Patent Grant
8073244
1. Technical Field
The present invention relates to pattern analysis, and more particularly, to a method for automated selection of operators and their parameters to achieve a pattern classification task.
2. Discussion of the Related Art
Mathematical morphology is a framework to extract information from images. Under this framework machine vision problems may be solved by image transformations that employ a sequence of parameterized operators. Pixel neighborhood level feature detection followed by a region level grouping and/or morphological filtering is a typical operation sequence in a variety of video and image analysis systems (e.g., document image analysis, video surveillance and monitoring, machine vision and inspection). However, the robustness of these systems is often questionable because of the use of operator/parameter combinations that are set by trial and error.
According to an aspect of the invention, there is provided a method for automated selection of a parameterized operator sequence to achieve a pattern classification task. A collection of labeled data patterns is input and statistical descriptions of the inputted labeled data patterns are then derived. Classifier performance for each of a plurality of candidate operator/parameter sequences is determined. Optimal classifier performance among the candidate classifier performances is identified. Performance metric information, including, for example, the selected operator sequence/parameter combination, is outputted. The operator sequences selected can be chosen from a default set of operators, or may be a user-defined set. The operators may include any morphological operators, such as erosion, dilation, closing, opening, close-open, and open-close. The operators may also include operators that map an input Boolean vector to an output Boolean vector. Additionally, the operators can include an operator that is defined as successive applications of a 1D filter in two orthogonal directions.
The collection of data patterns can include various patterns of interest. They can also include patterns of non-interest. The data patterns can include any gray-level data/color data transformed to a binary representation.
The statistical descriptions of the input patterns can be derived using any suitable probability model, such as, for example, a mixture of Hidden Markov Models (HMMs), a Bayesian network etc. A probability model that employs a non-parametric density representation may also be used.
The operator sequences selected can be chosen from a default set of operators, or may be a user-defined set.
The criteria for determining an optimal classifier performance can be defined by a user. The criteria will preferably relate to maximizing expected classifier performance. This may involve, for example, balancing the tradeoff of false alarm errors and miss detection errors.
The step of determining classifier performance may include, for each candidate operator sequence and corresponding parameter values and a description of the output statistic to the measured, and given derived statistical descriptions for the inputted data patterns,
According to another aspect of the invention, a method is provided for determining optimal classifier performance of a plurality of candidate operator sequences and corresponding parameter values. For each candidate operator sequence and corresponding parameter values, the method 1. Constructs an Embedded Markov Chain that uses the steps:
Numerous applications of the pattern classification framework can be realized, including any pattern analysis or classification task, e.g. document processing, license plate detection, text analysis from video, machine vision (object detection, localization and recognition tasks).
These and other aspects, features and advantages of the present invention will become apparent from the following detailed description of preferred embodiments, which is to be read in connection with the accompanying drawings.
According to the present invention, characterization of a morphological operator sequence may be viewed as the derivation of output image statistics as a function of the input image statistics, operator sequence and its tuning parameters. The difficulty is in defining the statistical models for the input data and the corresponding derivation of the output statistical models. In this invention, the input to the morphological algorithm is viewed as a binary random series. The output is viewed as another binary random series whose statistics (e.g. Hamming distance or run/gap length distributions) are functions of the input statistics.
For each particular morphological operator and its parameters, the corresponding Embeddable Markov Chain (EMC) model is built, and the EMC approach is applied to analyze the performance of the operator. The analyses gives insights about how one could automate the selection of a morphological operator sequence and its parameters to achieve a given error rate. License Plate detection is provided as a case study to illustrate the utility of the invention.
To facilitate a clear understanding of the present invention, illustrative examples are provided herein which describe certain applications of the invention (e.g., license plate detection). However, it is to be appreciated that these illustrations are not meant to limit the scope of the invention, and are provided herein to illustrate certain concepts associated with the invention.
It is also to be understood that the present invention may be implemented in various forms of hardware, software, firmware, special purpose processors, or a combination thereof. Preferably, the present invention is implemented in embedded code as a program tangibly embodied on a program storage device. In addition, various other peripheral devices may be connected to a computer platform such as an additional data storage device and a printing device, as well as various still and video imaging input devices.
Embeddable Markov Chain Approach
A system diagram for a typical pattern analysis system is illustrated in
Let f(t) denote the mapping from the pixel index set {1, 2, . . . N} to the gray level measurements R. Let b(t) denote the ideal unknown function representing the mapping that assigns the true labels (e.g., foreground/background (1/0)) for each index. The detection algorithm is viewed as a function that maps f(t) to the binary series {circumflex over (b)}d(t) by using a decision rule (that could be spatially varying). Define pf(t) as the conditional probability that the detector output {circumflex over (b)}d(t)=1 given that the true label b(t)=0. Let pm(t) be the conditional probability that {circumflex over (b)}d(t)=0 given b(t)=1.
A natural representation of the statistics {circumflex over (b)}d and {circumflex over (b)}g is the distribution of the run lengths as a function of the filter parameters used (e.g., structuring element sizes in a morphological operator). This representation is convenient in that it provides a natural way to interpret the results of morphological operations. The size distributions of shapes (granulometrics) have been used in the morphology literature to describe signal statistics. The derivation of the statistics of the run lengths may be viewed as a function of b(t), {circumflex over (b)}g(t), and the morphological operator parameter Tg. Let Ĉl denote the number of runs of length l in the output of the morphological algorithm. An objective is to derive the conditional distribution p(Ĉl;Tg,N|{pm(t)},{pf(t)}). Most research in theory of runs has addressed this problem by using combinatorial analysis. However, the prior art assumed stationary (i.e., pm(t)=pm and pf(t)=pf) while the present invention does not make any assumption about the form of pm(t) and pf(t).
Advantageously, the present invention uses a technique that embeds a discrete random variable into a finite Markov chain to numerically compute the probability mass function (pmf) of the discrete random variable. The pmf is essentially computed as a function of the N-step transition probabilities of the Embeddable Markov Chain (EMC). The main advantage of using this algorithm is that Monte-Carlo simulations are prohibitively slow when probabilities for unlikely events are being estimated.
For a given n, let Γn={0, 1, . . . , n} be an index set and Ω={a1, . . . , am} be a finite state space.
A nonnegative integer random variable Xn can be embedded into a finite Markov chain if:
Let Λt be the m×n transition probability matrix of the finite Markov chain ({Yt: tεΓn},Ω). Let Ur be a 1×m unit vector having 1 at the rth coordinate and 0 elsewhere, and let U′r be the transpose of Ur. Finally, for every Cx, define the 1×m vector U(Cx)=Σr:a
If Xn can be embedded into a finite Markov chain, then
where π0=[p(Y0=a1), . . . , p(Y0=am)] is the initial probability of the Markov chain.
If the Markov chain is homogenous, that is Λt=Λ for all tεΓn, then ∀x=0, . . . , l the exact distribution of the random variable Xn can be expressed by p(Xn=x)=π0ΛnU′(Cx).
In order to find the distribution of any embeddable random variable, one has to construct: (i) a proper state space Ω, (ii) a proper partition {Cx} for the state space. and (iii) the transition probability matrix Λt associated with the EMC. The exact process by which the state space is defined along with the partitioning is dependent on the nature of the statistic of interest and the operator used.
Statistics Calculation Using EMC Approach
Before the problem of deriving the run length distributions in the output of a morphological algorithm is addressed, the EMC approach is described. This approach can be used to derive the run length distribution of the observation of an uncorrelated random binary series. Here, the problem of calculation of the joint run length distribution, i.e., “What is the probability of having m runs with size M and n runs with size N?” is addressed.
State Space Construction
The state space construction for the computation of the distribution of run lengths is rather straightforward. View the sequence of binary observations up to pixel T as partial observations of the 0 and 1 runs. A variable xi is needed to denote the number of observations of runs of a given length i at pixel T and an indicator variable mi to denote the situation whether the preceding number of ones is exactly equal to i or not. Thus mi takes on value 1 if the last sequence of 1s is exactly equal to length i and 0 otherwise. The pair (X,M), X=[x1, . . . , xn, xn+], M=[m1, . . . , mn, mn+] denotes the states for the problem. Here xn+ denotes the number of runs larger than n and mn+ in the corresponding indicator variable. Given these states it is easy to see that the graph shown in
The partition of the state space corresponds to the singleton sets of C with assigned count values. The values for the probabilities in Λt are given by the following expressions. For example, the probability of observing a 0 at location t is given by the sum of two terms: the probability that the true value is 0 and there is no false alarm, and the probability that the true value is 1 and there is a misdetection.
qt(0)=pb(t){0}(1−pf(t))+pb(t){1}pm(t) (1)
qt(1)=pb(t){0}pf(t)+pb(t){1}(1−pm(t)) (2)
where pb(t){ } is the distribution of the ground truth.
It is clear that a large state space is needed for calculating the joint distribution when n is large. For example, when n=50, more than 1024 states are needed, thus requiring a large memory for implementation.
Statistics for One-Dimensional Morphology
It was shown that run-length statistics for a binary random series can be derived before the application of the morphological algorithm. The statistics can be derived for an uncorrelated binary series or a correlated series defined in terms of a homogenous or inhomogeneous Markov chain. Now we will use the probability of observing a given number of runs of length greater than equal to S,Gn,S, after the closing operation with closing parameter Tg=L as an example to illustrate how the output statistics of a morphological operator for binary series can be derived by using the EMCs. The trick again is to devise the appropriate EMC. Similar EMCs can be derived for openings, and openings followed by closings, etc.
To construct the state space we have to consider the property of the closing operation. Closing essentially fills gaps of sizes less than a given length L. At any given pixel the output {circumflex over (b)}g(t) is a 1 if and only if {circumflex over (b)}d(t)=1 or {circumflex over (b)}d(t)=0 and there exists two neighbors with indices t−i and t−j,i,j≧1, {circumflex over (b)}d(t−1)=1 and {circumflex over (b)}d(t+j)=1 with j+i<L. This implies that in addition to the number of runs of length greater than or equal to S, the state space has to include information about the run length of the last 1-run observed as well as the length of the last gap (if any 0-runs) to identify the partial state. One has to wait until the gap length is greater than L before deciding to terminate a previous run.
Formally, define the state variable to be (x,s,f) where
denotes the number of success runs of size greater than or equal to k,s=−1, 0, . . . , L is the number of failures in the last failure run, L is the size of the structuring element. The value −1 for f corresponds to a gap of length of greater than L that cannot be filled, while a value −1 for s corresponds to having a 1-run of length greater than or equal to S. The left graph of
The partition of the state space is based on the number of success runs, x. The values of transition probabilities are given by the qt and 1−qt where
qt=pb(t){0}pf(t)+pb(t){1}(1−pm(t)).
Statistics of Binary Series in Morphology Operator Output
The EMC approach is illustrated to show how it can be used to compute the probability distribution of the Hamming distance between the output closing operation and a ground truth signal perturbed by a stochastic process. After that, how one could derive similar statistics when multiple morphological operations are applied consecutively will be introduced.
Statistics for Closing Morphology Operator
The effect of the closing operator with parameter Tg when applied to the detector output {circumflex over (b)}d(t) to produce {circumflex over (b)}g(t) is analyzed. The deviation of {circumflex over (b)}g(t) from the ideal signal b(t) is measured by the Hamming distance (the number of locations where the two Boolean series differ in value). Let X be a discrete random variable corresponding to the Hamming distance between the binary random series {circumflex over (b)}g(t) and the ground truth b(t). In this analysis, b(t) is assumed to be composed of independent (but not necessarily identically distributed) random variables or spatially correlated random variables where the correlation is described by a Markov model. {circumflex over (b)}d(t) is a perturbed version of b(t) where the perturbation model parameters are described by pf(t) and pm(t).
Statistical Independent Case Example
Let the input binary series consist of Boolean random variables that are statistically independent with the probability, of the t-th pixel being value 0, qt. The EMC for calculation of the Hamming distance in this case is given below. The EMC for calculation of correlated binary sequences and the state transition diagram have been devised without providing details herein.
The State Space is:
Ω={(x,p,q): x=0, . . . , N; p=ω, 0, . . . , Tg; q=ω, 0, . . . , Tg}.
The state variable in the left graph is (x,p,q), where x is the Hamming distance, p is the number of 0s in the trailing run of the observed sequence and q is the number of 0s in the ground truth binary series in this trailing run's domain. The value p is needed since it provides the partial gap length observed so far and if this gap were closed, one would have to use the value of q to update the Hamming distance. The notation p=ω is used for an overflowing state that corresponds to the condition that a given gap will not be filled by the closing operation.
The Partition is:
Cx={(x,p,q): p=ω, 0, . . . , Tg; q=ω, 0, . . . , Tg}, x=0, . . . , N.
The State Transition Probabilities are specified by the following equations.
When b(t)==1 and {circumflex over (b)}d(t)==1, (correct detection):
pt(x+2q−p,0,0;x,p,q)=1−qt, p≠ω, q≠ω pt(x,0,0;x,ω,ω)=1−qt
When b(t)==0 and {circumflex over (b)}d(t)==0, (correct rejection):
pt(x,p+1,q+1;x,p,q)=1−qt, p≠ω, q≠ω pt(x,ω, ω;x,ω,ω)=1−qt
When b(t)==0 and {circumflex over (b)}d(t)==1, (false alarm error):
pt(x+2q−p+1,0,0;x,p,q)=qt, p≠ω, q≠ω pt(x+1,0,0;x,ω,ω)=qt
When b(t)==1 and {circumflex over (b)}d(t)==0, (miss detection error):
pt(x+1,p+1,q;x,p,q)=qt, p≠ω, q≠ω pt(x+1,ω,ω;x,ω,ω)=qt
The interpretation of the above equations is as follows. View the value x as the partial Hamming distance between b(t) and {circumflex over (b)}g(t) until the current pixel instant t. If the closing operation given the current observation results in an alteration of the output {circumflex over (b)}g(t) then we have to alter our estimated Hamming distance to correspond to the correct value. The state jumps are essentially of two types:
For example, when correct detection occurs, there are two possible cases to be considered:
In
Statistical Markov Case Example
The right graph in
Statistics for Close-Open Operator Sequences
Previously, a method was introduced to calculate the Hamming distance distribution after a morphological closing by using the EMC approach. Next, it will be shown how to generalize the EMC approach to get the statistics of close-open or open-close operator sequences. The main idea is to use more state variables to save all the temporary information and to extend the graph from a single layer representation to a multi-layered graph.
As one example to show how to generate the state space and transition probabilities matrix for the EMC when given an operator sequence, the Hamming distance after the closing-opening operator sequence-closing with parameter K1, opening with parameter K2, will be used. To simplify the description, the term “operator K1” is used to indicate the closing operator with closing parameter K1 and the “operator K2” to indicate the opening operator with opening parameter K2.
State Space Construction
View the sequence of binary observations up to a pixel instant T as partial observations of 0 and 1 runs. x is used to denote the Hamming distance and the state variable is (x,p2,p2a,q1,q1a). The components with the superscript a indicate how many correct detections in the corresponding numbers without the superscript. q1 is the number of 0's in the last 0-run and q1a is the number of 0's which are correct 0's in the last 0-run. p2 is the number of 1's in the last 1-run and p2a is the number of 1's which are correct 1's in the last 1-run. It is clear that there are q1−q1a errors in the last 0-run and p2−p2a errors in the last on run. Then if the closing operator is applied to q1, those q1a correct 0's in the last 0-run will become errors in the last 1-run. Those q1−q1a errors in the last 0-run will become correct 1's. Thus, the Hamming distance should be adjusted by q1a−(q1−q1a) Similar changes happen to the other component when the opening operator is applied.
Definition of State Transition Probabilities Λt: In the Hamming distance calculation, the events are:
When the event is I0,0, the state (x,p2,p2a,q1,q1a) moves to the state (x,p2p2a,q1+1,q1a+1). When the event is I1,0, the state (x,p2,p2a,q1,q1a) moves to the state (x+1,p2,p2a,q1+1,q1a).
Since the first operator in the operator sequence is a closing operator, no comparisons or judgments are necessary when the system gets a 0 (event I0,0 or I1,0). When the event is I0,1 or I1,1, operator effects need to be considered. Since the only differences between I0,1 and I1,1 are that I0,1 will increase the Hamming distance by 1 and reduce the correct 1's in p2a by 1, the event I1,1 is used to show state evolution. One of ordinary skill in the related art would be able to easily derive the states for event I0,1. Suppose the event is I1,1. If q1≦K1, the operator will close the last 0-run, then the state (x,p2,p2a,q1,q1a) will move to the state (x+2q1a−q1,p2+q1+1,p2a+q1−q1a+1,0,0). If q1>K1, the closing operator will have no impact to the sequence, and will have no impact to the state variable. Then, the operator K2 must be considered. If (q1>K1)&&(p2≦K2), the opening will take place, the state (x,p2,p2a,q1,q1a) will move to the state (x−p2+2p2a,1,1,0,0). If (q1>K1)&&(p2>K2), the morphology operator sequence will have no effect. The state (x,p2,p2a,q1,q1a) will move to the state (x,1,1,0,0).
Application
Numerous applications of the present invention can be realized, including pattern analysis or classification tasks, e.g., document processing, text analysis from video, machine vision. The following describes the use of the present invention in conjunction with license plate detection as one example to show how the present invention may be applied.
Hidden Markov Models (HMMs) have been extensively used in the document understanding literature for text extraction. In this example application, license plate and background binary pattern distributions are modeled as mixtures of HMMs. The main point is that while one way to implement the binary series classification might be by direct implementation of a mixture of HMM-based classifiers, a morphological operator sequence with associated parameters serves as an approximation to the classification mechanism. Denote the HMM parameters for the background binary series and license plate binary series as ΘB and ΘL respectively. The objective is to obtain the mapping from the (ΘB,ΘL) to a morphological operator sequence and its parameters (OS,ΘS). The major gain is in the computational performance as well as providing strong statistical justification for use of a morphological algorithm with associated parameters for the task.
In the application, pixels on the license plate are assumed to be 1s while non-plate pixels are assumed to be 0s. As before we use notations: ideal signal b(t), the detection result {circumflex over (b)}d(t), and the grouping result {circumflex over (b)}g(t), etc. The Hamming distance between b(t) and {circumflex over (b)}g(t) is used as the criterion function to evaluate the performance of the morphological sequence. Ideally, all the plate pixels should be detected and all of the non-plate pixels should be ignored. There are two types of errors. The type one error is the false alarm error, i.e., labeling the non-plate pixel (0) as a plate pixel (1), the type two error is the miss detection error, i.e., labeling a plate pixel (1) as a non-plate pixel (0). For example, in the binary images shown in
The license plate detection algorithm is comprised of the following steps. Initially, the image is inputted (
Experimentally, 60 license plate images were initially supplied. From the 60 license plates images, 50 images were randomly chosen as the training set and 10 images as the testing set. The size of the images were 768×512 and the size of the plate in the images was usually 135 pixels wide and 65 pixels high. Transition probabilities of the Markov model for the original size image were estimated. From the Markov property and the description of how the EMC approach works, it is apparent that if the morphology parameters for the original size image is calculated, the state space will be prohibitively large leading to high computational cost. Instead, the system was configured to downsample by a factor of 4 and use it as the output of detection stage (and also input for the morphology stage). Although downsampling increases the false alarm rate in the non-plate region and decreases the miss detection rate in the plate region, the main objective was just to illustrate the utility of EMC for this application.
From various experiments, it was concluded that the “optimum” morphological operator sequence for the horizontal direction was an open-close sequence with parameters 11 and 19 respectively. For the vertical direction, we found that the best operator sequence was a close-open sequence with parameters 2 and 5 respectively.
Although illustrative embodiments of the present invention have been described herein with reference to the accompanying drawings, it is to be understood that the invention is not limited to those precise embodiments, and that various other changes and modifications may be affected therein by one skilled in the art without departing from the scope or spirit of the invention.
This application is a continuation of U.S. application Ser. No. 10/339,433, which claims the benefit of U.S. Provisional Application Ser. No. 60/346,995, filed on Jan. 9, 2002, which are incorporated by reference herein in their entirety.
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