## Three-dimensional distortion-tolerant object recognition using photon-counting integral imaging

Optics Express, Vol. 15, Issue 4, pp. 1513-1533 (2007)

http://dx.doi.org/10.1364/OE.15.001513

Acrobat PDF (896 KB)

### Abstract

This paper addresses three-dimensional distortion-tolerant object recognition using photon-counting integral imaging (II). A photon-counting linear discriminant analysis (LDA) is proposed for classification photon-limited images. In the photon-counting LDA, classical irradiance images are used to train the classifier. The unknown objects used to test the classifier are labeled by the number of photons detected. The optimal solution of the Fisher’s LDA for photon-limited images is found to be different from the case when irradiance values are used. This difference results in one of the merits of a photon-counting LDA, namely that the high dimensionality of the image can be handled without preprocessing. Thus, the singularity problem of the Fisher’s LDA encountered in the use of irradiance images can be avoided. By using photon-counting II, we build a compact distortion tolerant recognition system that makes use of the multiple-perspective imaging of II to enhance the recognition performance. Experimental and simulation results are presented to classify out-of-plane rotated objects. The performance is analyzed in terms of mean-squared distance (MSD) between the irradiance images. It is shown that a low level of photons is sufficient in the proposed technique.

© 2007 Optical Society of America

## 1. Introduction

1. A. Mahalanobis and F. Goudail, “Methods for automatic target recognition by use of electro-optic sensors: introduction to the feature issue,” Appl. Opt. **43**,207–209 (2004). [CrossRef]

4. B. Javidi, ed., *Image Recognition and Classification: Algorithms, Systems, and Applications* (Marcel Dekker, New York, 2002). [CrossRef]

18. J.-S. Jang and B. Javidi, “Time-multiplexed integral imaging for 3D sensing and display,” Optics and Photonics News**15**,36–43 (2004). http://www.osa-opn.org/abstract.cfm?URI=OPN-15-4-36.

12. O. Matoba, E. Tajahuerce, and B. Javidi, “Real-time three-dimensional object recognition with multiple perspectives imaging,” Appl. Opt. **40**,3318–3325 (2001). [CrossRef]

21. E. Hecht, *Optics 4 ^{th} ed*. (Addison Wesley, 2001). [PubMed]

23. G. M. Morris, “Scene matching using photon-limited images,” J. Opt. Soc. Am. A. **1**,482–488 (1984). [CrossRef]

24. G. M. Morris, “Image correlation at low light levels: a computer simulation,” Appl. Opt. **23**,3152–3159 (1984). [CrossRef] [PubMed]

25. E. A. Watson and G. M. Morris, “Comparison of infrared upconversion methods for photon-limited imaging,” J. Appl. Phys. **67**,6075–6084 (1990). [CrossRef]

26. E. A. Watson and G. M. Morris, “Imaging thermal objects with photon-counting detector,” Appl. Opt. **31**,4751–4757 (1992). [CrossRef] [PubMed]

27. M. N. Wernick and G. M. Morris, “Image classification at low light levels” J. Opt. Soc. Am. A. **3**,2179–2187 (1986). [CrossRef]

28. L. A. Saaf and G. M. Morris, “Photon-limited image classification with a feedforward neural network,” Appl. Opt. **34**,3963–3970 (1995). [CrossRef] [PubMed]

29. D. Stucki, G. Ribordy, A. Stefanov, H. Zbinden, J. G. Rarity, and T. Wall, “Photon counting for quantum key distribution with Peltier cooled InGaAs/InP APDs,” J. Mod. Opt. **48**,1967–1981 (2001). [CrossRef]

33. M. Guillaume, P. Melon, and P. Refregier, “Maximum-likelihood estimation of an astronomical image from a sequence at low photon levels,” J. Opt. Soc. Am. A. **15**,2841–2848 (1998). [CrossRef]

34. K. E. Timmermann and R. D. Nowak, “Multiscale modeling and estimation of Poisson processes with application to photon-limited imaging,” IEEE Trans. Infor. Theor. **45**,846–862 (1999). [CrossRef]

35. Ph. Refregier, F. Goudail, and G. Delyon, “Photon noise effect on detection in coherent active images,” Opt. Lett. **29**,162–164 (2004). [CrossRef] [PubMed]

16. S. Yeom, B. Javidi, and E. Watson, “Photon counting passive 3D image sensing for automatic target recognition,” Opt. Express **13**,9310–9330 (2005). [CrossRef] [PubMed]

10. D. L. Swets and J. Weng, “Using discriminant eigenfeatures for image retrieval,” IEEE Trans. Pattern. Anal. Mach. Intell. **18**,831–836 (1996). [CrossRef]

15. S. Yeom and B. Javidi, “Three-dimensional distortion tolerant object recognition using integral imaging,” Opt. Express **12**,5795–5809 (2004). [CrossRef] [PubMed]

7. R. O. Duda, P. E. Hart, and D. G. Stork, *Pattern Classification 2 ^{nd} ed*. (Wiley Interscience, New York, 2001). [PubMed]

36. K-R. Muller, S. Mika, G. Ratsch, K. Tsuda, and B. Scholkopf, “An introduction to kernel-based learning algorithms,” IEEE Trans. Neural Networks **12**,181–201 (2001). [CrossRef]

37. A. Ruiz and P. E. Lopez-de-Teruel, “Nonlinear kernel-based statistical pattern analysis,” IEEE Trans. Neural Networks **12**,16–32 (2001). [CrossRef]

*the curse of dimensionality*(the required number of training data sets increases exponentially as the dimension increases). However, this notion becomes problematic not in the photo-counting LDA which is the parametric learning system, but in the non-parametric learning system [7

7. R. O. Duda, P. E. Hart, and D. G. Stork, *Pattern Classification 2 ^{nd} ed*. (Wiley Interscience, New York, 2001). [PubMed]

8. K. Fukunaga, *Introduction to Statistical Pattern Recognition 2 ^{nd} ed*. (Academic Press, Boston, 1990). [PubMed]

## 2. Overview of integral imaging

12. O. Matoba, E. Tajahuerce, and B. Javidi, “Real-time three-dimensional object recognition with multiple perspectives imaging,” Appl. Opt. **40**,3318–3325 (2001). [CrossRef]

## 3. Photon-counting detection model

*y*photons in a time interval

*τ*can be shown to be Poisson distributed [21

21. E. Hecht, *Optics 4 ^{th} ed*. (Addison Wesley, 2001). [PubMed]

*y*is the number of photon-counts produced by a detector during a time interval

*τ*; and

*a*is a rate parameter. The rate parameter can be given by

*η*is the quantum efficiency of the detection process;

*P*is the optical power incident on the detector;

_{o}*h*is Plank’s constant; and

*ν*̄ is the mean frequency of the quasi-monochromatic light source. The mean of photo-counts

*n*is given by [38

_{p}38. A. Papoulis, *Probability, Random Variables, and Stochastic Processes 3 ^{rd} ed*. (McGraw-Hill, Inc. 1991). [PubMed]

*E*(·) denotes the expectation operator.

16. S. Yeom, B. Javidi, and E. Watson, “Photon counting passive 3D image sensing for automatic target recognition,” Opt. Express **13**,9310–9330 (2005). [CrossRef] [PubMed]

*i*can be given by

*y*is the number of photons detected at pixel

_{i}*i*. The parameter

*n*(

_{p}*i*) is given by

*N*is an expected number of photon-counts in the scene;

_{P}*x*is normalized irradiance at NT pixel

_{i}*i*such that

*N*=

_{T}*N*×

_{x}*N*; and

_{y}*N*and

_{x}*N*are the size of the image in the

_{y}*x*and

*y*directions, respectively.

**y**with the number of photon-counts of all pixels as

**y**follows independent Poisson distribution in Eq. (4); and the superscript

*t*denotes transpose. The conditional mean vector and conditional covariance matrix of

**y**given

**x**, respectively become

*diag*(·) denotes the diagonal matrix operator.

## 4. Classification of photon-limited images

### 4.1 Overview of Fisher’s LDA

**x**∈

**R**

^{d×1}, where

**R**

^{d×1}is

*d*-dimensional Euclidean space; and

*d*is the same with the number of pixels (

*N*). The Fisher’s LDA maximizes the ratio of determinant of between-class covariance matrix to determinant of within-class covariance matrix. The within-class covariance is a measure of the concentration of each class, which is defined as

_{T}*w*represents the event that the random vector

_{j}**x**is a member of class

*j*; and

**μ**

_{x|wj}is the class-conditional mean vector of

**x**:

**μ**

*is the mean vector of*

_{x}**x**:

_{xx}=∑

*+∑*

^{W}_{xx}*, where ∑*

^{B}_{xx}*is the covariance matrix of*

_{xx}**x**:

*W*∈

_{F}*R*

^{d×r}satisfies the following Fisher’s criterion:

*W*are eigenvectors of (∑

_{F}*)*

^{W}_{xx}^{-1}∑

*corresponding to non-zero eigenvalues of (∑*

^{B}_{xx}*)*

^{W}_{xx}^{-1}∑

*. It is noted that rank(∑*

^{B}_{xx}*)≤min[*

_{B}*n*-1,

_{c}*d*]. Therefore

*W*is composed of at most

_{F}*n*-1 orthogonal vectors when

_{c}*d*is larger than

*n*-1. The maximum value of Eq. (15), |

_{c}*W*∑

^{t}_{F}*|/|*

^{B}_{xx}W_{F}*W*∑

^{t}_{F}*| is equal to the summation of non-zero eigenvalues of (∑*

^{W}_{xx}W_{F}*)*

^{W}_{xx}^{-1}∑

*.*

^{B}_{xx}### 4.2 Photon-counting LDA

**y**is composed of numbers of photons detected. Let one realization of the random vector

**y**be photon events at all pixels as in Eq. (6). Each component of

**y**follows independent Poisson distribution. We find the following relationships of the first and the second moments between the irradiance random vector

**x**and the photon event random vector

**y**[See Appendix A]:

**y**are, respectively derived as [See Appendix A]:

*w*maximizes the following criterion:

_{P}*W*are eigenvectors of (∑

_{F}*)*

^{W}_{yy}^{-1}∑

*corresponding to non-zero eigenvalues of (∑*

^{B}_{yy}*)*

^{W}_{yy}^{-1}∑

*. The rank of ∑*

^{B}_{yy}*is the same with that of ∑*

^{B}_{yy}*. The maximum value of Eq. (21), |*

^{B}_{xx}*W*∑

^{t}_{p}*|/|*

^{B}_{yy}W_{p}*W*∑

^{t}_{p}*| is equal to the summation of non-zero eigenvalues of {*

^{W}_{yy}W_{p}*W*[diag(

^{t}_{p}**μ**

*)/*

_{x}*N*+∑

_{p}*]*

^{W}_{xx}*W*}

_{p}^{-1}∑

*. Also note that as*

^{B}_{xx}*N*becomes large, Eq. (21) becomes the same as Eq. (15).

_{p}### 4.3 Parameter estimation

**x**is available (true for many of the practical classification examples we encounter), needed parameters should be estimated in a proper way. In the following, the conventional estimators of the within-class covariance matrix and the between-class covariance matrix of the Fisher’s LDA are described, and then they are applied to the photon-counting LDA.

*, which is derived in Appendix B:*

^{W}_{xx}**x**

*(*

_{j}*n*) is the

*n*-th realization of the object that belongs to class

*j*;

*n*is the number of classes;

_{c}*n*is the number of realizations in class

_{j}*j*; and

*n*is the total number of realizations used to train the classifier, i.e.,

_{t}**μ**̂

_{x|wj}is the class-conditional sample mean vector for class

*j*:

*[See Appendix B]:*

^{B}_{xx}**μ**̂

*. is the sample mean vector of*

_{x}**x**:

*n*is usually smaller than the dimension

_{t}*d*of the image. Therefore,

*is a singular matrix since rank(*

^{W}_{xx}*)≤min[*

^{W}_{xx}*n*-

_{t}*n*,

_{c}*d*]. To overcome the singularity problem of

*, dimension reduction techniques such as principal component analysis (PCA) can be applied prior to the Fisher’s LDA [10*

^{W}_{xx}10. D. L. Swets and J. Weng, “Using discriminant eigenfeatures for image retrieval,” IEEE Trans. Pattern. Anal. Mach. Intell. **18**,831–836 (1996). [CrossRef]

11. P. N. Belhumer, J. P. Hespanha, and D. J. Kriegman, “Eigenfaces vs. Fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pattern. Anal. Mach. Intell. **19**,711–720 (1997). [CrossRef]

15. S. Yeom and B. Javidi, “Three-dimensional distortion tolerant object recognition using integral imaging,” Opt. Express **12**,5795–5809 (2004). [CrossRef] [PubMed]

39. Y. Cheng, Y. Zhuang, and J. Yang, “Optimal Fisher discriminant analysis using the rank decomposition,” Patt. Recog. **25**,101–111 (1992). [CrossRef]

*is nonsingular with the non-zero components of*

^{W}_{yy}**μ**̂

*, therefore no additional process is required to avoid the singularity problem.*

_{x}_{xx|wj}in Appendix B may not be a proper estimator when

*n*≤

_{j}*d*since

_{xx|wj}is not a positive definite matrix which is the nature of the covariance matrix of any nonsingular random vector

**x**. In the literature, there have been efforts to estimate the covariance matrix with a small sample (available training data) size and a large scale (dimension) [40

40. J. Schäfer and K. Strimmer, “A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics,” Statistical applications in genetics and molecular biology ,**4**, 32.1–30 (2005). [CrossRef]

*in Eq. (27) is more governed by the diagonal components since it is anticipated that the covariance between the irradiance values of two different components (pixels) is much smaller than the mean values and closer to zero when they are further apart. Therefore, one may postulate that the following approximation holds:*

^{W}_{yy}*x*

_{j,i}(

*n*) and

*μ*Į

_{x|wj,i}. are the

*i*-th component of the

*n*-th training vector

**x**

*(*

_{j}*n*) and the class-conditional sample mean vector

**μ**̂

_{x|wj}, respectively, that is, ∑

*in Eq. (28) is approximated by merely adopting the diagonal components of*

^{W}_{yy}*in Eq. (27). It is noted that the computational complexity of the inversion of Eq. (28) is significantly lower than that of Eq. (27) when*

^{W}_{yy}*d*is large. In the experiments described in Section 6 we compare the performance of the photon-counting LDA between using Eq. (28) and Eq. (27).

## 5. Decision rule and performance evaluation

**x**and

**y**, respectively are the irradiance and the photon event vectors corresponding to one elemental image. Therefore, multiple-perspective training data of a specific orientation can be constituted by a single exposure. During the test, the multiple photon event vectors are used to take advantage of the multiple perspective imaging, thus, the test vector for an unknown input scene is

**y**

*(*

_{test}*n*) is the photon event vector corresponding to the

*n*-th photon-limited elemental image and

*n*is the number of elemental images tested. The images used for testing are gathered from unknown rotations of object.

_{test}**z**

*as the member of class*

_{test}*j*̂ if

**μ**̂

_{z|wj}is the estimate of the class-conditional mean vector. Assuming the distribution of

**y**

*is of the same as the distribution of the images*

_{test}**y**used for training, we can show that

**x**

*is the irradiance random vector for training representing class*

_{j}*j*;

**x**

*(*

_{j}*n*) is the

*n*-th realization of

**x**

*;*

_{j}**x**

*(*

_{test}*s*,

*i*) is the irradiance random vector for testing labeled as class s and rotation angle

_{s}*i*; and

_{s}**x**

*(*

_{test}*m*;

*s*,

*i*) is the

_{s}*m*-th realization of

**x**

*(*

_{test}*s*,

*i*). In the experiments, the rotation angle for training is fixed. It is noted that the irradiance vectors in Eq. (36) are the normalized elemental images used for training and testing due to the normalized condition of the irradiance image in Eq. (5).

_{s}## 6. Experimental and simulation results

### 6.1 Integral imaging acquisition and preprocessing

*mm*, the focal length of the imaging lens is 50

*mm*, and the

*f*-number of the imaging lens is 2.5. The imaging lens is placed between the lenslet array and the CCD camera due to the short focal length of the lenslets. Three classes of toy cars are used in the experiments as shown in Fig. 4. The size of three toy cars is about 2.5 cm×2.5 cm×4.5 cm. The distance between the CCD camera and the imaging lens is about 7.2 cm, and the distance between the micro-lenslet array and the imaging lens is about 2.9 cm. Integral images of the toy cars are gathered at rotation angles of 30°, 33°, 36°, 39°, 42°, and 45°. Rotation is with respect to the perpendicular to the optical axis of the micro-lenslet array. Thus, six integral images for each toy car are obtained; one at each of the six different out-of-plane rotation angles. Captured irradiance images are the same ones in [15

15. S. Yeom and B. Javidi, “Three-dimensional distortion tolerant object recognition using integral imaging,” Opt. Express **12**,5795–5809 (2004). [CrossRef] [PubMed]

**12**,5795–5809 (2004). [CrossRef] [PubMed]

*d*) of the vectors

**x**and

**y**is 7500 (=60×125). The sizes of the integral image in the row and the column directions are 300 (=60×5) and 750 (=125×6), respectively. Figure 5 shows the movies of 6 integral images (frames) for each toy car with the out-of-plane rotation. In a practical situation, photon-limited elemental images can be aligned and cropped by a pre-process like non-linear correlation filtering of photon events [16

16. S. Yeom, B. Javidi, and E. Watson, “Photon counting passive 3D image sensing for automatic target recognition,” Opt. Express **13**,9310–9330 (2005). [CrossRef] [PubMed]

### 6.2 Classification results and performance analysis

*n*) associated with training from each integral image is 30. The number of classes (

_{j}*n*) is 3 so the total number of vectors (

_{c}*n*) used in training the estimator is 90 (=30×3).

_{t}*N*= 3 for each elemental image normalized. The number of test elemental images,

_{p}*n*in Eqs. (30) and (36) is 30 since each integral image is composed of 30 elemental images. Therefore, the mean number of photon-counts (

_{test}*N*×

_{p}*n*) in the entire scene (integral image) is 90 (=3×30). The averaged classification results are illustrated in Fig. 6.

_{test}*N*= 5 and 10 when the corresponding mean numbers of photon-counts are 150 and 300, respectively. The averaged classification results are presented in Figs. 7 and 8, respectively. As illustrated in Figs. 6 to 8, a low level of photons can classify the distorted objects. The averaged correct classification rates increase when a larger number of photons are used while the averaged false classification rates decrease. Figure 9 shows an example of the test input scene (photon-limited integral image) with

_{p}*N*= 10. Figure 9 is generated from the integral image of the class 1 with the rotation angle 30° which corresponds to the first frame of the movie in Fig. 5(a). The actual number of photons in Fig. 9 is 289.

_{p}### 6.3 Classification results with the average irradiance variation

*N*= 10 over 1000 runs, that is, the mean photon number in the test input scene is 300. Figure 12 shows the averaged classification results and Fig. 13 shows the MSD between the elemental images used for training and the elemental images used for testing. As shown in Fig. 12, the performance is degraded compared with Fig. 8. The smaller MSD presented in Fig. 13 than in Fig. 10 may cause the performance degradation. Figure 14 shows the classification results with the approximated within-class covariance estimator in Eq. (28), showing very similar results with Fig. 12.

_{p}*N*= 3, 15, 30, 150, and 300, respectively. As illustrated in Fig. 15, when the mean photon number is 900, reliable classification results (more than about 0.95 for the correct classification rate and less than about 0.05 for the false classification rate) are obtained for all the classes. It is noted that the mean photon number 900 in the entire scene can be considered a very low level of photons (900 photons at an average wavelength of 4000nm represents an energy of less than 10

_{p}^{-16}J incident on the focal plane). The experimental results with the approximated within-class covariance matrix estimator in Eq. (28) are shown in Fig. 16 showing very similar results with Fig. 15. From Figs. 15 and 16, the photon-counting LDA is shown to overcome the unknown average irradiance problem with a moderate number of photon-counts. It can be thought that with a considerably large expected photon number

*N*, the photon-counting LDA criterion in Eq. (21) becomes closer to the Fisher’s criterion in Eq. (15), and the effect of the average irradiance variation can be approximately compensated by the ratio of the between-class covariance matrix to the within-class covariance matrix in Eq. (15). However, because the class-condition mean vectors in the decision rule in Eq. (31) are estimated without the average irradiance variation, the performance can be degraded as shown in Figs. 15 and 16.

_{p}## 7. Conclusions

## Appendix A

**y**in Eq. (16) as follows:

**μ**

_{y|x}=

*N*

_{p}**x**in Eq. (7). Similarly, for the class-conditional mean vector of

**y**, it can be shown that

**y**in Eq. (17) can be proved as follows:

**x**is given. It is noted that, for the deterministic value of

**x**, Eqs. (7) and (8) are the same with Eqs. (A1) and (A3), respectively since

**μ**

*=*

_{x}**x**and ∑

*= 0. Similarly, for the class-conditional covariance matrix of*

_{xx}**y**, we can show that

**y**in Eq. (18) can be derived as follows:

**μ**

*=*

_{x}*E*(

_{wj}**μ**

_{x|wj}) and ∑

*=*

^{W}_{xx}*E*(∑

_{wj}_{xx|wj}).

**y**in Eq. (19), we can prove that

## Appendix B

**x**by the sample mean vector as

**x**can be estimated by the class-conditional sample covariance matrix as

*j*[41–43].

**x**, we can show that

*π*is the probability mass function of the event

_{j}*W*. which is the rate of observations of class

_{j}*j*:

*is*

^{W}_{xx}**x**, the following relationships hold, respectively:

**μ**

*and ∑*

_{x}*can be, respectively derived by Eqs. (B1), (B4), (B6) and (B7), and the invariant property of MLE as*

^{B}_{xx}## Acknowledgments

## References and links

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38. | A. Papoulis, |

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**OCIS Codes**

(000.0000) General : General

(030.5260) Coherence and statistical optics : Photon counting

(100.5010) Image processing : Pattern recognition

(100.5760) Image processing : Rotation-invariant pattern recognition

(100.6890) Image processing : Three-dimensional image processing

(110.6880) Imaging systems : Three-dimensional image acquisition

**ToC Category:**

Image Processing

**History**

Original Manuscript: November 15, 2006

Revised Manuscript: January 18, 2007

Manuscript Accepted: January 23, 2007

Published: February 19, 2007

**Citation**

Seokwon Yeom, Bahram Javidi, and Edward Watson, "Three-dimensional distortion-tolerant object recognition using photon-counting integral imaging," Opt. Express **15**, 1513-1533 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-4-1513

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