binary principal component analysis

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Figure 8: B-PCA base vectors and its corresponding box functions for vehicle modeling.

could be similar. The test database comprises 64 aligned face images in the FERETdatabase. Each image is projected to the PCA and B-PCA subspace and the coefcientsare used as features. A nearest neighbor classier is used to nd the best match as therecognition result. We compared the PCA with B-PCA using the original projection

process which has the pseudo-inverse (ΨT

Ψ )− 1

ΨT

x and also that of direct dot prod-uct (DNP) Ψ T x for recognition, the results are shown in Fig. 9 and Fig. 10, in Fig. 9,the approximation threshold ζ is set to 0.9, in Fig. 10, ζ is 0.1. As can be observed,when ζ = 0.1, which means the B-PCA base vectors are more orthogonal, the differencebetween recognition rates using pseudo-inverse and DNP is very small. The DNP, Ψ T x,is computationally much cheaper because it only needs several additions. The differencebetween (Ψ T Ψ )− 1Ψ T x and Ψ T x is small when the B-PCA base vectors are more orthog-onal (smaller ζ ), and it is large when the base vectors are less orthogonal (larger ζ ). Aswe can observe that when the approximation threshold ζ is small, the difference betweenB-PCA and PCA base vectors is small, the difference between DNP and pseudo-inverseprojection is also small.

Figure 9: Recognition performance for facedataset with ζ = 0.9.

Figure 10: Recognition performance for facedataset with ζ = 0.1.

4.3 Speed Improvement of B-PCA

Suppose the image size is m × n , T PCA denotes the time for computing the PCA subspaceprojection coefcients. N denotes the number of PCA base vectors. It will take m × n × N oating point multiplications and N × (m × n − 1) + ( N − 1) oating point additions to

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perform the projection operation, orT PCA = N × m × n × T f m + [ N × (m × n − 1) + ( N − 1)] × T f a (4)

where T f m is the time for a single oating point multiplication, T f a is the time for a singleoating point addition.

For B-PCA, the time for a single projection is denoted as T BPCA . It consists of twoparts, one is T ii , the time to construct the integral image, for a m × n image, it will takem × n × 2 integer additions with recursive implementation. This is performed only oncefor each image. The other is the time for the projection operation PΨ (x) = ( Ψ T Ψ )− 1Ψ T x.When the bases are nearly orthogonal to each other, we can approximate the projectioncoefcient using direct dot-product Ψ T x. The B-PCA base vector ψ i (1 ≤ i ≤ N ) is repre-sented as a linear combination of N i box functions, ψ i = ∑ N i

j= 1 c jb j. The projection of x to

ψ i can be written as ψ i, x = ∑ N i j= 1 ci, j b j, x . Each box function b j has n j boxes, wheren j can be one or two. The b j, x can be performed using 3 × n j integer additions. Sincec j is oating point, ψ i, x needs N i oating point multiplications and N i − 1 oating pointadditions.

T BPCA = ∑ N i= 1 ∑ N i

j= 1 (3 × n j × T ia + N i × T f m + ( N i − 1) × T f a) (5)

where T ia is the time for one integer addition. As we can observe, T BPCA is only dependenton the number of binary box functions which is often much smaller than the dimensionof the image. While for PCA, the time is proportional to the image dimension. Sincethe number of operations in B-PCA is much smaller than PCA, T BPCA is much faster thanT PCA , and the speed up is more dramatic when the dimension of data is higher.

Table 2: Comparison of the computational cost between PCA and B-PCA projectionoperation

Approximation threshold: ζ 0.4 0.6 0.85# binary box functions 409 214 68

T PCA (a single subspace projection (s)) 3 .15 × 10− 4

T ii (integral image computation (s)) 4 .72 × 10− 5

T BPCA (a single subspace projection (s)) 8 .11 × 10− 6 4.29 × 10− 6 1.45 × 10− 6

Speedup ( T PCAT BPCA + T ii / N ) 27.97 42.34 68.47

Suppose m = n = 24, N = 15, T PCA needs 24 × 24 × 15 = 8640 oating point multipli-cations to compute the projection coefcients. Suppose the total number of NBS basevectors used to represent all the B-PCA base vectors is 200, that is, ∑ N

i= 1 N i = 200, theB-PCA needs only 2 × ∑ N

i= 1 N i − 1 = 399 oating point operations. The speed up is sig-nicant.

The experiment for speed improvement is carried out on a Pentium 4, 3.2GHz, 1GRAM machine, using C++ code. We compute the 15 PCA base vectors and 15 B-PCAbase vectors, and evaluate the computation time for an image to project onto the PCAsubspace and B-PCA subspace. We tested the PCA projection and B-PCA projection with

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1000 images and the time for a single projection is computed as the average. In Table-IV, we can observe, the B-PCA projection process Ψ T x is much faster than direct PCAprojection operation. The improvement is even more dramatic if more base vectors areused for comparison. Note: 1) for PCA projection test, the image data is put in continuousmemory space to avoid unnecessary memory access overhead; 2) the T ii is distributed intoeach base vector projection operation to make the comparison fair, because integral imageis only needed to be computed once.

5 Conclusion and future work

A novel compact and efcient representation called B-PCA is presented in this paper. Itcan capture the main structure of the dataset while take advantages of the computational

efciency of NBS. A PCA-embedded OOMP method is proposed to obtain the B-PCAbasis. We have applied the B-PCA method to the image reconstruction and recognitiontasks. Promising results are demonstrated in this paper. Future work may use more so-phisticated classication methods to enhance the recognition performance.

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binary principal component analysis

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