This invention relates to a method and apparatus for preparing a coffee drink and particularly to replicating a coffee drink of a desired flavor
Coffee drinks are popular beverages in many parts of the world and have been consumed for thousands of years. A “coffee drink,” as used herein, refers to a liquid that is prepared using coffee beans and water, often for consumption as a beverage.
Numerous methods and devices exist for placing the ground coffee beans in contact with hot water (S12) and separating the remaining solid grounds (S13) to produce coffee drinks. Devices include, for example, percolators, vacuum pots, the French press, drip coffee makers, and espresso makers.
In the AeroPress® device 22 of
To make an espresso drink, as shown in
In the process for preparing coffee, as shown in
Additionally, because coffee beans are grown and roasted under a wide variety of conditions, each batch of roasted coffee beans may require different conditions to prepare high quality, flavorful coffee. Properly adjusting the fineness of the coffee grinds and other brewing parameters to achieve the desired flavor generally requires trial and error, and often a trained “barista.” A way to predictably reproduce the desired coffee flavor without spending significant time and effort is desired.
In one aspect, the inventive concept pertains to a computer-implemented method for controlling brewing of a coffee drink. The method entails providing instructions to receive an identifier for the coffee drink, the identifier associated with a set of brew parameters used to make the coffee drink; and retrieving the set of brew parameters from a database storing the set of brew parameters using the identifier.
In another aspect, the inventive concept pertains to a computer-readable medium storing instructions for controlling brewing of a coffee drink, the instructions including instructions to receive an identifier for the coffee drink, the identifier associated with a set of brew parameters used to make the coffee drink and including a grind size parameter; and to retrieve the set of brew parameters from a database storing the set of brew parameters using the identifier.
In yet another aspect of the inventive concept, the inventive concept pertains to a coffee grinder. The coffee grinder includes a grinder for grinding coffee beans; a grind control unit for controlling a ground size of the ground coffee beans; and a computer processor connected to the grind control unit.
Preparation of coffee drinks has remained largely an art in the sense that efficient and accurate reproduction of a desired flavor is extremely difficult. Generally, it requires a trained person (e.g., a barista) with a refined palate to taste a coffee drink and make another cup that has the same flavor, especially with a different batch of beans. While many people make and drink coffee drinks as part of their daily routine without much thought or complaint, even an average coffee drinker would admit that some coffee drinks taste better than others. The inventive concept disclosed herein allows a coffee drinker to reproduce the flavor of a model coffee drink in the convenience of his own home, without having to travel to a remote coffee shop or spending a lot of time. This reproducibility is achieved by characterizing a model coffee drink with a set of brew parameters and making those parameters available to people who make coffee.
In one aspect, a method for preparing a coffee drink having consistent taste characteristics includes characterizing a coffee drink with a set of brew parameters that are specific to batch of roasted coffee beans. As will be disclosed in more detail below, brew parameters include grind size, ratio of water to grounds, packing pressure (for espresso coffee drinks), water temperature, dwell time, extraction pressure, and agitation time, among others. A small portion of the roasted batch of beans may be used for the characterization to determine the brew parameters, with the assumption that the same characterization applies to the “batch” of beans that came from the same source (e.g., field) and were roasted together. The brew parameters may hence be used with portions of the batch that were not directly characterized.
In another aspect, brew parameters for a batch of coffee beans may be stored in a database and associated with an identifier assigned to the particular batch of coffee beans. A user can use the identifier to access the brew parameters that are stored. For example, the user may enter the identifier affixed to a bag of coffee into a website to retrieve the brew parameters. Alternatively, the user may use a portable device (e.g., a smartphone) to read the identifier that is encoded on the bag of beans and obtain brew parameters.
In some embodiments, the brew parameters are encoded directly onto a bag of beans so that a user can read the code to obtain the brew parameters. In other embodiments, a software would receive input from the user regarding the type of coffee maker he is using or the type of drink he wants to make (e.g., a cappuccino) and provide brew conditions that are obtained using the brew parameters. “Brew conditions” include parameters other than brew parameters that affect the brewing, such as settings for particular grinders in the market and recipes that may be used to make the desired drink. The “brew conditions” are determined based on the brew parameters that characterize the beans.
The specific taste characteristic of brewed coffee is the result of the total dissolved solids (“TDS”) that dissolve from the ground roasted coffee beans into the water. Another component that affects a coffee drink's taste characteristic is the type and quantity of compounds extracted that manifest as TDS. For example, hotter water and longer steep times will tend to release more chlorogenic acid lactones and phenylindanes, while cooler water and shorter steep times will tend to release more acidic compounds. The taste is affected by the quantity of TDS compared to the amount of water (the “TDS ratio”) and also the chemical composition of the TDS. The TDS ratio and chemical composition are influenced by the characteristics of the roasted coffee beans themselves and the process used for preparing coffee, i.e., the process used for extracting the TDS. The characteristics of the roasted coffee beans can be influenced by factors such as the growing conditions and roasting conditions that can be difficult to reproducibly control. However, for a given batch of roasted coffee beans, if the process for preparing the coffee which is an aqueous extraction process for the TDS—can be reproduced, the same taste characteristics for coffee resulting from the batch of roasted coffee beans can be reproduced.
Once the “model coffee drink” is selected, its properties are translated to brew parameters (S34) that will allow consumers to replicate its flavor. The aqueous extraction process for extracting the TDS from roasted coffee beans into a coffee drink is controlled by a set of brew parameters, which include the following:
The brew parameters listed above are not intended to be exhaustive, and the inventive concept disclosed herein is not limited to any particular combination of brew parameters. For example, in some embodiments, the following parameters may be included in the brew parameters:
Each brewing method requires managing the set of parameters that are relevant to the particular device. For example, for espresso extractions, the primary parameters necessary to control are the grind, pack ratio, temperature, dwell and pressure. These brew parameters result in the extraction time and volume, and the resulting quantity and chemical composition of the TDS. This relationship can be expressed as the following relationship (1):
{grind,pack,ratio,temp,dwell,pressure}{extraction time,volume}=infusion compound (1).
For an AeroPress® extraction, the primary parameters necessary to control are the grind, ratio, temperature, agitation, dwell, and pressure. These primary parameters result in the extraction time, and the resulting quantity and chemical composition of the TDS. With an Aeropress® as compared to espresso, there is no “puck” so the pack is not relevant. However, the agitation time is important. For an Aeropress®, the relationship can be expressed as follows:
{grind,ratio,temp,agitation,dwell,pressure}{extraction time}=infusion compound (2).
For a French press, the primary variables are grind, ratio, temperature, agitation and dwell. With a French press no pressure is applied and also there is no “puck,” so no pack pressure is required, and the relationship can be expressed as follows:
{grind,ratio,temp,agitation,dwell}=infusion compound (3).
Traditionally, coffee brewers have treated the brew variables as an informally managed group. This is largely because the grind, i.e. particle size distribution of the ground roasted coffee beans, has been impractical to measure, and thus needed to be determined via trial and error and validated each time by tasting the result. At the same time, the grind has a high impact on the resulting taste characteristics of the coffee. Thus, of all the variable parameters in the brewing process, the ability to accurately measure and control the grind has a great effect on the quality of the coffee.
With the ability to measure grind accurately, it becomes possible to treat grind as the main primary parameter, and fix the remainder of the primary parameters based on the grind to produce predictable results in the secondary group and consequently create consistent characteristics from the extraction. A novel method for determining particle size distribution for ground roasted coffee beans is described below with respect to
An example of encoded brew parameters may look like the following:
Batch ID: 875QQT-CR
Method: AeroPress
Coffee Weight: 35 g
Water Weight: 200 g
Water Temp.: 175f
Grind: 80% @203 μm, SD<1.4
Pre-Infusion Time: 4 sec
Agitation Time: 10 sec
Dwell Time: 5 sec
Extraction Time: 20 sec Extraction Volume: 185 g
These brew parameters, which are for use with an Aeropress® device, includes the settings a user can use to produce a coffee drink that tastes like the “model flavor” that was selected by the model determination process described above. In this particular example, the grind brew parameter indicates that to reproduce the selected flavor, 80% of the grind particles have a size of 203 μm or less (within a standard deviation of 1.4).
The brew parameters affect each other, such that when one parameter changes, one or more of the other parameters may be adjusted to “work with” the change and obtain the same result. For example, if the grind is too coarse, the water temperature and/or dwell time can be adjusted (e.g., increased) to compensate for the coarseness and achieve the same desired flavor. Similarly, if the grind is too fine, the water temperature and dwell time can be adjusted (e.g., decreased) accordingly to compensate for it.
Once the brew parameters are determined in S34, the brew parameters are made available to consumers (S36). In one embodiment, the brew parameters are encoded and associated with the batch of beans. For example, where the beans are packaged into one-pound bags, the brew parameters may be encoded and affixed to the packaging. In one embodiment, the brew parameters for different types of coffee makers may be printed on a sheet of paper or sticker and affixed to a bag of coffee beans. A user who purchases the coffee beans would automatically receive the brew parameters for reproducing the “model coffee drink” using those beans.
In another embodiment, the brew parameters are encoded and affixed or printed onto the packaging of the coffee beans. A user would scan the code using a scanner or a code reader (which may be implemented as an application for a smartphone) to obtain the encoded brew parameters. The parameters could also be encoded on an RFID or NFC device.
In one embodiment, the selection of the model coffee flavor (S32) is done for a number of cateuories, such as “strong and robust,” “medium body,” and “light and smooth,” with a model coffee drink (and its brew parameters) selected for each category.
In yet another embodiment, the brew parameters are stored in a database and accessed by the users. Each batch of beans that is characterized is assigned an identifier. For each coffee bean batch identifier, brew parameters for different types of coffee makers (e.g., French press, Aeropress®, espresso) are entered for the different flavor categories into a database. The brew parameter database may include other data, such as a list of commonly used coffee makers and any brew parameter adjustments to be used with each of the different-brands and models. Also, if professional tasters were used to determine the “model coffee drink,” instead of agreeing on one set of best brew parameters, the parameters most liked by each professional taster could be stored separately and identified. This way, with experience, someone making the coffee would know which taster's selection he usually prefers and choose the brew parameters associated with that taster's model drink. Additionally, the database may include dates and timetables so that, if the brew parameters change over time, for instance as the roasted beans become less fresh, corrected brew parameters can be accessed.
In addition to the brew parameters associated with each brew condition, a related table of settings, for instance for particular devices, may be stored (e.g., in a look-up table).
Additionally, the identifier may be linked to other information related to the batch of roasted coffee beans, such as, for example, varietal, origin, batch and roast dates, additional descriptive content (i.e. color and size of beans), “use by” dates, additional recommendation regarding the roasted coffee beans, suggested brew methods and these information may be stored or linked to the brew parameter database. The database can be updated as needed.
The parameter-based coffee replication process 40 may be implemented as a software or firmware residing in different machines. For example, the software may be implemented as an application for a smartphone that can be downloaded. The software may also be implemented as a web-based application in which the user interacts with a web-based interface. The software may also be implemented, in total or partially, as part of an automatic coffee grinding and/or brewing device, and used to run the device as described below.
In one embodiment, the grind control unit 52 may be implemented using the particle size determination method that is described below with respect to
The smart coffee grinder 50 may be equipped with a local memory 56 that stores the coffee bean identifier and the grind settings associated with that identifier. Hence, if a user wants to make the same type of coffee drink for several days using the same bag of coffee, the smart coffee grinder 50 would not have to repeat the entire process of
Once the coffee is brewed, the software can, optionally, request and incorporate feedback regarding the coffee (not shown). This feedback may be, for example, a chemical analysis of either specific chemicals or of the resulting coffee in total, to compare to results of a chemical analysis described above, such as to provide a cross-check on the resulting coffee. This feedback may also be, for example, provided by the users. For example, as the community gains experience with a roast batch they may provide variations on the brew parameters to achieve different results. These updates can be published and rated by other members of the community.
The control unit 750 controls the nodes independently in a coordinated manner. The control unit 750 has access to the brew parameter database, and retrieves the parameters for each node from the database. It then sets the node parameters according to the retrieved parameters (S760) and initiates the node operation (S770). The node operations are then performed (S780) in a coordinated manner, such that the steps of the process happen in the right order. With each step, the control unit 750 confirms that the operation has been completed before moving on to the next step (S790). There is a timer built into or read by the control unit 750 so that the control unit 750 can set the runtimes for various processes (e.g., grinding, agitating) according to the brew parameters.
The method of determining the particle size distribution of ground coffee beans using an image is described with respect to
The method entails generating or otherwise obtaining an image (e.g., a bitmap) that includes the small particles of the ground coffee beans and a calibration pattern, wherein the dimensions of the calibration pattern are known. The image may be produced, for example, using any conventional camera to image the small particles positioned on or around a calibration pattern. Where particle size is small (e.g., in the order of 10−6 inches), corrections are made to account for distortions caused by the camera lens.
In one embodiment of the method that is disclosed, the particle size of the ground coffee beans is determined using a calibration pattern that includes marks of known dimensions. The calibration pattern and the particles are imaged together, with the assumption that the same distortions will apply to both the calibration pattern and the particles. Using the known dimensions of the calibration pattern, a transformation is generated for removing the distortion effects and converting the image to a corrected image that is free of distortion effects. The transformation is then applied to determine particle size distribution from the corrected image.
Advantageously, any camera (or other device capable of producing a bitmap) can be used to record the bitmap, and specialized imaging or recording equipment is not required. Furthermore, because the calibration pattern and particles are recorded together in a single image, the camera, lens and other image parameters do not need to be known to correct for distortions and extract accurate particle sizes. Capturing the particles and the calibration pattern together in one image eliminates the need to record additional, separate calibration images to obtain a set of parameters to correct for distortions before recording the measurement image. In the method that is disclosed, the information necessary to accurately determine the sizes of the ground coffee beans is embedded in a single image.
Any technology for capturing or generating two-dimensional or three-dimensional images may be used in step S810, including techniques that utilize photographic (i.e., visible light), ultrasound, x-ray, or radar. The inventive concept is not limited to any particular way of capturing the bitmap. The calibration pattern, such as calibration grid 54, usefully includes a background and calibration marks. For example, the calibration marks of calibration grid 54 are the outlines of squares having consistent dimensions x and y, which are drawn with lines of a substantially constant thickness and repeated at a consistent interval w in rows and columns. For example,
The calibration pattern, for example calibration grid 54, is most useful if the main color or shade of the background has a high contrast to the color of the particles to be measured and the color of the calibration marks. It is also useful to have the color of the calibration marks be different than the color of the particles whose size is to be determined. For example, if the particles are brown or black, the background may be white and the calibration patterns may have a blue hue. In general, any material may be used for the calibration pattern, and it may be useful if the surface of the calibration pattern is impervious to the material of the particles, so that the particles do not damage the calibration pattern. In one example, the calibration pattern is a pattern printed on a sheet of paper and the particles of the ground coffee beans are sprinkled on it. A digital image of the calibration pattern may be obtained by the user and printed at home.
The smallest particle size of the coffee grounds that can be determined using the technique disclosed herein depends on a number of factors, one of which is the ratio between the smallest measurable object's pixel coverage and the calibration mark's pixel coverage. There also should be enough camera resolution to capture one or more calibration marks and enough particles of the sample to gather statistics. The camera that is part of an iPhone that is currently available in the market would allow determination of particle size as small as 1×10−6 inches. In one embodiment, the dimensions of the calibration marks depend on the digital resolution of the camera and the size of the smallest particle to be measured. For example, for a camera having a resolution of 3264×2448 pixels, a square of 2 pixels by 2 pixels is needed to measure a particle with a diameter of 25×10−6 inches, and a calibration pattern that is 400 pixels per side (i.e., surrounds an area of 160,000 pixels) is used.
Patterns may include repeating squares of a grid, and also patterns other than the repeating squares may be used for the calibration patterns. However, using a regular pattern that is mathematically easy to model, e.g., that includes orthogonal lines, for the calibration patterns may simplify the processing of the recorded image. Additionally, the calibrations mark uses maximum use of the color that has a high contrast to the color of the particles whose size is being determined. A pattern such as a “checkerboard” pattern, for example, may not work as efficiently because half of the boxes would be dark and, where the particles are dark, too many of the particles would lie on the dark areas where there is not much contrast between the particles and the surface they are on.
At step S820, the image of calibration grid 55 is corrected to remove distortions. One of the problems of using imaging for determining particle size is that various distortions in the recorded image will lead to inaccuracies in the size determination. These distortions are particularly deleterious when trying to make accurate measurements of small particles, in the size ranges of, for example, 10−6 inches. Such distortions can be the result of imperfections in lens geometry or alignment, which may cause, for example, straight lines to be captured in the image as non-straight lines. Distortion can also be the result of perspective distortion, which occurs when the camera's optical axis is not perpendicular to the center of the object being imaged, causing parallel lines to appear non-parallel in the image. As particle sizes are determined using the dimensions of the calibration marks, distortion in the image of the calibration marks will likely result in error in size determination. Hence, distortions in the calibration marks are corrected before particle size determination is done. At the end of step S820, a “corrected image” is produced that includes the distortion-free calibration marks and the original (uncorrected) image of the particles.
In one approach to removing distortions in step S820, calibration marks are extracted from the image. A domain filtering technique may be used for this extraction. For example, where the calibration marks are blue, a technique that uses a chroma range to extract patterns in the blue hue range may be used (the background of the calibration pattern is a different color from the marks). The result is a calibration mark object array that is built by extracting contours from the blue portions of the image. The calibration mark object array is used to generate a uniform scale that is useful for mapping the image, as in orthophotographs. This generation of a uniform scale is an ortho-transformation, which may be reused with the same calibration pattern repeatedly once it is generated.
At step S830, particle sizes of ground coffee beans are determined using the corrected image of the calibration pattern. This process entails extracting the particles from the corrected image using domain filtering. For example, domain filtering may be accomplished using a chroma range to extract the particles that are known to be in the brown/black color range. A particle object array is generated by extracting contours of the particles from the processed image. Each element of the particle object array can then be measured to determine particles dimensions, for example the extent (diameter), area and circularity of each particle. From these measurements, a size distribution of the measured particles is obtained. In measuring the particle sizes, the calibration pattern acts as a dimensional reference.
As mentioned above, the dimensions of the calibration marks in the physical world (as opposed to in the image) are known. Optionally, the accuracy of the calibration mark distortion correction used in S820 can be cross-checked by measuring the calibration marks in the corrected image and calculating the discrepancy from known sizes of the calibration marks in the calibration pattern.
As step S840, the measurements of the particle object array are characterized. A size distribution histogram, an area distribution histogram, volume distribution histogram, minimum, maximum and standard deviations and distribution peak analysis may be used to determine a set of size parameters that characterize the size and shape of the particles of the ground coffee beans. A “set” of size parameters, as used herein, is intended to mean at least one size parameter.
In some cases, the particle size determination may be made with the objective of processing (e.g., grinding) the particles to achieve a desired size. In these cases, there may be a profile of target particles that is available, and the profile may be defined in terms of the set of size parameters that is used in step S840. Based on a comparison of the measured parameters (result of step S840) against the profile of the target particles, one may determine what action needs to be taken to bring the two profiles closer. This determination may be made automatically by a processor or by a person. Where the determination is made automatically, the processor may indicate to an operator (e.g., visually and/or by sound) that further grinding would bring the measurement closer to the target profile.
When brewing coffee as shown in
Now, further details will be provided about the distortion correction step S820 of
The technique proposed by Zhang uses the camera to observe a planar pattern shown at a few (at least two) different orientations. The pattern can be printed on a laser printer and attached to a “reasonable” planar surface (e.g., a hard book cover). Either the camera or the planar pattern may be moved and it is not necessary for the details of the motion to be known. The proposed approach lies between the photogrammetric calibration and self-calibration because 2D metric information is used rather than 3D or purely implicit one. Both computer simulation and real data have been used to test the technique. The technique described herein advances 3D computer vision from laboratory environments to the real world.
Constraints on the camera's intrinsic parameters are provided by observing a single plane. A 2D point is denoted by m=[u,ν]T. A 3D point is denoted by M=[X, Y, Z]T. The symbol {tilde over ( )} denotes the augmented vector by adding 1 as the last element: {tilde over (m)}=[u, ν, 1]T and {tilde over (M)}=[X, Y, Z, 1]T. A camera is modeled by the usual pinhole: the relationship between a 3D point M and its image projection m is given by:
s{tilde over (m)}=A[R t]{tilde over (M)} Equation (1)
where s is an arbitrary scale factor; (R, t), called the extrinsic parameters, is the rotation and translation which relates the world coordinate system to the camera coordinate system; and A, called the camera intrinsic matrix, is given by:
with (u0, υ0) the coordinates of the principal point, α and β the scale factors in image u and υ axes, and γ the parameter describing the skewness of the two image axes. The abbreviation A−T is used to represent (A−1)T or (AT)−1.
The model plane may be assumed as being on Z=0 of the world coordinate system. The ith column of the rotation matrix R will be denoted as ri. From Equation (1), we have:
The symbol M is still used to denote a point on the model plane, but M=[X, Y]T since Z is equal to 0. In turn, M−=[X, Y, l]T. Therefore, a model point M and its image m is related by a homography H:
s{tilde over (m)}=HM{tilde over ( )} with H=A[r1 r2 t]. Eq. (2)
As is clear, the 3×3 matrix H is defined up to a scale factor.
Given an image of the model plane, a homography can be estimated. Denoting the homography by H=[h1 h2 h3], from Equation (2), we have
[h1 h2 h3]=λA[r1 r2 t]
where λ is an arbitrary scalar. Using the knowledge that r1 and r2 are orthonormal, the following constraints are obtained:
h1TA−TA−1h2=0 Eq. (3)
h1TA−TA−1h1=h2TA−TA−1h2 Eq. (4)
These are the two basic constraints on the intrinsic parameters, given one homography. Because a homography has 8 degrees of freedom and there are 6 extrinsic parameters (3 for rotation and 3 for translation), 2 constraints are obtained on the intrinsic parameters. The parameter A−TA−1 actually describes the image of the absolute conic. A geometric interpretation will now be provided.
The model plane, under the convention used herein, is described in the camera coordinate system by the following equation:
where w=0 for points at infinity and w=1 otherwise. This plane intersects the plane at infinity at a line, and we can easily see that
and
are two particular points on that line. Any point on it is a linear combination of these two points, i.e.,
Computing the intersection of the above line with the absolute conic, and knowing that by definition, the point x∞ (the circular point) satisfies x∞T x∞=0, i.e.,
(ar1+br2)T(ar1+br2)=0, or a2+b2=0.
The solution is b=±ai, where i2=−1. That is, the two intersection points are
Their projection in the image plane is then given, up to a scale factor, by
{tilde over (m)}∞=A(r1±ir2)=h1±ih2.
Point {tilde over (m)}∞ is on the image of the absolute conic, described by A−T A−1. This gives
(h1±ih2)T A−1 A−1(h1±ih2)=0.
requiring that both real and imaginary parts be zero yields (3) and (4).
The details on how to effectively solve the camera calibration problem will now be provided. An analytical solution will be presented, followed by a nonlinear optimization technique based on the maximum likelihood criterion. Finally, both analytical and nonlinear solutions will be provided, taking lens distortion into account.
Consider the following Equation (5):
Noting that B is symmetric and defined by a 6D vector
b=[B11,B12,B22,B13,B23,B33]T. Equation (6)
Let the ith column vector of H be hi=[hi1, hi2, hi3]T. Then, we have
hiTBhj=vijTb Equation (7)
with
vijhi1hj1,hi1hj2+hi2hj1,hi2hj2,hi3hj1+hi1hj3,hi3hj2+hi2hj3,hi3hj3]T.
Therefore, the two fundamental constraints (3) and (4), from a given homography, can be rewritten as 2 homogeneous equations in b, as shown below in Equation (8):
If n images of the model plane are observed, by stacking n such equations as (8), the result can be expressed as:
Vb=0. Equation (9)
where V is a 2π×6 matrix. If n≥3, generally, a unique solution b defined up to a scale factor is obtained. If n=2, the skewless constraint γ=0, i.e., [0, 1, 0, 0, 0, 0]b=0, can be imposed and added as an additional equation to Equation (9). (If n=1, two camera intrinsic parameters, e.g., α and β, may be solved assuming u0 and υ0 are known (e.g., at the image center) and γ=0. The solution to Equation (9) is well-known as the eigenvector of VTV associated with the smallest eigenvalue (equivalently, the right singular vector of V associated with the smallest singular value).
Once b is estimated, values in camera intrinsic matrix A can be calculated. Once A is known, the extrinsic parameters for each image may be computed. Using Equation (2), for example, the following may be obtained:
r1=λA−1h1
r2=λA−1h2
r3=r1×r2
t=λA−1h3
with λ=1/∥A−1h1∥=1/∥A−1h2∥. Due to the presence of noise in data, the so-computed matrix R=[r1, r2, r3] does not satisfy the properties of a rotation matrix.
Suppose there are n images of a model plane and m points on the model plane. Suppose also that the image points are corrupted by independent and identically distributed noise. The maximum likelihood estimate can be obtained by minimizing the following functional:
Σi=1nΣj=1m∥mij−{circumflex over (m)}(A,Ri,ti,Mj)∥2, Value (10)
where {circumflex over (m)} (A, Ri, ti, Mj) is the projection of point Mj in image i, according to Equation (2). A rotation R is parameterized by a vector of 3 parameters, denoted by r, which is parallel to the rotation axis and whose magnitude is equal to the rotation angle. R and r are related by the Rodrigues formula. Minimizing Value (10) is a nonlinear minimization problem, which may be solved with the Alevenberg-Marquardt Algorithm. It uses an initial guess of A, {Ri, ti|i=1 . . . n} which can be obtained using the technique described above.
The solutions above do not consider lens distortion of a camera. However, a desktop camera usually exhibits significant lens distortion, especially radial distortion. Now, first two terms of radial distortion will be discussed. It is likely that the distortion function is dominated by the radial components, and especially by the first term.
Let (u, υ) be the ideal (distortion-free) pixel image coordinates, and ({hacek over (u)}, {hacek over (υ)}) the corresponding real observed image coordinates. The ideal points are the projection of the model points according to the pinhole model. Similarly, (x, y) and ({hacek over (x)}, {hacek over (y)}) are the ideal (distortion-free) and real (distorted) normalized image coordinates.
{hacek over (x)}=x+x[k1(x2+y2)+k2(x2+y2)2]
{hacek over (y)}=y+y[k1(x2+y2)+k2(x2+y2)2],
where k1 and k2 are the coefficients of the radial distortion. The center of the radial distortion is the same as the principal point. From
{hacek over (u)}=u+(u−u0)[k1(x2+y2)+k2(x2+y2)2] Equation (11)
{hacek over (υ)}=υ+(υ−υ0)[k1(x2+y2)+k2(x2+y2)2]. Equation (12)
Estimating Radial Distortion by Alternation.
As the radial distortion is expected to be small, one would expect to estimate the other five intrinsic parameters, using the technique described above reasonably well by simply ignoring distortion. One strategy is then to estimate k1 and k2 after having estimated the other parameters, which will give the ideal pixel coordinates (u, υ). Then, from Equations (11) and (12), we have two equations for each point in each image:
Given m points in n images, we can stack all equations together to obtain in total Zinn equations, or in matrix form as Dk=d, where k=[k1, k2]T. The linear least-squares solution is given by
k=(DTD)−1DTd. Equation (13)
Once k1 and k2 are estimated, one can refine the estimate of the other parameters by solving Equation (10) with {circumflex over (m)}(A, Ri, ti, Mj) replaced by Equations (11) and (12). We can alternate these two procedures until convergence.
The convergence of the above alternation technique may be slow. A natural extension to Equation (10) is to estimate the complete set of parameters by minimizing the following functional:
Σi=1nΣj=1m∥mij−{hacek over (m)}(A,k1,k2,Ri,ti,Mj)∥2, Value (14)
where {hacek over (m)} (A, k1, k2, Ri, ti, Mj) is the projection of point Mj in image i according to Equation (2), followed by distortion according to Equations (11) and (12). This is a nonlinear minimization problem, which is solved with the Levenberg-Marquardt Algorithm. A rotation is again parameterized by a 3-vector r, as disclosed above. An initial guess of A and {Ri, ti|i=1 . . . n} can be obtained using the technique described above. An initial guess of k1 and k2 can be obtained with the radial distortion solutions described above, or simply by setting them to 0.
At step S930, the lens distortion parameters k1 and k2 are then used on the original image to obtain an image that is corrected for lens distortion. Alternatively, instead of using the image, a particle object array that describes the particle's geometries, may be corrected. The lens distortion parameters k1 and k2 are used to correct the image for lens distortion using the following equations (15) and (16).
Let (xcorrect, ycorrect) represent the correct locations in the image if there were no distortion due to the lens. Then:
xdistorted=xcorrect+(xcorrect−u0)(k1r2+k2r4) (15)
ydistorted=ycorrect+(ycorrect−ν0)(k1r2+k2r4) (16)
where r2=(xcorrect−u0)2+(ycorrect−ν0)2, and u0 and ν0 are principal points (i.e., the intersection points of the cameral optical axis and image plane).
The lens distortion can then be corrected by warping the recorded bitmap with a reverse distortion. For each pixel in the corrected image, its corresponding location is mapped in the distorted image using equation (15) and (16) above. The section below titled Geometric Image Transformations describes how the two-dimensional image coordinates and the three-dimensional world coordinates are related by a camera's intrinsic parameters (e.g., focal lens, principal points, distortion coefficients) and extrinsic parameters (rotation and translation matrix). For each integer pixel coordinates in the destination (corrected bitmap), trace back to the source (recorded bitmap) and find the corresponding float coordinates, using the surrounding integer pixels to interpolate the float coordinates. Bilinear interpolation may be used in this process.
In Summary, the lens-distortion correction procedure proposed here, part of which incorporate Zhang's technique, is as follows:
At step S940, correction for perspective distortion may be achieved using the four corner points of a rectangular calibration mark to solve the homology H using the following process:
Suppose we have a point in world coordinates
and we write this in homogeneous coordinates as
Similarly, the corresponding point in image coordinates is
and we write this in homogenous coordinates as
The relationship between these two can be expressed in the following equation:
where
is the Homography we want to solve.
Expanding both sides of the equation using matrix multiplication, we get:
kximg=k(H11xworld+H12yworld+H13)
k3yimg=k(H21xworld+H22yworld+H23)
k1=k(H31xworld+H32yworld+1)
Plug in the third equation to the first two, we get two equations from this pair of points:
ximg=H11xworld+H12xworld+H13−H21ximgxworld−H32ximgyworld
yimg=H21xworld+H22yworld+H23−H31yimgxworld−H22yimgyworld
Since there are eight unknowns in H, we need 4 pairs of points to solve H. We write the eight equations in matrix form:
Thus, for each image suffers from the projective distortion, we pick four points in this image, and given the world coordinates of these four points, we are able to solve H.
The “k” in the above equation is a scalar for the homogeneous representation of the two-dimensional coordinates, different from the lens coefficients k1, k2. Four points (e.g., four corners where the shape is rectangular) of multiple calibration marks may be used to account for nonuniformity of distortions. The starting point for perspective correction in this approach is not the original image, but the corrected image where lens distortion effects have been removed. The homography H is determined using four points from a set of calibration marks identified in the image corrected for lens distortion, and then is applied to the corrected image to determine a bitmap (the corrected, or true-size, bitmap) that has been corrected for both lens and perspective distortion.
A perspective distortion usually occurs when the camera's optical axis is not perpendicular to the center of the object. Using the image of particles captured on a grid-patterned background, multiple (e.g., five) pairs of orthogonal lines in the scene may be used to find the homography to correct the perspective distortion. This correction will often make parallel lines in the physical world also parallel in the image, orthogonal lines in the physical world also orthogonal in the image, squares in the physical world have unit aspect ratio in the image, and/or circles in the physical world circular in the image.
To summarize the perspective distortion correction process detailed above, the process entails the following steps:
The lens distortion correction and the projection distortion correction maybe tested separately and cascaded, so that Shepard's interpolation may only be done once.
The method illustrated in
This section will now talk about some known image transformation functions that may be used to correct and manipulate the image of the particle. More specifically, the functions in this section perform various geometrical transformations of 2D images. They do not change the image content but deform the pixel grid and map this deformed grid to the destination image. In fact, to avoid sampling artifacts, the mapping is done in the reverse order, from destination to the source. That is, for each pixel (x, y) of the destination image, the functions compute coordinates of the corresponding “donor” pixel in the source image and copy the pixel value:
dst(x,y)=src(fx(x,y),fy(x,y))
In case the forward mapping is specified as <gx, gy>:src→dst, the functions described below compute the corresponding inverse first mapping, <fx, fy>:dst→src and then use the above formula.
The actual implementations of the geometrical transformations, from the most generic Remap and to the simplest and the fastest Resize, need to solve two main problems with the above formula.
This function calculates an affine matrix of 2D rotation. Some parameters used for this process are as follows:
The transformation maps the rotation center to itself. If this is not the target, the shift should be adjusted.
Get Affine Transform
This function calculates the affine transform from 3 corresponding points. Some parameters used for this process are as follows:
Get Perspective Transform
This function calculates a perspective transform from four pairs of the corresponding points. Some parameters used for this process are as follows:
Get Quadrangle SubPix
This process retrieves a pixel rectangle from an image with sub-pixel accuracy. Some parameters used for this process are as follows:
The values of the pixels at non-integer coordinates are retrieved using bilinear interpolation. When the function needs pixels outside of the image, it uses replication border mode to reconstruct the values. Every channel of multiple-channel images is processed independently.
GetRectSubPix
This function retrieves the pixel rectangle from an image with sub-pixel accuracy.
This function remaps an image to a log-polar space.
Inverse transformation (CV_WARP_INVERSE_MAP is set):
dst(x,y)=src(ϕ,ρ)
where
ρ=M·log √{square root over (x2+y2)},ϕ=a tan(y/x)
The function emulates the human “foveal” vision and can be used for fast scale and rotation invariant template matching, for object tracking and so forth. The function cannot operate in-place.
Remap
This function applies a generic geometrical transformation to the image.
This function resizes an image.
To shrink an image, it will generally look best with INTER_AREA interpolation, whereas to enlarge an image, it will generally look best with INTER_CUBIC (slow) or INTER_LINEAR (faster but still looks OK).
WarpAffine
This function applies an affine transformation to an image.
The function is similar to GetQuadrangleSubPix but they are not exactly the same. WarpAffine requires input and output image have the same data type, has larger overhead (so it is not quite suitable for small images) and can leave part of destination image unchanged. While GetQuadrangleSubPix may extract quadrangles from 8-bit images into floating-point buffer, has smaller overhead and always changes the whole destination image content. The function cannot operate in-place.
WarpPerspective
This function applies a perspective transformation to an image. Parameters useful for this function include the following:
if CV_WARP_INVERSE_MAP is not set
otherwise
Note that the function cannot operate in-place.
While some of the embodiments are described in terms of a method or technique, it should be understood that the disclosure may also cover an article of manufacture that includes a non-transitory computer readable medium on which computer-readable instructions for carrying out embodiments of the method are stored. The computer readable medium may include, for example, semiconductor, magnetic, opto-magnetic, optical, or other forms of computer readable medium for storing computer readable code. Further, the disclosure may also cover apparatuses for practicing embodiments. Such apparatus may include circuits, dedicated and/or programmable, to carry out operations pertaining to embodiments.
Examples of such apparatus include a general purpose computer and/or a dedicated computing device when appropriately programmed and may include a combination of a computer/computing device and dedicated/programmable hardware circuits (such as electrical, mechanical, and/or optical circuits) adapted for the various operations pertaining to the embodiments.
For example, the brew parameter database may be stored in a computer readable medium and accessed directly with computer-readable instructions stored on the computer readable medium, or via computer-readable instructions provided through an internet connection, or some combination thereof. Additionally, the workflow through the nodes may be accomplished, for example, on a general purpose computer or on a dedicated computing device integrated with the device to grind and brew coffee.
This application claims the benefit of U.S. Provisional Application No. 61/776,619 filed on Mar. 11, 2013.
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