Bayro-Corrochano E., Scheuermann G. Geometric Algebra Computing: in Engineering and Computer Science

Подождите немного. Документ загружается.

268 Y. Xu et al.

Fig. 16 The scheme of the proposed image fusion method

Fig. 17 Comparison of image fusion results

components of image pairs much better in contrast to those of QWT scheme. The

potential of QWT might be further enhanced by selecting proper fusion methods.

Here we mainly focus on the practicability of QWT in image fusion task.

6 The Potential Use of QWT in Color Image Recognition

A major motivation for the study of quaternion Fourier transforms (QFT) is that they

handle color image pixels as vectors and thus offer scope to process color images

holistically, rather than by separating luminance and chrominance or color space

components. Nowadays QFT ﬁnds its extensive use in color image processing, such

as hypercomplex spectrum ﬁltering [5], color image registration [23], and color wa-

termarking [24]. Similar to complex Fourier transform, QFT looses the notion of

QWT: Retrospective and New Applications 269

chronology in the frequency domain. The researchers hold high hopes on develop-

ing QWT into a useful tool for local color image analysis. Until now the invertible

QWT for multiscale vector-valued color image analysis is still an open issue [8]. The

most commonly used quaternion wavelets in color image processing are quaternion

Gabors, which can achieve joint spatial-frequency representation along a given di-

rection in color space [6, 7]. In this section, we show how to use quaternion Gabors

to establish invariant color features, which have been successfully applied in our

color face recognition system [7].

There are many variants for the deﬁnition of quaternionic Gabors. In Jones’s

work [6], it is formulated as



x



=μ







−







x

2σ



cos





x



−e

−



(16)

so that it maps to pure quaternionic values, conceptually pointing in a particular

direction in three-channel color space. In (16), x is the 2D spatial location vector,



indicates the direction of the sinusoidal oscillation of the ﬁlter, and μ is a unit

pure quaternion which represents a speciﬁc direction in RGB color space. To ex-

tract multiscale and multioriented features, we deﬁne a quaternion Gabor set with

constant octave band,

u,v

(x, y) =μ

−

u+1

)



√

(x cos

+y sin

)

−e

−



, (17)

where parameters σ and f

respectively denote the standard deviation of the

Gaussian envelope and the center frequency at the ﬁnest scale. Here we select

u = 0, 1,...,4 and v = 0, 1,...,7 to uniformly partition the scale and the orien-

tation scope, whereby it results in image decomposition at ﬁve scales and eight

orientations. Because of the substantial power in natural signals at low frequencies,

this DC sensitivity is eliminated by the term in the square bracket to avoid a positive

bias of the response. In order to ﬁnd an adaptive selection of μ and thus realize ho-

mogeneous projection in the color vector space, the computation of μ depends on

quaternion principal component analysis (QPCA) and is summarized as follows:

Step 1. Provide the covariance matrix S for the three color planes in the original

image,

S =









−











, (18)

where M is the number of color pixels; P

, P

, and P

are respectively the nor-

malized pixel value matrix of red, green, and blue channel.

Step 2. Impose QPCA on the covariance matrix S and determine the pure quater-

nion μ as the principal eigenvector of S. Once the color axis μ is determined, quater-

nion Gabor Features are extracted by the convolution of the original color image

(x, y) and the given quaternionic Gabor kernel F

u,v

(x, y). Here color pixel val-

ues are represented as pure quaternion I

(x, y) =I

(x, y)i +I

(x, y)j +I

(x, y)k,

270 Y. Xu et al.

Fig. 18 Quaternion Gabor Feature images: (a) magnitude; (b) argument

where the r, g, and b subscripts denote the red, green, and blue color channels,

respectively. No matter what transformation happens in the RGB vector space be-

tween two corresponding color face images, we always depict the convolution out-

put O

(x, y) =Iq(x,y)⊗F

u,v

(x, y) =a +b ·i +c ·j +d ·k as one scalar part and

one vector part, i.e., the real part O

μ

(x, y) and the imaginary part O

⊥μ

(x, y),

where the former one is the negative projection of O

(x, y) to the principal color

component μ of the original image, while the latter one is orthogonal to color axis μ.

Two kinds of quaternion Gabor feature images are established according to the

convolution result. The magnitude color image is the one-to-one mapping from

imaginary coefﬁcients b, c, and d to red, green, and blue channels. Another color

image is set up from the arguments between the imaginary parts and the real part,

that is,

=arctan(b/a), (19)

=arctan(c/a), (20)

=arctan(d/a), (21)

where arctan(·) ∈[−π,π). One example of quaternion Gabor feature images is il-

lustrated in Fig. 18. There are eight orientations along each row and ﬁve scales along

each column for two kinds of images.

It can be deduced that argument image is robust to the illumination contrast

variations. Since nonparametric histogram method could enhance the robustness of

features against variations of pose, illumination, and expressions in face dataset,

we use Local Binary Pattern (LBP) histogram [25] to establish local invariant de-

scriptors for quaternion Gabor features. These descriptors are named here as local

quaternionic Gabor binary pattern (LQGBP) descriptors that encode the relation-

ship between the quaternion Gabor features. We unfold the construction procedure

of LQGBP descriptors as follows:

Step 1. Generate quaternion Gabor feature images at ﬁve scales and eight orien-

tations, as shown in Fig. 18.

QWT: Retrospective and New Applications 271

Fig. 19 The conﬁguration of LQGBP-based color face recognition system

Step 2. Each quaternion Gabor feature image f(c) is further mapped to f



(c),

where f



A(c

) =



1iff(c

) ≥f(c)and c

⊂Ω

0 otherwise,

(22)





A(c

) ×2

, (23)

where c indicates the position of image pixel, and Ω denotes a neighborhood sys-

tem.

Step 3. The output quaternion Gabor feature images after local binary mapping

are divided into 8 ×8 blocks and represented by the stacks of the local histograms

of these subregions. These local histograms contribute to a global invariant descrip-

tor of quaternion Gabor features. For two magnitude/argument quaternion Gabor

feature images, the similarity measure E is deﬁned as follows:

E =



s,o,c

∩H

)/N, (24)

N = 6N

×N

, (25)

where the subscript of H is used to indicate the local histogram from different face

images; symbols s, o, and c further point out at which scale, orientation, and po-

sition histogram H is computed; N

is the number of pixels in the original image;

and N

respectively compute the number of scales and orientations; the oper-

ator ∩ retains the smaller individual of the two matrices. The ﬂowchart shown in

272 Y. Xu et al.

Fig. 20 Color face recognition results

Fig. 19 delineates the proposed face recognition method, which is formulated as an

automatic face feature extraction module and a feature matching procedure. It makes

full use of the interrelationship among different color channels and achieves promis-

ing performance against noises, variations of lighting, and expressions in the open

face dataset downloaded from http://cswww.wssex.ac.uk/mv/allfaces/index.html,as

shown in Fig. 20. Detailed quantitative analysis can be referred to [7].

7Conclusion

In this paper, we give an overview of the development of QWT and investigate

its potential applications in scalar image processing and vector image processing.

QWT is natural to deal with the operation in color/vector space, but nowadays the

construction of invertible multiscale QWT for vector-valued color images is still a

far-reaching issue. Several important problems remain open: constraints of quater-

nion wavelet needed to reconstruct a color signal, the deﬁnition of sparse quaternion

wavelet transform in color space, the selection of color space, and so on.

Acknowledgements This work was supported by Research Fund for the Doctoral Program

of Higher Education of China (200802481006), National Natural Science Foundation of China

(60702044, 60902073), NCET-06-0409, and the Cultivation Fund of the Key Scientiﬁc and Tech-

nical Innovation Project, Ministry of Education of China (706022).

References

1. Bülow, T.: Hypercomplex spectral signal representations for the processing and analysis of

images. Dissertation, Kiel: Institut für Informatik und Praktische Mathematik der Christian-

Albrechts-Universitat zu Kiel (1999)

QWT: Retrospective and New Applications 273

2. Corrochano, E.B.: The theory and use of the quaternion wavelet transform. J. Math. Imaging

Vis. 24, 19–35 (2006)

3. Chan, W.L., Choi, H., Baraniuk, R.: Quaternion wavelets for image analysis and processing.

Proc. IEEE Int. Conf. Image Process. 5, 3057–3060 (2004)

4. Xu, Y., Zhou, J., Yang, X.: Quaternion wavelet phase based stereo matching for uncalibrated

images. Pattern Recogn. Lett. 28(12), 1509–1522 (2007)

5. Sangwine, S., Ell, T.: Hypercomplex Fourier transforms of color images. IEEE Trans. Image

Process. 16(1), 22–35 (2007)

6. Jones, C., Abbott, A.: Color face recognition by hypercomplex Gabor analysis. In: 7th Inter-

national Conference on Automatic Face and Gesture Recognition, pp. 126–131 (2006)

7. Lu, W., Xu, X.Y.Y., Song, L.: Local quaternionic Gabor binary patterns for color face recogni-

tion. In: International Conference on Acoustics, Speech, and Signal Processing, pp. 741–744

(2008)

8. Carré, P., Denis, P.: Quaternionic wavelet transform for colour images. Proc. SPIE 6383,

638 301/1–638 301/15 (2006). Invited paper

9. Hamilton, W.: Elements of Quaternions. Longman, Harlow (1866)

10. Felsberg, M.: Optimized fast algorithms for the quaternionic Fourier transform. In: Proc. 8th

International Conference on Computer Analysis of Images and Patterns, vol. 1689, pp. 209–

216 (1999)

11. Pei, S., Ding, J., Chang, J.: Efﬁcient implementation of quaternion Fourier transform, convo-

lution, and correlation by 2-D complex FFT. IEEE Trans. Signal Process. 49(11), 2783–2797

(2001)

12. Kingsbury, N.: Complex wavelets for shift invariant analysis and ﬁltering of signals. Appl.

Comput. Harmon. 10(3), 234–253 (2001)

13. da Silva, E.A.B., Ghanbari, M.: On the performance of linear phase wavelet transform in low

bitrate coding. IEEE Trans. Image Process. 5(5), 689–704 (1996)

14. Gabor, D.: Theory of communication. J. IEE 93, 429–457 (1946)

15. Fleet, D., Jepson, A.: Stability of phase information. IEEE Trans. Pattern Anal. Mach. Intell.

15(12), 1253–1268 (1993)

16. Lindeberg, T.: Feature detection with automatic scale selection. Int. J. Comput. Vis. 30(2),

79–116 (1998)

17. Mikolajczyk, K., Tuytelaars, T.: A comparison of afﬁne region detectors. Int. J. Comput. Vis.

65(1/2), 43–72 (2005)

18. Kovesi, P.: Invariant feature measures of image features from phase information. Ph.D. Dis-

sertation, University of Western Australia (1996)

19. Morrone, M., Owens, R.: Feature detection from local energy. Pattern Recogn. Lett. 6, 303–

313 (1987)

20. Mikolajczyk, K., Schmid, C.: A performance evaluation of local descriptors. IEEE Trans.

Pattern Anal. Mach. Intell. 27(10), 1615–1630 (2005)

21. Zhang, Z., Deriche, R., Faugeras, O., Luong, Q.: A robust technique for matching two uncal-

ibrated images through the recovery of the unknown epipolar geometry. Artif. Intell. J. 78,

87–119 (1995)

22. Do, M., Vetterli, M.: Contourlets. Beyond wavelets, pp. 1–27 (2001)

23. Lu, Z., Xu, Y., Yang, X., Song, L., Traversoni, L.: 2D quaternion Fourier transform: the spec-

trum properties and its application in color image registration. In: International Conference on

Multimedia and Expo, pp. 1715–1718 (2007)

24. Ma, X., Xu, Y., Song, L., Yang, X., Burkhardt, H.: Color image watermarking using local

quaternion Fourier spectral analysis. In: International Conference on Multimedia and Expo,

pp. 233–236 (2008)

25. Ahonen, T., Hadid, A., Pietikainen, M.: Face description with local binary patterns: application

to face recognition. IEEE Trans. Pattern Anal. Mach. Intell. 28(12), 2037–2041 (2006)

Part IV

Computer Vision

Image Sensor Model Using Geometric Algebra:

From Calibration to Motion Estimation

Thibaud Debaecker, Ryad Benosman,

and Sio H. Ieng

Abstract In computer vision image sensors have universally been deﬁned as the

nonparametric association of projection rays in the 3D world to pixels in the im-

ages. If the pixels’ physical topology can be often neglected in the case of per-

spective cameras, this approximation is no longer valid in the case of variant scale

sensors, which are now widely used in robotics. Neglecting the nonnull pixel area

and then the pixel volumic ﬁeld of view implies that geometric reconstruction prob-

lems are solved by minimizing a cost function that combines the reprojection errors

in the 2D images. This paper provides a complete and realistic cone-pixel camera

model that equally ﬁts constant or variant scale resolution together with a protocol

to calibrate such a sensor. The proposed model involves a new characterization of

pixel correspondences with 3D-cone intersections computed using convex hull and

twists in Conformal Geometric Algebra. Simulated experiments show that standard

methods and especially Bundle Adjustment are sometimes unable to reach the cor-

rect motion, because of their ray-pixel approach and the choice of reprojection error

as a cost function which does not particularly ﬁt the physical reality. This problem

can be solved using a nonprojective cone intersection cost function as introduced

below.

1 Introduction

A large amount of work has been carried out on perspective cameras introducing the

pinhole model and the use of projective geometry. This model turns out to be very

efﬁcient in most cases, and it is still widely used within the computer vision com-

munity. Several computation improvements have been introduced [5]; nevertheless,

this model has limitations, notably that it can only be applied to projective sensors.

The pixel sampling and slight ﬂuctuations in the calibration process lead to the fact

T. Debaecker (



)

Institut des Systèmes Intelligents et de Robotique (ISIR), 4 place Jussieu, Paris, France

e-mail: thibaud.debaecker@isir.jussieu.fr

E. Bayro-Corrochano, G. Scheuermann (eds.), Geometric Algebra Computing,

277