Abstract - BMVC 2010
"How old are you?" : Age Estimation with Tensors of Binary Gaussian Receptive Maps INRIA Grenoble Rhones-Alpes Research Center and Laboratoire d’Informatique de Grenoble (LIG) 655 avenue de l’Europe 38 334 Saint Ismier Cedex, France
John A. Ruiz-Hernandez [email protected]
James L. Crowley [email protected]
Augustin Lux [email protected]
1
Binary Gaussian Receptive Maps Half-Octave Gaussian Pyramid
Gm k =
x2 +y2 2σ 2
Binary Gaussian Receptive Maps
∗ I(x, y) Image Representation
Relevance Vector Machines (Regression)
Age Estimation
Tensorial Representation (1st Order)
MPCA
Tensorial Representation (2nd Order)
MPCA
Tensorial Representation (3rd Order)
MPCA
Tensorial Representation (4th Order)
MPCA
(B)
(1)
∂ I(x,y,σ ) ∂ k x∂ m y
MPCA
(A)
Half-Octave Gaussian Pyramid
1 2σ π e
Tensorial Representation (all Orders)
Image Representation
Following the terminology in [7], we refer to the use of Gaussian derivatives for images analysis as Gaussian receptive Maps. A local Gaussian scale space is computed as: I(x, y, σ ) = G(x, y, σ ) ∗ I(x, y) =
Binary Gaussian Receptive Maps
Relevance Vector Machines (Regression)
2 5 8 10 14
19 22 28 29 33
Age Estimation
Figure 1: Proposed Configurations for Age Estimation. (a) The tensors Where σ is the size of the support in terms of the second moment are fused into one before applying MPCA (b) MPCA is applied at each (or variance), I is the image and ∗ is the convolution operator. From tensor and then the resulting vectors are fused into one the preceeding equation, the steerable filter [4] reponse for the Gaussian derivative up to fourth order for an arbitrary orientation θ could be defined This representation may be projected to a more compact uncorrelated as follows: vectorial representation, by performing Multilinear Principal Component Analysis (MPCA) [6] over each tensor. Gσ1 (θ ) = cos(θ )G01 + sin(θ )G10
3
Gσ2 (θ ) = cos2 (θ )G02 − 2 sin(θ ) cos(θ )G11 + sin2 (θ )G20
Optimal Configurations for Age Estimation
We have compared two possible algorithms for automatic age estimation using T (BGσ (θ ))n . The first algorithm concatenate the tensors T (BGσ (θ ))1 , T (BGσ (θ ))2 , T (BGσ (θ ))3 and T (BGσ (θ ))4 to form a 4-D tensor TT as shown in the next equation:
Gσ3 (θ ) = cos3 (θ )G03 − 3 sin(θ ) cos2 (θ )G12 +3 sin2 (θ ) cos(θ )G21 − sin3 (θ )G30 Gσ4 (θ ) = cos4 (θ )G04 − 4 cos3 (θ ) sin(θ )G13 + 6 sin2 (θ ) cos2 (θ )G22
TT = [T (BGσ (θ ))n ] n = 1, 2, 3, 4 TT ∈ RNn ×Nθ ×Npos ×Nbins
−4 cos(θ ) sin3 (θ )G31 + sin4 (θ )G40
(5)
(2) MPCA is then applied to this tensor TT to obtain a vector YT ∈ Rm≤M . Then to compute Binary Gaussian Receptive Maps, a Local Binary Pattern (LBP) [2] is applied over each Gaussian derivative to assign a YT = MPCA (TT ) YT ∈ Rm (6) label to each pixel of the image by thresholding the 3 × 3 neighborhood The second algorithm computes MPCA over each tensor separetly to of each pixel with the center pixel value and considering the result as a 2 , y3 and y4 , these vectors are concatenated to form binary or decimal number. The mathematic expression for this operation obtain the vectors y1 , y4×m a single vector Y ∈ R , where m ≤ M. F is: BGσn (θ ) = LBP(Gσn (θ ))
(3)
YF = [yn ] YF ∈ R4×m
n = 1, 2, 3, 4
(7)
Finally, we use Relevance Vector Machines (RVM) [8] rather than Where BGσn (θ ) is the Binary Gaussian Receptive map, n is the order of the Gaussian derivative, σ is the scale factor, θ is the steering angle Support Vector Machines (SVM) as a regression algorithm. and LBP is the Local Binary Pattern. The Gaussian Receptive Maps could be efficiently computed at half- 4 Experimental Evaluation octave scales using a linear complexity O(N) algorithm for computing a We have performed several experiments to compare different approachs Gaussian pyramid [3]. for estimating age from facial images. Two publicly available databases have been used in our experiments: The FG-NET [1] database and the σ 2 Tensorial Representation of BGn (θ ) MORPH [5] database. To compute the Tensorial representation, first we divide each BGσn (θ ) of each image into non-overlapping rectangular sub-regions with a specific size. A set of histograms is then computed for each sub-region and finally each histogram is organized in four different 3-D tensors, where each tensor corresponds to an specific derivative order of the Binary Gaussian Receptive maps. The characteristic equation of T (BGσn (θ )) is shown as follows: T (BGσ (θ ))n ∈ RNθ ×Npos ×Nbins
n = 1, 2, 3, 4
(4)
Where n and Nθ are the order and orientation angles for the gaussian derivatives respectively, N pos is the number of non-overlapping positions in the map and Nbins is the number of bins used in the construction of each local histogram.
[1] The fg-net aging database. http://www.fgnet.rsunit.com/. accessed: April 2010. [2] T. Ahonen, A. Hadid, and M. Pietikainen. Face description with local binary patterns: Application to face recognition. IEEE PAMI, 28(12):2037–2041, 2006. [3] J.L. Crowley and O. Riff. Fast computation of scale normalised gaussian receptive fields. In Proc. Scale Space Methods in Computer Vision, pages 584–598, 2003. [4] W. T. Freeman and E. H. Adelson Y. The design and use of steerable filters. IEEE PAMI, 13:891–906, 1991. [5] Karl Ricanek Jr. and Tamirat Tesafaye. Morph: A longitudinal image database of normal adult age-progression. IEEE FG, 2006. [6] Haiping Lu, Konstantinos N. Plataniotis, and Anastasios N. Venetsanopoulos. Mpca: Multilinear principal component analysis of tensor objects. IEEE TNN, 19(1):18–39, 2008. [7] B. Schiele and J.L. Crowley. Recognition without correspondence using multidimensional receptive field histograms. IJCV, 36:31–50, 2000. [8] Michael E. Tipping. Sparse bayesian learning and the relevance vector machine. Journal of Machine Learning Research, 1:211–244, 2001.
John A. Ruiz-Hernandez [email protected]
James L. Crowley [email protected]
Augustin Lux [email protected]
1
Binary Gaussian Receptive Maps Half-Octave Gaussian Pyramid
Gm k =
x2 +y2 2σ 2
Binary Gaussian Receptive Maps
∗ I(x, y) Image Representation
Relevance Vector Machines (Regression)
Age Estimation
Tensorial Representation (1st Order)
MPCA
Tensorial Representation (2nd Order)
MPCA
Tensorial Representation (3rd Order)
MPCA
Tensorial Representation (4th Order)
MPCA
(B)
(1)
∂ I(x,y,σ ) ∂ k x∂ m y
MPCA
(A)
Half-Octave Gaussian Pyramid
1 2σ π e
Tensorial Representation (all Orders)
Image Representation
Following the terminology in [7], we refer to the use of Gaussian derivatives for images analysis as Gaussian receptive Maps. A local Gaussian scale space is computed as: I(x, y, σ ) = G(x, y, σ ) ∗ I(x, y) =
Binary Gaussian Receptive Maps
Relevance Vector Machines (Regression)
2 5 8 10 14
19 22 28 29 33
Age Estimation
Figure 1: Proposed Configurations for Age Estimation. (a) The tensors Where σ is the size of the support in terms of the second moment are fused into one before applying MPCA (b) MPCA is applied at each (or variance), I is the image and ∗ is the convolution operator. From tensor and then the resulting vectors are fused into one the preceeding equation, the steerable filter [4] reponse for the Gaussian derivative up to fourth order for an arbitrary orientation θ could be defined This representation may be projected to a more compact uncorrelated as follows: vectorial representation, by performing Multilinear Principal Component Analysis (MPCA) [6] over each tensor. Gσ1 (θ ) = cos(θ )G01 + sin(θ )G10
3
Gσ2 (θ ) = cos2 (θ )G02 − 2 sin(θ ) cos(θ )G11 + sin2 (θ )G20
Optimal Configurations for Age Estimation
We have compared two possible algorithms for automatic age estimation using T (BGσ (θ ))n . The first algorithm concatenate the tensors T (BGσ (θ ))1 , T (BGσ (θ ))2 , T (BGσ (θ ))3 and T (BGσ (θ ))4 to form a 4-D tensor TT as shown in the next equation:
Gσ3 (θ ) = cos3 (θ )G03 − 3 sin(θ ) cos2 (θ )G12 +3 sin2 (θ ) cos(θ )G21 − sin3 (θ )G30 Gσ4 (θ ) = cos4 (θ )G04 − 4 cos3 (θ ) sin(θ )G13 + 6 sin2 (θ ) cos2 (θ )G22
TT = [T (BGσ (θ ))n ] n = 1, 2, 3, 4 TT ∈ RNn ×Nθ ×Npos ×Nbins
−4 cos(θ ) sin3 (θ )G31 + sin4 (θ )G40
(5)
(2) MPCA is then applied to this tensor TT to obtain a vector YT ∈ Rm≤M . Then to compute Binary Gaussian Receptive Maps, a Local Binary Pattern (LBP) [2] is applied over each Gaussian derivative to assign a YT = MPCA (TT ) YT ∈ Rm (6) label to each pixel of the image by thresholding the 3 × 3 neighborhood The second algorithm computes MPCA over each tensor separetly to of each pixel with the center pixel value and considering the result as a 2 , y3 and y4 , these vectors are concatenated to form binary or decimal number. The mathematic expression for this operation obtain the vectors y1 , y4×m a single vector Y ∈ R , where m ≤ M. F is: BGσn (θ ) = LBP(Gσn (θ ))
(3)
YF = [yn ] YF ∈ R4×m
n = 1, 2, 3, 4
(7)
Finally, we use Relevance Vector Machines (RVM) [8] rather than Where BGσn (θ ) is the Binary Gaussian Receptive map, n is the order of the Gaussian derivative, σ is the scale factor, θ is the steering angle Support Vector Machines (SVM) as a regression algorithm. and LBP is the Local Binary Pattern. The Gaussian Receptive Maps could be efficiently computed at half- 4 Experimental Evaluation octave scales using a linear complexity O(N) algorithm for computing a We have performed several experiments to compare different approachs Gaussian pyramid [3]. for estimating age from facial images. Two publicly available databases have been used in our experiments: The FG-NET [1] database and the σ 2 Tensorial Representation of BGn (θ ) MORPH [5] database. To compute the Tensorial representation, first we divide each BGσn (θ ) of each image into non-overlapping rectangular sub-regions with a specific size. A set of histograms is then computed for each sub-region and finally each histogram is organized in four different 3-D tensors, where each tensor corresponds to an specific derivative order of the Binary Gaussian Receptive maps. The characteristic equation of T (BGσn (θ )) is shown as follows: T (BGσ (θ ))n ∈ RNθ ×Npos ×Nbins
n = 1, 2, 3, 4
(4)
Where n and Nθ are the order and orientation angles for the gaussian derivatives respectively, N pos is the number of non-overlapping positions in the map and Nbins is the number of bins used in the construction of each local histogram.
[1] The fg-net aging database. http://www.fgnet.rsunit.com/. accessed: April 2010. [2] T. Ahonen, A. Hadid, and M. Pietikainen. Face description with local binary patterns: Application to face recognition. IEEE PAMI, 28(12):2037–2041, 2006. [3] J.L. Crowley and O. Riff. Fast computation of scale normalised gaussian receptive fields. In Proc. Scale Space Methods in Computer Vision, pages 584–598, 2003. [4] W. T. Freeman and E. H. Adelson Y. The design and use of steerable filters. IEEE PAMI, 13:891–906, 1991. [5] Karl Ricanek Jr. and Tamirat Tesafaye. Morph: A longitudinal image database of normal adult age-progression. IEEE FG, 2006. [6] Haiping Lu, Konstantinos N. Plataniotis, and Anastasios N. Venetsanopoulos. Mpca: Multilinear principal component analysis of tensor objects. IEEE TNN, 19(1):18–39, 2008. [7] B. Schiele and J.L. Crowley. Recognition without correspondence using multidimensional receptive field histograms. IJCV, 36:31–50, 2000. [8] Michael E. Tipping. Sparse bayesian learning and the relevance vector machine. Journal of Machine Learning Research, 1:211–244, 2001.
