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Acquiring linear subspaces for face recognition under variable lighting

Kuang-chih LeeJ. HoDavid Kriegman

article2005TPAMI2,495 citations

Shows that low-dimensional linear subspaces for face recognition under variable lighting can be constructed directly from five to nine real images taken under specific point-source directions, eliminating the need for 3D reconstruction or large training datasets.

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Lighting variation poses a central challenge for robust face recognition systems, as images of the same face can differ dramatically depending on the number, direction, and intensity of light sources. Prior theoretical work established that these variations can be modeled by low-dimensional linear subspaces, including the illumination cone and its nine-dimensional harmonic approximation, yet computing or acquiring the required basis images has typically demanded either dense image sets, 3D surface reconstruction, or physically unrealizable lighting.

The article set out to determine whether small numbers of physically realizable single distant light sources could be chosen so that the images they produce span a subspace that closely approximates the illumination cone and supports accurate recognition.

The authors developed two greedy selection algorithms that operate on a large set of candidate directions sampled uniformly over the sphere or hemisphere. One algorithm maximizes a similarity measure based on principal angles between the candidate subspace and the harmonic subspace; the second additionally maximizes the solid angle of the intersection between the candidate subspace and the illumination cone. Both algorithms were run on 3D face models from the Yale Face Database B to produce nested sequences of subspaces from one to nine dimensions. The resulting per-person configurations proved qualitatively similar, allowing a single “universal” configuration of nine directions to be computed by averaging the objective function across individuals.

When subspaces spanned by images taken under the universal configuration were used for recognition on the Yale Face Database B, error rates fell to zero with nine images and remained below 1 percent with as few as five images. On the larger extended Yale set the five-dimensional universal subspace produced a 0.2 percent error rate, an order of magnitude better than the median performance of 16,000 randomly chosen five-light configurations. Comparable results held on the CMU PIE database. Subspaces of dimension five and higher performed well even when test images contained strong ambient illumination, and performance improved further as additional ambient sources were introduced.

These findings indicate that carefully chosen sets of five to nine real images can replace both dense training collections and intermediate 3D reconstruction steps, thereby lowering the cost and complexity of enrolling individuals in controlled environments such as security checkpoints or licensing offices. The approach also suggests that, when lighting statistics are known in advance, even fewer training images may suffice.

The principal limitations are the assumption of approximately Lambertian reflectance, the use of convex-face models that omit some cast-shadow effects, and reliance on greedy rather than exhaustive search for the optimal directions. Nevertheless, the consistency of the discovered configurations across subjects and databases, together with the large performance gap relative to random selections, supports high confidence that the reported universal configurations are effective and practical for real-world face recognition under variable lighting.

University of California, San DiegoUniversity of FloridaUniversity of Illinois Urbana-Champaignlighting conditionsfacial imagesfeature spaceface recognitionlinear subspacesvariable lighting
Lee et al (2005).pdf
  • Paper: Robust Face Recognition via Sparse Representation, John Wright et al. (2009). This work extends linear subspace modeling for face recognition by formulating sparse representation over illumination dictionaries to achieve robustness against extreme lighting and occlusion.
  • Paper: Face recognition using Laplacianfaces, Xiaofei He et al. (2005). This paper advances linear subspace face recognition under lighting variations by preserving local manifold geometry through locality preserving projections.
  • Paper: Incremental Learning for Robust Visual Tracking, David A. Ross et al. (2008). It applies the concept of low-dimensional appearance subspaces dynamically by incrementally updating the linear basis to track objects under continuous lighting changes.
  • Paper: Deep Retinex Decomposition for Low-Light Enhancement, Chen Wei et al. (2018). This work generalizes physical lighting decomposition by using deep Retinex networks to separate illumination and reflectance components under severe low-light conditions.
Cover for Acquiring linear subspaces for face recognition under variable lighting

Abstract

Previous work has demonstrated that the image variation of many objects (human faces in particular) under variable lighting can be effectively modeled by low-dimensional linear spaces, even when there are multiple light sources and shadowing. Basis images spanning this space are usually obtained in one of three ways: A large set of images of the object under different lighting conditions is acquired, and principal component analysis (PCA) is used to estimate a subspace. Alternatively, synthetic images are rendered from a 3D model (perhaps reconstructed from images) under point sources and, again, PCA is used to estimate a subspace. Finally, images rendered from a 3D model under diffuse lighting based on spherical harmonics are directly used as basis images. In this paper, we show how to arrange physical lighting so that the acquired images of each object can be directly used as the basis vectors of a low-dimensional linear space and that this subspace is close to those acquired by the other methods. More specifically, there exist configurations of k point light source directions, with k typically ranging from 5 to 9, such that, by taking k images of an object under these single sources, the resulting subspace is an effective representation for recognition under a wide range of lighting conditions. Since the subspace is generated directly from real images, potentially complex and/or brittle intermediate steps such as 3D reconstruction can be completely avoided; nor is it necessary to acquire large numbers of training images or to physically construct complex diffuse (harmonic) light fields. We validate the use of subspaces constructed in this fashion within the context of face recognition.

Topics

lighting conditionsfacial imagesfeature space

Keywords

face recognitionlinear subspacesvariable lightingillumination conesbasis imagesprincipal component analysislow-dimensional linear approximation

Table of Contents

  • 2 PRELIMINARIES
  • 2.1 Illumination Cone
  • 2.2 Lambertian Reflection and Spherical Harmonics
  • 2.3 Harmonic Images
  • 2.4 Motivations
  • 3 LOW-DIMENSIONAL LINEAR APPROXIMATIONS OF ILLUMINATION CONE
  • 3.1 Computing Linear Subspace R
  • 3.2 Preliminary Experiments
  • 3.3 An Explicit Calculation
  • 3.4 Maximizing the Intersection Volume
  • 3.5 Discussion and Results
  • 3.6 Computing a Universal Configuration
  • 4 EXPERIMENTS AND RESULTS
  • 4.1 Recognition Experiments with Nine Points of Light
  • 4.2 Recognition with Lower Dimensional Subspaces
  • 4.3 Recognition Experiments with Randomly Generated Lighting Configurations
  • 4.4 Recognition with Ambient Lighting
  • 5 CONCLUSIONS
  • ACKNOWLEDGMENTS
  • REFERENCES

Knowls

  1. Knowl 1 — Universal Configuration of Lighting Directions for Face Recognition

    model/method

    The universal configuration consists of a fixed set of kk single distant isotropic point light source directions chosen such that the kk-dimensional linear subspace spanned by images taken under these directions serves as an accurate, subject-independent approximation to the illumination cone of any human face.

    In spherical coordinates (θ,ϕ)(\theta, \phi)—where θ∈[0∘,180∘]\theta \in [0^\circ, 180^\circ] represents the elevation angle from the optical/normal zz-axis (with θ=0∘\theta = 0^\circ corresponding to frontal illumination), and ϕ∈[−180∘,180∘]\phi \in [-180^\circ, 180^\circ] denotes the azimuth angle around the image plane—the universal configuration of 9 lighting directions computed across frontal hemisphere samples is:

    {(0∘,0∘),(68∘,−90∘),(74∘,108∘),(80∘,52∘),(85∘,−42∘),(85∘,−137∘),(85∘,146∘),(85∘,−4∘),(51∘,67∘)}\{(0^\circ, 0^\circ), (68^\circ, -90^\circ), (74^\circ, 108^\circ), (80^\circ, 52^\circ), (85^\circ, -42^\circ), (85^\circ, -137^\circ), (85^\circ, 146^\circ), (85^\circ, -4^\circ), (51^\circ, 67^\circ)\}

    Geometrically, this configuration exhibits a characteristic structure:

    1. One to two direct or near-frontal illumination directions (θ≈0∘\theta \approx 0^\circ to 50∘50^\circ), which align closely with the 9-dimensional harmonic subspace.
    2. Four to five lateral illumination directions (θ≈70∘\theta \approx 70^\circ to 85∘85^\circ) distributed quasi-uniformly around the lateral rim, which maximize the solid angle (volume) spanned by the subspace within the illumination cone.

    Acquiring kk images of an individual under the first kk directions of this sequence directly provides a kk-dimensional linear basis for face recognition without requiring 3D surface reconstruction, albedo estimation, or training image decomposition.

  2. Knowl 2 — Multi-Model Greedy Optimization for Universal Lighting Configuration

    algorithm

    To identify a subject-independent sequence of dd lighting directions {x1,x2,…,xd}\{x_1, x_2, \dots, x_d\} from a discrete set of candidate directions Ω⊂S2\Omega \subset S^2, an optimization criterion averages the ratio of subspace distance to harmonic plane distance across a collection of MM 3D face models.

    Input: Discrete set of candidate lighting directions Ω⊂S2\Omega \subset S^2, target dimension dd, training face models m∈{1,…,M}m \in \{1, \dots, M\} with precomputed 9D harmonic subspaces HmH^m and rendered single-source images xmx^m for each x∈Ωx \in \Omega
    Output: Sequence of lighting directions x1,…,xdx_1, \dots, x_d and nested subspaces RimR_i^m for each model mm
    Ω0←Ω\Omega_0 \leftarrow \Omega
    for each model m∈{1,…,M}m \in \{1, \dots, M\} do
        R0m←{0}R_0^m \leftarrow \{0\}
    end for
    for i=1i = 1 to dd do
        xi←arg⁡max⁡x∈Ωi−1∑m=1Mdist(xm,Ri−1m)dist(xm,Hm)x_i \leftarrow \arg\max_{x \in \Omega_{i-1}} \sum_{m=1}^M \frac{\text{dist}(x^m, R_{i-1}^m)}{\text{dist}(x^m, H^m)}
        Ωi←Ωi−1∖{xi}\Omega_i \leftarrow \Omega_{i-1} \setminus \{x_i\}
        for each model m∈{1,…,M}m \in \{1, \dots, M\} do
            Rim←Ri−1m⊕span(xim)R_i^m \leftarrow R_{i-1}^m \oplus \text{span}(x_i^m)
        end for
    end for
    return x1,…,xdx_1, \dots, x_d

    Here, dist(xm,S)\text{dist}(x^m, S) denotes the L2L^2 Euclidean distance from the image vector xm∈Rnx^m \in \mathbb{R}^n to a linear subspace S⊂RnS \subset \mathbb{R}^n. For the initial step i=1i=1, dist(xm,R0m)≡1\text{dist}(x^m, R_0^m) \equiv 1, selecting the ray closest to the harmonic subspace HmH^m. The numerator dist(xm,Ri−1m)\text{dist}(x^m, R_{i-1}^m) maximizes the volume (orthogonality/solid angle) spanned by the new basis image relative to existing ones, while the denominator dist(xm,Hm)\text{dist}(x^m, H^m) prevents the subspace from deviating away from the harmonic plane.

  3. Knowl 3 — Face Recognition Error Rates on Subsets of the Yale Face Database B

    data/table

    Face recognition performance was evaluated on the Yale Face Database B (10 subjects, 450 total images under 45 lighting conditions per subject divided into four subsets of increasing lighting angle relative to the camera axis: Subset 1 & 2 (0∘0^\circ–25∘25^\circ), Subset 3 (25∘25^\circ–50∘50^\circ), and Subset 4 (50∘50^\circ–77∘77^\circ)). Recognition is performed by projecting each test image onto the candidate linear subspaces or calculating nearest distances in image space, assigning identity based on the minimal L2L^2 residual distance.

    The Nine Points of Light (9PL) method with real basis images achieves a 0.0%0.0\% error rate across all lighting conditions, matching the top-performing Illumination Cones method (with cast shadows) while eliminating the need for 3D model reconstruction or offline eigenvalue decomposition.

    Method Error Rate (%) vs. Illumination
    Subset 12 Subset 3 Subset 4 Total
    Correlation 0.0 23.3 73.6 29.1
    Eigenfaces 0.0 25.8 75.7 30.4
    Eigenfaces w/o 1st 3 0.0 19.2 66.4 25.8
    Nearest Neighbor (9 images) 13.8 54.6 7.0 22.6
    Linear Subspace (3D) 0.0 0.0 15.0 4.6
    Cones-attached 0.0 0.0 8.6 2.7
    9PL (simulated images) 0.0 0.0 2.8 0.87
    Harmonic Images (no cast shadow) 0.0 0.0 3.571 1.1
    Harmonic Images-cast (with cast shadows) 0.0 0.0 2.7 0.85
    Gradient Angle 0.0 0.0 1.4 0.44
    Cones-cast 0.0 0.0 0.0 0.0
    9PL (real images) 0.0 0.0 0.0 0.0

    The Nearest Neighbor baseline using the same 9 images demonstrates that forming the 9D subspace is essential: Nearest Neighbor fails to generalize to frontal and intermediate subsets (yielding a 54.6%54.6\% error rate on Subset 3) because individual sample points cannot capture the continuous variation of illumination.

  4. Knowl 4 — Recognition Performance Across Varying Subspace Dimensions on Extended Yale and CMU PIE

    empirical result

    Using nested linear subspaces R1⊂R2⊂⋯⊂Rk⊂⋯⊂RdR_1 \subset R_2 \subset \dots \subset R_k \subset \dots \subset R_d generated by the initial kk lighting directions of the universal configuration, face recognition error rates decrease sharply as subspace dimension kk increases:

    1. On the Extended Yale Face Database (1,710 images of 38 individuals under 45 lighting conditions), error rates decline from approximately 90%90\% at dimension k=1k=1 (single frontal image) to under 1%1\% at k=5k=5 (specifically 0.2%0.2\% error, or 4 misclassifications out of 1,710 images), and reach near 0.0%0.0\% for dimensions k=6k=6 through 99.
    2. On the CMU PIE Database (1,587 images of 69 individuals across 23 lighting conditions), testing with a 7-dimensional subspace formed by the closest available database directions yields a 1.9%1.9\% error rate under directional lighting without background illumination, and a 2.8%2.8\% error rate with ambient background illumination.

    These results establish that a 5-dimensional to 7-dimensional linear subspace spanned by carefully configured real images provides an accurate model of variable lighting on human faces.

  5. Knowl 5 — Linear Independence Criterion for Illumination Cone and Subspace Intersection

    theoretical result

    Let C⊂RnC \subset \mathbb{R}^n be the illumination cone formed by all non-negative linear combinations of extreme rays of a convex Lambertian object under point sources at infinity. Let Ω⊂Rn\Omega \subset \mathbb{R}^n be a finite set of extreme rays, and let R=span{x1,…,xk}R = \text{span}\{x_1, \dots, x_k\} be a kk-dimensional linear subspace spanned by basis rays x1,…,xk∈Ωx_1, \dots, x_k \in \Omega.

    Define the subcone RC⊆C∩RR_C \subseteq C \cap R generated by the non-negative linear combinations of these basis elements:

    RC={x∈R  |  x=∑i=1kαixi,  αi≥0}R_C = \left\{ x \in R \;\middle|\; x = \sum_{i=1}^k \alpha_i x_i, \; \alpha_i \ge 0 \right\}

    If C∩R≠RCC \cap R \neq R_C, then there exists a non-trivial linear relation among the extreme rays in Ω\Omega that contains at least one of the basis elements {x1,…,xk}\{x_1, \dots, x_k\}.

    Because the image space dimension nn is very large (e.g., n>30,000n > 30,000 pixels) relative to the number of sampled rays in Ω\Omega (e.g., ∣Ω∣≈500|\Omega| \approx 500 to 1,0001,000), the extreme rays in Ω\Omega are almost entirely linearly independent (e.g., spanning 497 dimensions out of 500 rays). Consequently, when kk is small (k≤9k \le 9), C∩R=RCC \cap R = R_C holds in practice, and maximizing the unit volume vol(C∩R)\text{vol}(C \cap R) is equivalent to maximizing the solid angle subtended by the cone RCR_C.

  6. Knowl 6 — Optimization of Lighting Directions via Principal Angles with the Harmonic Subspace

    algorithm

    To approximate the 9-dimensional harmonic plane HH using images taken under single directional light sources, the similarity between a candidate kk-dimensional subspace RR and HH is formulated using the principal angles between them.

    Let A∈Rn×kA \in \mathbb{R}^{n \times k} and B∈Rn×9B \in \mathbb{R}^{n \times 9} be matrices whose columns form orthonormal bases for RR and HH, respectively. The singular values σ1,…,σk\sigma_1, \dots, \sigma_k of BTAB^T A represent the cosines of the principal angles between RR and HH. The subspace similarity is defined as:

    Sim(H,R)=∑i=1kσi2\text{Sim}(H, R) = \sum_{i=1}^k \sigma_i^2

    where 0≤σi≤10 \le \sigma_i \le 1, and Sim(H,R)=k\text{Sim}(H, R) = k if and only if R⊆HR \subseteq H.

    A greedy algorithm incrementally constructs nested subspaces R1⊂R2⊂⋯⊂R9R_1 \subset R_2 \subset \dots \subset R_9 from a candidate set Ω⊂S2\Omega \subset S^2:

    Input: Finite set of sampled lighting directions Ω⊂S2\Omega \subset S^2, 9D harmonic subspace HH
    Output: Nested sequence of subspaces R1⊂⋯⊂R9R_1 \subset \dots \subset R_9 and basis rays x1,…,x9x_1, \dots, x_9
    Ω0←Ω\Omega_0 \leftarrow \Omega
    R0←{0}R_0 \leftarrow \{0\}
    for i=1i = 1 to 9 do
        xi←arg⁡max⁡x∈Ωi−1Sim(H,Ri−1⊕span(x))x_i \leftarrow \arg\max_{x \in \Omega_{i-1}} \text{Sim}(H, R_{i-1} \oplus \text{span}(x))
        Ri←Ri−1⊕span(xi)R_i \leftarrow R_{i-1} \oplus \text{span}(x_i)
        Ωi←Ωi−1∖{xi}\Omega_i \leftarrow \Omega_{i-1} \setminus \{x_i\}
    end for
    return R9,{x1,…,x9}R_9, \{x_1, \dots, x_9\}

    When applied to face models, this procedure selects frontal illumination directions first (θ≈0∘\theta \approx 0^\circ), followed by lateral directions (θ≈90∘\theta \approx 90^\circ) distributed around the rim, a rear direction (θ>90∘\theta > 90^\circ), and a final direction.

  7. Knowl 7 — Universal 5-Light Subspace vs. Random Lighting Subspaces

    empirical result

    On the Extended Yale Face Database (1,710 images of 38 individuals under 45 lighting directions per person), the universal 5-light subspace (R5R_5) was evaluated against 16,000 randomly selected 5-light configurations drawn from the (455)=1,221,759\binom{45}{5} = 1,221,759 possible combinations (using 5 images for training and 40 images for testing per person):

    1. The universal 5-light configuration achieves an error rate of 0.2%0.2\% (only 4 misclassifications out of 1,710 images).
    2. Randomly generated 5-light configurations yield a mean error rate of just under 10%10\% (165 misclassified images) with a median error rate of approximately 4%4\%.
    3. Out of the 16,000 random configurations tested, only 3 configurations (0.01875%0.01875\%) outperformed the universal configuration. All 3 top-performing random configurations shared the exact spatial distribution pattern of the universal configuration: one frontal lighting direction coupled with four near-lateral lighting directions.

  8. Knowl 8 — Effect of Ambient Illumination and Multiple Light Sources on Recognition Complexity

    empirical result

    Face images captured under ambient illumination or multiple simultaneous light sources lie in the interior of the illumination cone, where cast and attached shadows soften, whereas images produced by single isolated point sources lie on the extreme boundary of the illumination cone. Consequently, ambient and multi-source lighting conditions reduce face recognition difficulty:

    1. In synthetic multi-source experiments on the Yale Face Database B, composite testing sequences Lk={l1,…,lk}L^k = \{l_1, \dots, l_k\} were created by averaging kk real images (ILk=1k∑i=1kIliI_{L^k} = \frac{1}{k}\sum_{i=1}^k I_{l_i}) for k=1,…,12k = 1, \dots, 12 across 50 random sequences. For linear subspaces of dimension 5 to 9, the recognition error rate drops to almost 0.0%0.0\% across all k≥1k \ge 1. For a 1D linear subspace trained on only a single frontal image, the error rate drops below 10%10\% once approximately 7 or more light sources are active simultaneously.
    2. On the CMU PIE database with ambient illumination, a 1D subspace trained solely on a single frontal image achieves approximately 50%50\% recognition accuracy across 69 subjects (compared to random chance accuracy of 1.5%1.5\%).

  9. Knowl 9 — Canonical Symmetric 5-Point Lighting Configuration via Exhaustive Search

    model/method

    Direct exhaustive maximization of the subspace similarity metric Sim(R,H)=∑i=15σi2\text{Sim}(R, H) = \sum_{i=1}^5 \sigma_i^2 over all (204)=4,845\binom{20}{4} = 4,845 combinations of five directions containing the frontal direction θ=0∘\theta = 0^\circ (drawn from a 21-point sampled hemisphere IU′IU' with circles at θ=0∘,45∘,90∘,125∘\theta = 0^\circ, 45^\circ, 90^\circ, 125^\circ) identifies an identical, globally optimal 5-point configuration for all 10 subjects in the Yale Face Database B.

    The resulting 5 lighting directions in spherical coordinates (θ,ϕ)(\theta, \phi)—with elevation θ∈[0∘,180∘]\theta \in [0^\circ, 180^\circ] and azimuth ϕ∈[−180∘,180∘]\phi \in [-180^\circ, 180^\circ]—are:

    {(0∘,0∘),(90∘,−60∘),(90∘,−120∘),(90∘,120∘),(90∘,60∘)}\{(0^\circ, 0^\circ), (90^\circ, -60^\circ), (90^\circ, -120^\circ), (90^\circ, 120^\circ), (90^\circ, 60^\circ)\}

    This configuration is symmetric with respect to the vertical facial symmetry axis and combines one direct frontal view with four lateral views situated at right angles (θ=90∘\theta = 90^\circ) covering left, right, upper-lateral, and lower-lateral lighting angles.

Coverage note — None was omitted; all key algorithms, theoretical properties, universal light configurations, and empirical benchmarks were covered.

References

  1. 1.Y. Adini, Y. Moses, and S. Ullman, “Face Recognition: The Problem Of Compensating For Changes In Illumination Direction,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 19, no. 7, pp. 721-732, July 1997.
  2. 2.R. Basri and D. Jacobs, “Lambertian Reflectance and Linear Subspaces,” Proc. Int’l Conf. Computer Vision, vol. 2, pp. 383-390, 2001.
  3. 3.P. Belhumeur and D. Kriegman, “What Is the Set of Images of an Object under All Possible Lighting Conditions,” Int’l J. Computer Vision, vol. 28, pp. 245-260, 1998.
  4. 4.H. Chen, P. Belhumeur, and D. Jacobs, “In Search of Illumination Invariants,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1-8, 2000.
  5. 5.R. Epstein, P. Hallinan, and A. Yuille, “5+/-2 Eigenimages Suffice: An Empirical Investigation of Low-Dimensional Lighting Models,” Proc. IEEE Workshop Physics-Based Modeling in Computer Vision, pp. 108-116, 1995.
  6. 6.A. Georghiades, D. Kriegman, and P. Belhumeur, “Illumination Cones for Recognition under Variable Lighting: Faces,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 52-58, 1998.
  7. 7.A. Georghiades, D. Kriegman, and P. Belhumeur, “From Few to Many: Generative Models for Recognition under Variable Pose and Illumination,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 40, pp. 643-660, 2001.
  8. 8.G. Golub and C. vanLoan, Matrix Computation. The John Hopkins Univ. Press 1989.
  9. 9.P. Hallinan, “A Low-Dimensional Representation of Human Faces for Arbitrary Lighting Conditions,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 995-999, 1994.
  10. 10.J. Ho, K. Lee, and D. Kriegman, “On Reducing the Complexity of Illumination Cones for Face Recognition,” Proc. IEEE CVPR Workshop Identifying Objects Across Variations in Lighting: Psychophysics and Computation, pp. 56-63, 2001.
  11. 11.K. Lee, J. Ho, and D. Kriegman, “Nine Points of Lights: Acquiring Subspaces for Face Recognition under Variable Lighting,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 519-526, 2001.
  12. 12.S. Nayar and H. Murase, “Dimensionality of Illumination in Appearance Matching, ” Proc. IEEE Conf. Robotics and Automation, pp. 1326-1332, 1996.
  13. 13.R. Ramamoorthi, “Analytic PCA Construction for Theoretical Analysis of Lighting Variability in Images of a Lambertian Object,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 10, pp. 1-12, Oct. 2002.
  14. 14.R. Ramamoorthi and P. Hanrahan, “An Efficient Representation for Irradiance Environment,” Proc. SIGGRAPH Conf., pp. 497-500, 2001.
  15. 15.R. Ramamoorthi and P. Hanrahan, “A Signal-Processing Framework for Inverse Rendering,” Proc. SIGGRAPH Conf., pp. 117-228, 2001.
  16. 16.Y.Y. Schechner, S.K. Nayar, and P.N. Belhumeur, “A Theory of Multiplexed Illumination,” Proc. Int’l Conf. Computer Vision, pp. 808-815, 2003.
  17. 17.A. Shashua, “On Photometric Issues in 3D Visual Recognition form a Single Image,” Int’l J. Computer Vision, vol. 21, pp. 99-122, 1997.
  18. 18.T. Sim, S. Baker, and M. Bsat, “The CMU Pose, Illumination, and Expression (Pie) Database,” Proc. IEEE Int’l Conf. Automatic Face and Gesture Recognition, pp. 53-58, 2002.
  19. 19.P. Sloan, J. Kautz, and J. Snyder, “Precomputed Radiance Transfer for Real-Time Rendering in Dynamic, Low-Frequency Lighting Environments,” Proc. SIGGRAPH Conf., pp. 527-536, 2002.
  20. 20.W. Strauss, Partial Differential Equations. John Wiley and Sons 1992.
  21. 21.M. Turk and A. Pentland, “Eigenfaces for Recognition,” J. Cognitive Neuroscience, vol. 3, no. 1, pp. 71-96, 1991.
  22. 22.L. Zhao and Y. Yang, “Theoretical Analysis of Illumination in PCA-Based Vision Systems,” Pattern Recognition, vol. 32, no. 4, pp. 547-564, 1999.
  23. 23.W.Y. Zhao and R. Chellappa, “Symmetric Shape-From-Shading Using Self-Ratio Image,” Int’l J. Computer Vision, vol. 45, no. 1, pp. 55-75, 2001.

Citation

MLA
Kuang-Chih Lee, et al. “Acquiring Linear Subspaces for Face Recognition Under Variable Lighting”. IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 27, no. 5, 2005, pp. 684–98, https://doi.org/10.1109/TPAMI.2005.92.
APA
Kuang-Chih Lee, Ho, J., & Kriegman, D. J. (2005). Acquiring linear subspaces for face recognition under variable lighting. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27(5), 684–698. https://doi.org/10.1109/TPAMI.2005.92
Chicago
Kuang-Chih Lee, J. Ho, and D. J. Kriegman. 2005. “Acquiring Linear Subspaces for Face Recognition Under Variable Lighting”. IEEE Transactions on Pattern Analysis and Machine Intelligence 27 (5): 684–98. https://doi.org/10.1109/TPAMI.2005.92.
Harvard
Kuang-Chih Lee, Ho, J. and Kriegman, D.J. (2005) “Acquiring linear subspaces for face recognition under variable lighting”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 27(5), pp. 684–698. Available at: https://doi.org/10.1109/TPAMI.2005.92.
Vancouver
1. Kuang-Chih Lee, Ho J, Kriegman DJ (2005) Acquiring linear subspaces for face recognition under variable lighting. IEEE Transactions on Pattern Analysis and Machine Intelligence 27:684–698

BibTeX

@article{Kuang_Chih_Lee_2005, title={Acquiring linear subspaces for face recognition under variable lighting}, volume={27}, ISSN={0162-8828}, url={http://dx.doi.org/10.1109/TPAMI.2005.92}, DOI={10.1109/tpami.2005.92}, number={5}, journal={IEEE Transactions on Pattern Analysis and Machine Intelligence}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Kuang-Chih Lee and Ho, J. and Kriegman, D.J.}, year={2005}, month=May, pages={684–698} }
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