaged intensity autocorrelation. In certain aspects, the inverse module may include a machine learning module, a gradient descent module, a non-linear solver, a curve-fitting module, or a differential evolution algorithm module. In further aspects, the inverse module may be configured to generate the PSD. In other aspects, the inverse module may be configured to generate a cumulative distribution function (CDF) and to differentiate the CDF to generate the PSD.
In certain aspects of the second example embodiment, the inverse module may include the machine learning module. For example, the machine learning module may include a neural network. In some aspects, the neural network may be configured to determine the PSD. The neural network may be a convolutional neural network. In further aspects, the convolutional neural network may include at least one skip connection. In yet further aspects, the convolutional neural network may include a plurality of stages. Each stage of the plurality of stages may include at least one skip connection and at least one batch-normalization and activation layer. In additional aspects, the convolutional neural network may include a linear layer.
In certain aspects of the second example embodiment, the device may include a grinder adapted to grind the plurality of particles. The grinder may be adapted to transmit the at least partially coherent beam for illumination of the plurality of particles. In additional aspects, the device may include an agitator adapted to agitate the plurality of particles. The agitator may be adapted to transmit the at least partially coherent beam for illumination of the plurality of particles.
In a third example embodiment, a computer program product for monitoring a particle size distribution (PSD) is provided. The computer program product includes a computer readable storage medium having program instructions embodied therewith. The program instructions are executable by a processor to cause the processor to perform a method defined herein above with respect to the first example embodiments and various aspects thereof. For example, a plurality of particles having a particle size distribution (PSD) is illuminated by an at least partially coherent beam to produce scattered light. The scattered light is captured by a pixelated photoelectric detector, thereby creating an ensemble of raw speckled images. Based on the ensemble of the raw speckled images, an ensemble-averaged intensity correlation is computed. The ensemble-averaged intensity correlation is provided to an inverse module. The inverse module is configured to determine the PSD based on the ensemble-averaged intensity correlation. The PSD is obtained from the inverse module.
In various embodiments, the devices and methods disclosed herein employ machine learning (ML)-assisted speckle analysis to quantitative real-time monitoring of mixing and drying processes. In various embodiments, one application of such processes may be the drying of powder suspensions and wet powders (“cakes”) in the pharmaceutical industry, where quantifying the PSD is of particular interest. The system and method disclosed herein may not, however, be limited to such processes; other possible applications may include other pharmaceutical processes such as milling, and powder blending, rheological measurements in complex fluids or emulsions, characterization, and characterization of cell growth processes for personalized medicine, and/or other applications.
The speckle is an encoding of spatially variant patterns on the cake surface, which can include particle locations and sizes. However, conventionally no explicit relationships exist to allow an easy inverse mapping from the speckle to the statistical properties of the spatially variant particle features. Disclosed herein are (i) methods and systems that produce a speckle from light scattered by a powder in a chemical drying process that carries sufficient information to extract the PSD in the cake; and (ii) methods and systems for processing the speckle registered on a pixelated photoelectric detector, such as a CCD or CMOS camera, to obtain the PSD quantitatively.
In various embodiments, methods are provided. In the disclosed methods, a cake may be illuminated by a laser beam. The light scattered from the cake surface may be captured by a pixelated photoelectric detector, such as a CCD or CMOS camera. These captured image data may be referred to as the “raw speckle image.” The raw speckle image may be processed. Using the optical propagation equation, the relationship between the PSD and the ensemble-averaged autocorrelation of captured raw speckle images may be established. The camera may capture the raw speckle images, and may transmit them to a dedicated digital computer, in-house or in the cloud, which may perform the computations necessary to obtain the PSD quantitatively. The methods disclosed herein may be used both for training a machine learning (ML) algorithm, which may be equivalently referred to as an ML model/module, and/or for regular operation in the industrial or laboratory environment. The digital computer may first compute the ensemble-averaged intensity autocorrelation function of the raw speckle image and forward it to an ML model/module. If the ML model/module, such as a DNN, is in training mode, ground truth data may be used to specify the weights in the ML model/module such that it outputs the correct PSD matching the ground truth training data. After training, the ML model/module, such as a DNN, may be capable of receiving the ensemble-averaged autocorrelations as input and producing the correct PSD as output, reliably and with fast computation. Such methods are capable of increased interpretability, compared to conventional “black box” deep learning approaches, since the input to the ML model/module, such as a DNN, may be shaped by physical law, namely the ensemble-averaged autocorrelation of the raw speckle images. Such methods, as presented herein, have been validated using a commercial particle size analyzer to independently establish the PSD and match it to the results of the disclosed ML approach. Additional details about the disclosed methods and possible embodiments are included in section 3, below.
In various embodiments, systems are provided. In the disclosed systems, in various embodiments, the laser and CCD camera (or alternative photoelectric detector) are mounted on a frame such that the light scattered from the cake is effectively captured. The “common path” configuration, described in detail in section 3 below, may be preferred because it maximizes the light capturing efficiency. The camera may capture the raw speckle images, and may transmit them to a dedicated digital computer, in-house or in the cloud, which may performs the methods and/or computations described herein, such as in the previous paragraph. The disclosed system may be used both for training the ML algorithm and/or for regular operation in the industrial or laboratory environment. The digital computer may first compute the ensemble-averaged intensity autocorrelation function of the raw speckle image and may forward it to the ML model/module. If the ML model/module, such as a DNN, is in training mode, ground truth data may be used to specify the weights in the ML model/module such that it outputs the correct PSD matching the ground truth training data. After training, the ML model/module, such as a DNN, may be capable of receiving the ensemble-averaged autocorrelations as input and producing the correct PSD as output, reliably and with fast computation. Such systems are capable of increased interpretability, compared to conventional “black box” deep learning approaches since the input to the ML model/module, such as a DNN, may be shaped by physical law, namely the ensemble-averaged autocorrelation of the raw speckle images. Such systems, as presented herein, have been validated with a laboratory prototype operating under conditions closely emulating the conditions encountered in industry, using a commercial particle size analyzer to independently establish the PSD and match it to the results of the disclosed ML approach. Additional details about the disclosed methods and possible preferred embodiments are included in section 3, below. One purpose of the methods and systems, as disclosed herein, may be to overcome limitations of conventional techniques by providing a real-time, minimally invasive, and easily deployable in the industrial setting instrument to monitor particle size distributions quantitatively. The methods and systems disclosed herein are based on physics and algorithms, so they do not involve human intervention and/or human subjectivity. The algorithms operate faster and more efficiently as compared to conventional techniques because the algorithms include a component based on ML, which after training may not be computationally demanding. The ML algorithm may perform computations during the training phase, which may occur during the design of the algorithm, and which may insignificantly or not impact the user.
The methods and systems, as described herein, use one or more images of a speckle, which may be an encoding of the particle sizes. A speckle image results from propagation of a wavefront whose phase has been strongly modulated by spatially variant features across a surface and/or a volume. According to the Huygens principle, the influence of each feature upon its neighbors expands as the propagation distance increases. Interference from light scattered by each feature with its neighbors results in a “salt and pepper” appearance of a speckle at any observation screen downstream, such as the viewer's retina or a digital camera. For a fixed rough surface, a speckle may technically be deterministic, because it may entirely be determined by the surface morphology. However, because the surface morphology is seldom known and/or very difficult to determine, and it is difficult to compute the scattered light exactly, traditional analysis of a speckle treats the surface morphology as a random process in the spatial domain. As long as the morphology statistics are invariant, it may be straightforward to relate statistical moments of the surface to the statistical moments of the speckle itself.
The speckle has long been used to characterize surface roughness, but the conventional techniques to characterize surface roughness only work when the surface height fluctuation, or equivalently, typical particle size, is smaller or comparable than the light wavelength. This limits its application to surfaces encountered in many industrial processes, such as pharmaceuticals manufacturing. Electronic Speckle Pattern Interferometry (ESPI) is another conventional technique that can measure the surface motion distribution even at nanometer scales. However, ESPI requires a reference beam, and it is disadvantageously is a two-step measurement that cannot detect absolute height. Another conventional technique, laser speckle contrast imaging, is qualitative and does not yield quantitative results on the surface roughness. Yet another conventional technique, interferometric particle imaging, can measure the particle size and shape, but only works for a single particle or sparse distributed particles.
The methods and systems disclosed herein treat the speckle as a nonlinear superposition of roughness whose statistics vary with position on the surface. The map includes the complexities of mixing-drying dynamics and speckle formation, as noted earlier; thus, it is may not always be practical to invert it directly. Instead, an inverse module, which may include a machine learning (ML) algorithm assisted by the physics of speckle formation, may be applied.
In recent years, ML algorithms have been used in imaging through scattering media and in speckle suppression. Speckle surface analysis, which focuses on the scattering media itself may be assisted by an ML algorithm, such by performing classifications to distinguish different materials. Due to the surface randomness and the phase sensitivity, it may be difficult for an ML algorithm, such as a neural network to capture key features. In the methods and systems disclosed herein, the speckle image may be preprocessed according to the beam propagation equations to enhance the essential features to reduce the burden on the ML module and/or model. From the theory and equations, such as equation (24), disclosed herein, the ensemble averaged spatial-integral autocorrelation function can be used to build a forward model from the statistics of powder surface, the PSD, to the processed image. In various embodiments, the ML model/module, which may use a DNN that is a deep convolution neural network, may be used to learn the inverse mapping from the averaged autocorrelation image to the PSD. The forward model is made more interpretive using physics, as seen below. The forward model may allow for the bounds of the prediction ability to be known. In some cases, this may be a difficult to determine using black-box algorithms, such as those using ML models.
Drying cakes made up of multiple particles of varying sizes, such as powders, typically include surfaces that are rough, which may be due to the presence of powders. As a cake is agitated and gradually dries, the particles tend to agglomerate to a wide range of sizes, from a few micrometers to a few millimeters. These particles are distributed all across the cake surface and move rapidly due to the agitation. As a result, the surface roughness statistics are also strongly variant across the cake surface and with time. The process of speckle formation described herein may be used to encode the complex dynamics of mixing and drying of the cake and/or particles.
Although the descriptions herein may refer to ML models/modules performing various operations, in various embodiments, these models/modules may be included within an inverse module, which may alternatively or additionally include a gradient descent module, a non-linear solver, a curve-fitting module, or a differential evolution algorithm module as components. In various embodiments, the operations, functionality, and/or configurations of the ML models/modules described herein can be equivalently attributed to the inverse module and/or any one or more components within the inverse module without departing from the scope and spirit of what is described herein.
One or more computers can be used to implement such a computational pipeline, using one or more general-purpose computers, such as client devices including mobile devices and client computers, one or more server computers, or one or more database computers, or combinations of any two or more of these, which can be programmed to implement the functionality such as described in the example implementations.
Examples of such general-purpose computers include, but are not limited to, larger computer systems such as server computers, database computers, desktop computers, laptop, and notebook computers, as well as mobile or handheld computing devices, such as a tablet computer, handheld computer, smart phone, media player, personal data assistant, audio and/or video recorder, or wearable computing device.
With reference to
A computer storage medium is any medium in which data can be stored in and retrieved from addressable physical storage locations by the computer. Computer storage media includes volatile and nonvolatile memory devices, and removable and non-removable storage devices. Memory 504, removable storage 508 and non-removable storage 510 are all examples of computer storage media. Some examples of computer storage media are RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optically or magneto-optically recorded storage device, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices. Computer storage media and communication media are mutually exclusive categories of media.
The computer 500 may also include communications connection(s) 512 that allow the computer to communicate with other devices over a communication medium. Communication media typically transmit computer program code, data structures, program modules or other data over a wired or wireless substance by propagating a modulated data signal such as a carrier wave or other transport mechanism over the substance. The term “modulated data signal” means a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal, thereby changing the configuration or state of the receiving device of the signal. By way of example, and not limitation, communication media includes wired media such as a wired network or direct-wired connection, and wireless media include any non-wired communication media that allows propagation of signals, such as acoustic, electromagnetic, electrical, optical, infrared, radio frequency and other signals. Communications connections 512 are devices, such as a network interface or radio transmitter, that interface with the communication media to transmit data over and receive data from signals propagated through communication media.
The communications connections can include one or more radio transmitters for telephonic communications over cellular telephone networks, and/or a wireless communication interface for wireless connection to a computer network. For example, a cellular connection, a Wi-Fi connection, a Bluetooth connection, and other connections may be present in the computer. Such connections support communication with other devices, such as to support voice or data communications.
The computer 500 may have various input device(s) 514 such as various pointer devices (whether single pointer or multi-pointer), such as a mouse, tablet and pen, touchpad and other touch-based input devices, stylus, image input devices, such as still and motion cameras, audio input devices, such as a microphone. The compute may have various output device(s) 516 such as a display, speakers, printers, and so on, also may be included. These devices are well known in the art and need not be discussed at length here.
The various storage 510, communication connections 512, output devices 516 and input devices 514 can be integrated within a housing of the computer or can be connected through various input/output interface devices on the computer, in which case the reference numbers 510, 512, 514 and 516 can indicate either the interface for connection to a device or the device itself as the case may be.
An operating system of the computer typically includes computer programs, commonly called drivers, which manage access to the various storage 510, communication connections 512, output devices 516 and input devices 514. Such access generally includes managing inputs from and outputs to these devices. In the case of communication connections, the operating system also may include one or more computer programs for implementing communication protocols used to communicate information between computers and devices through the communication connections 512.
Any of the foregoing aspects may be embodied as a computer system, as any individual component of such a computer system, as a process performed by such a computer system or any individual component of such a computer system, or as an article of manufacture including computer storage in which computer program code is stored and which, when processed by the processing system(s) of one or more computers, configures the processing system(s) of the one or more computers to provide such a computer system or individual component of such a computer system.
Each component (which also may be called a “module” or “engine” or “computational model” or the like), of a computer system such as described herein, and which operates on one or more computers, can be implemented as computer program code processed by the processing system(s) of one or more computers. Computer program code includes computer-executable instructions and/or computer-interpreted instructions, such as program modules, which instructions are processed by a processing system of a computer. Generally, such instructions define routines, programs, objects, components, data structures, and so on, that, when processed by a processing system, instruct the processing system to perform operations on data or configure the processor or computer to implement various components or data structures in computer storage. A data structure is defined in a computer program and specifies how data is organized in computer storage, such as in a memory device or a storage device, so that the data can accessed, manipulated, and stored by a processing system of a computer.
To find the relationship between the speckle pattern and the PSD, a simplified one-dimensional (1D) analytical model was built.
The particle radius ri is interpreted as a random variable distributed according to the PSD p(r). The particle location xi is a random variable uniformly distributed across the object plane. H(x) describes the surface height resulting from the randomly placed particles. The corresponding phase of the scattered light is
From optical propagation theory, the electric field E(x) at the imaging plane is
and f3 is the focal length of lens L3. The intensity collected by the CCD camera is
We now define the spatial-integral autocorrelation of the speckle image as
Substituting equations (2) and (3) into equation (4),
After integrating over x,
The delta function is non-zero only when ξ2=ξ1+η1-η2. Then the equation can be further simplified as,
From Equation (7), we do the variable substitution with η=η2, σ=η1−η2 and τ=ξ1−η2.
After substituting S with equation (2),
In the internal integral over σ, the phase term
varies much faster than a(σ) and ejuσ, so the rotating wave approximation can be applied to move w(σ)w*(σ+τ) out of the integral over σ.
Where W(τ)=(w(σ)w*(σ+τ))σ, is the spatial average of w(σ)w*(σ+τ) over σ. W(τ) describes spatial correlation of the surface phase, W(τ) will drop to 0 if τ is out of the correlation length. For the ideal rough surface, the correlation length is infinitely small, so that W(τ) degrades into δ(t). a(x) is the mask of particles, and a(x) can be defined using the following formulas.
Because the particles cannot overlap, and the correlation length is well smaller than the particle size, it can approximately be assumed that
After substituting equation (14) to equation (11),
The internal integral is the Fourier transform,
If the assumption that W(t)=δ(τ) is applied, equation (17) can be transferred into,
i is the index of the i-th particle, and ri and xi are the radius and the position for the i-th particle, respectively. If there are enough particles in the field of view, equation (18) can be reformulated as
The term
results from the granular feature in
The third equal sign comes from the fact that r and x are independent. The radius ri follows the probability distribution p(r) which is invariant. The xi is randomly distributed in the space, and it is ergodic, so the ensemble average over x is equal to the spatial average,
D is the laser spot diameter
term determines the average speckle size, which is consistent with the traditional results from a textbook.
If equations (21) and (22) are substituted back to equation (20),
The particle size of a sample may vary from ˜50 μm to ˜1000 μm, which is much larger than the wavelength 532 nm. Thus, the criterion λ<<H(x) is met, implying that the phase correlation length is much smaller than the particle size. So, the assumption W(τ)=δ(τ) can be adopted safely. Then,
Here, ·
denotes the ensemble average and D is the beam spot diameter.
Equation (24) is an intuitive yet approximate forward relationship between the raw speckle images and the PSD p(r) through the speckle autocorrelation function A(u). The goal is to invert this relationship, or rather eq. (24) and obtain p(r) from A(u), which is described below. To gain some insight into this inverse problem, in
term decays faster vs. u as the radius r grows. The intensity of the lobes is modulated by the
term of Equation (24). The intensity of the first-order lobe is higher on the upper panel than on the lower panel. Moreover, the second-order lobe can be seen in the upper panel, but not in the lower panel.
To perform an inverse of Equation (24), inverse techniques may be used by an inverse module, which may also include one or more modules to perform one or more aspects of the inverse techniques. The inverse module may include a machine learning module that utilizes a machine learning model, a gradient descent module that uses a gradient descent technique, a non-linear solver, a curve-fitting module that performs a curve-fitting technique, or a differential evolution algorithm module that performs a differential evolution algorithm. Parameterization of the particle size distribution can help reduce the number of parameters to be determined, and accelerate the convergence of the technique used to determine the inverse. Parameterization of the particle size distribution may require prior knowledge regarding the distribution. For example, one way to parameterize distributions is as a “sum of humps,” such as a sum of Gaussians, or a sum of Lorentzians, and/or the like. For these ways to parameterize distributions may include parameters such as the amplitudes, widths, and locations of the hump functions. In various examples and figures described herein, an inverse module, which may include an ML model/module, takes one or more images as input and outputs cumulative distribution functions (CDFs) and/or particle size distributions (PSDs). In various examples and figures described herein, an ML model/module takes one or more images as input and outputs cumulative distribution functions (CDFs) and/or particle size distributions (PSDs). As such, for this machine learning model/module, the number of parameters equals the number of bins in the distribution.
The forward relationship in Equation (24) is highly nonlinear; moreover, some information for the PSD may be in the sidelobes and, hence, the solution may be disproportionately susceptible to detection noise and other disturbances. A wide variety of techniques exist for dealing with such inversions, such as those performed by an inverse module, including gradient methods, maximum likelihood, and others. As a general strategy to stabilize the solution against disturbances, some form of regularization may be employed. In typical situations, the most effective regularizing priors may be derived from sparsity arguments, which may also be referred to as compressed sensing. That is, the unknown function may first expressed in a set of basis functions where it becomes sparse, meaning that very few coefficients are non-zero. For linear inverse problems, sparse formulations lead to convex functionals that may be numerically dealt with using methods such as TwIST and ADMM.
From Equation (24), some information may be in the sidelobes of the PSD, namely when the agglomerate sizes become very large. This suggests that sparsifying the PSD as, for example, a superposition of radial basis functions, may be risky. Instead, an inverse module approach, such as one using machine learning, using data from the experiment and the independent particle size analyzer may be used to determine and/or learn the regularizing prior. The algorithm(s) used by the inverse module may not be purely data-driven—it may take the physical forward model of Equation (24) explicitly into account as disclosed below. In various embodiments, the inverse module may use a machine learning algorithm/model/module, such as a deep neural network (DNN) which can be a deep convolutional neural network, to determine the regularizing prior. In various embodiments, the use of a physical forward model such as the one in Equation (24) as a preprocessor to an inverse module, such one using a neural network to perform the inverse may be described as an Approximant. In such embodiments, the learning approach simplifies extending the simple 1D analysis to two dimensions (2D), taking into account that particles may overlap along the longitudinal direction, and rectifying the deviation induced by the finite spatial integral and finite frames average in the ensemble autocorrelation calculation.
An example inverse module implementation of a neural network module using a deep learning architecture is shown in
There are two reasons the cumulative distribution function may be used herein rather than the PSD directly as the output of the neural network. First, the cumulative distribution is monotonic from 0 to 1, which discourages overfitting the fluctuations that would inevitably appear in the PSD. Second, the cumulative distribution may easily be derived from Equation (24) as
where rect(x)=1 when
otherwise=0 is the boxcar function. The physical meaning of the right-hand term in (25) is shown in
To train the neural network shown in
To evaluate the generalization ability of the DNN model, shown in
Although deep neural network and deep convolutional neural network ML algorithms/models/modules are referenced herein, it should be understood that any ML algorithm/model/module may be used without departing from the scope and spirit of what is described herein. In particular, in various embodiments, the ML algorithms/models/modules may include a feedforward neural network, a radial basis function network, a self-organizing map, learning vector quantization, a recurrent neural network, a Hopfield network, a Boltzmann machine, an echo state network, long short term memory, a bi-directional recurrent neural network, a hierarchical recurrent neural network, a stochastic neural network, a modular neural network, an associative neural network, a deep neural network, a deep belief network, a convolutional neural network, a convolutional deep belief network, a large memory storage and retrieval neural network, a deep Boltzmann machine, a deep stacking network, a tensor deep stacking network, a spike and slab restricted Boltzmann machine, a compound hierarchical-deep model, a deep coding network, a multilayer kernel machine, a deep Q-network, and/or the like. The ML algorithms/models/modules described herein may additionally or alternatively comprise weak learning models, linear discriminant algorithms, logistic regression, and the like. The ML algorithms/models/modules described herein may include supervised learning algorithms, unsupervised learning algorithms, reinforcement learning algorithms, and/or a hybrid of these algorithms.
The experiments described below with reference to the figures were conducted using devices and methods described herein.
Two samples of KCl powder were tested using the devices and methods described herein: (i) a sample powder having particle size of about 180-250 μm and (ii) a sample powder having particle size of about 425-500 μm.
In various embodiments, an air filter dryer (AFD) typical of those used in the pharmaceutical industry may be used to dry a wet solid, such as a cake. The wet solid may be sealed in the dryer, and it may be monitored non-invasively with a laser beam delivered through a glass window. An agitator placed along the axis of the container may impel the sample at a rotation speed of 5 rpm.
A laser beam operating at a particular wavelength, such as at a 532 nm wavelength, may illuminate the wet solid. In various embodiments, the laser model may need a sufficient output power and temperature rising rate, which may be low enough to not disturb the drying process for the duration of observation. In addition, the coherence length of the laser and the corresponding temporal bandwidth may be set to produce a sufficient temporal coherence and thus to produce sharp speckles. Additionally, the angle of incidence on the surface may be chosen so as to avoid specular back reflection from the window onto the camera. For example, the laser model may be an Excelsior 532 Single Mode with 300 mW output power, inducing a 1.2 mK/s temperature rising rate, which may be low enough to not disturb the drying process for the duration of observation. As another example, the coherence length of the laser may be 25 m, which corresponds to a temporal bandwidth of 0.01 pm. This may result in sufficient temporal coherence to produce sharp speckles. The angle of incidence on the surface is chosen to be approximately 10 degrees, so as to avoid specular back reflection from the window onto the camera.
An example optical train is shown in
Lens L3 may concentrate the scattered light so that an angular range that is as large as possible can be captured by the CCD.
The data processing strategy for the raw speckle images collected by the apparatus of
Some examples of test set results are shown in
approaches 0 as r approaches 0, making it harder to distinguish at small radii.
As described herein, methods and systems have been provided that decode the size information in the speckle pattern quantitatively with the help of a physics-encoded inverse module, such as one using a deep neural network (DNN). The physical forward operator takes the form of the ensemble-averaged autocorrelation and it serves as Approximant input to the inverse module, such as a neural network. Such systems and/or methods method may be used to facilitate speckle processing in other applications of diffuse light imaging.
It should be understood that the subject matter defined in the appended claims is not necessarily limited to the specific implementations described above. The specific implementations described above are disclosed as examples only.
This application claims the benefit of U.S. Provisional Application No. 63/318,880 filed Mar. 11, 2022, which is hereby incorporated by reference in its entirety.
Filing Document | Filing Date | Country | Kind |
---|---|---|---|
PCT/US2022/050045 | 11/16/2022 | WO |
Number | Date | Country | |
---|---|---|---|
63318880 | Mar 2022 | US |