Stereo Disparity Estimation in Moment Space

Transcription

1 Stereo Disarity Estimation in oment Sace Angeline Pang Faculty of Information Technology, ultimedia University, 63 Cyberjaya, alaysia. R. ukundan Deartment of Comuter Science, University of Canterbury, Christchurch, ew Zealand. g Liang Shing Faculty of Engineering, ultimedia University, 63 Cyberjaya, alaysia. Abstract This aer exlores various ways by which ixel disarities along eiolar rows of a stereo image air could be obtained using Chebyshev moments of the corresonding D images. Chebyshev moments allow us to reresent the image features based on orthogonal basis functions, and hence to reconstruct the intensity distribution from a set of moments fairly accurately. Disarity estimates can be obtained either analytically from the inverse moment transform, or by comaring the reconstructed intensity values. The aer also resents some reliminary results obtained using both synthetic and real images. Keywords: Stereo Image Processing, Disarity Estimation, Discrete Orthogonal oments, Chebyshev Polynomials. Introduction oment functions such as geometric and Zernike moments, are used in several comuter vision alications for reresenting global and invariant shae characteristics of image features. Orthogonal moments (Zernike, Pseudo-Zernike etc.) have been roved to be less sensitive to image noise when comared to geometric moments, and ossess far better feature reresentation caabilities [-3]. For examle, the information redundancy measure is minimum in an orthogonal moment set. Recently, discrete orthogonal moments based on Chebyshev (Tchebichef) olynomials were introduced [4-6], which can also be used as image descritors with minimum information redundancy. Chebyshev moments are orthogonal in the image coordinate sace itself, and thus rovide more accurate reconstructions of the original image; while the comutation of Zernike moments involves discrete aroximations and coordinate transformations which severely affect their orthogonality roerties in the image sace. Since the image intensity distribution can be very accurately reresented by Chebyshev moments, the matching of intensity values in alications such as stereosis, can also be attemted in the moment sace. Along eiolar stereo image rows, the corresondence between two intensity values I(x) and I(x+d x ) is reresented by the disarity d x at ixel x. Using discrete orthogonal moments, we can have an accurate olynomial aroximation of the intensity distribution I(x) in terms of the kernel functions (Chebyshev olynomials of degree ) t (x). Thus, in the moment sace, the corresondence between two ixels in a stereo air can be exressed as a relation involving terms t (x+d x ) of different orders, where the coefficients are moments comuted from the two images. The analytical characteristics of the kernel functions are well known. We can therefore try to 26 Image and Vision Comuting Z

2 imlement some novel aroaches to disarity estimation, such as obtaining the values of d x by comaring moment terms rather than by comaring intensity values. deriving analytical solutions for d x from moment equations. The above considerations were the main motivation behind the work reorted in this aer. The next section outlines the framework of Chebychev moments and their inverse moment transforms. A few analytical derivations of the disarity values from moment equations are given in Section 3. Exerimental results and a discussion on the advantages and limitations of moment based stereosis are given in Section 4. t (x)= 2 ( ) ( 2 ) t( x) t ( x) ( ) t 2 2, > (5) The image intensity function f(x, y) has a olynomial reresentation given by f(x, y) = T qt tq ( y) (6) = q= where the coefficients T q are the Chebyshev moments defined in (2). 2 Chebyshev oments Given an x image, discrete orthogonal olynomials {t n (x)} over the image coordinate sace satisfy the condition tm t n = ρ ( n, ) δ, m, n =,,. () mn Left Image Right Image where n,) is the squared norm of the olynomial set t n. The classical discrete Chebyshev olynomials [7,8] satisfy the above roerty of orthogonality (). For image analysis alications, we can aroriately scale the above olynomials to avoid numerical instabilities in the comutation of large-order moments. Some of the ossible sets of scale factors are discussed in [4]. Alternatively, we can also make the olynomial functions orthonormal with n,)=, (which also amounts to introducing a scale factor) and minimize the roagation of errors using the rocess of renormalization [9]. The Chebyshev moments of order +q are defined as [4-6] T q = t tq ( y) f ( x, y),, ) q, ) y=, q =,,. (2) and, one set of scaled Chebyshev olynomials t (x) can be comuted using the following recurrence relation: t (x) =. (3) t (x) = 2x + (4) Row 3 of L-image Reconstructed Images Row 3 of R-image Figure : Original and reconstructed intensity distributions on an image row of a stereo air. We can easily write the one-dimensional versions of equations (2), (6) as Palmerston orth, ovember 23 26

3 T = t, ) f, =,,. (7) and, f(x) = T t. (8) = Figure gives an examle of a synthetic stereo image air, and the original and reconstructed intensity distributions on a articular row. 3 Disarity Values and oment Equations In this section, we attemt to derive analytical relationshis between disarity values and the intensity moments using the equations given above. If we denote the one-dimensional intensity distributions along eiolar lines of the left and the right image by f l (x) and f r (x) resectively, then a disarity value d x at the ixel x establishes a corresondence in the intensity domain given by f l (x) = f r (x+d x ) (9) If we use the olynomial reresentation of the intensity function as given in (8), we get = ( t ( x + d ) T t ). T () x (l) (r) where T, T denote the th order Chebychev moments of intensity functions (7) along the left and the right eiolar image rows, and is a sufficiently large value (tyically /2 or greater) denoting the maximum order of moments. Several analytical reresentations for the disarity function d x can be derived from (). For examle, we can write d x = ( T T ) t = = T t () where t (x) denotes the derivative of t (x). A th order olynomial will have zeros, and hence for large values of, the derivative of the function t (x) will have many oints where the value is either zero or is very large. Thus () will not, in many cases, rovide any meaningful results. An alternative aroach is to comute the coarse disarity estimates using low-order moments (using(3),(4)) as d x = ( T T ) + ( T T ) 2T 2x + (2) and then develo an iterative udate scheme for d x using higher order moments, alying the constraint (). The lot of d x for a fixed value of x is given in figure 2. d = d = = = Figure 2: Variation of disarity with resect to coarse-to-fine reconstruction using moments. 4 Exerimental Analysis Instead of using the analytical aroaches detailed in the revious section, we could also attemt to estimate disarity values in the moment sace, by minimizing the difference between reconstructed intensity values: In other words, for each x, we seek a value d x which minimizes λ x = ( T t x + d ) T t ) = ( (3) x Assuming that the disarity function is continuous in a neighborhood [x k, x+k] of x for small values of k, we can redefine the cost function as: k λ x = ( T t x + d + i) T t ( x + i) ) i= k = 2 2 ( x (4) The disarity values obtained for the 3 th row of the image air in figure., and for the whole image, are shown below. 262 Image and Vision Comuting Z

4 d = 5 d = Row 3 Whole image Figure 3: Disarity ma for the stereo air in figure. Left Image Right Image Figure 5. Real images used in exerimental analysis. The rimary advantage of moment based stereo disarity estimation is that the disarity values can be reresented as offsets in the arguments of kernel functions, whose characteristics are well known (unlike the intensity distribution). This allows us to choose cost functions that deend on the maximum order of moments used. The reconstructed olynomial aroximation of the intensity values, can exhibit Gibb s henomenon (as in the case of inverse Fourier transform) near regions of large intensity gradients. Such effects will have to be eliminated for imroving the accuracy of the estimates. Figure 4 shows the reconstruction of row 42 of the left-image in figure, and the effect of the Gibb s henomenon on disarity estimates. Figure 6. Disarity ma obtained using images in figure 5. Some ossible extensions of the work reorted in this aer are listed below. A rogressive udate of disarity values using (2) with local smoothness constraints. Combining the minimization roblem (3) with a dynamic rogramming aroach [] for refining the disarity ma. Row 42 Disarity Figure 4: Effect of Gibb s henomenon on disarity values. The olynomial aroximation of the intensity function can reduce the effects of image noise. However, since an entire row is reresented by a single function, background intensities will not be searable from those of the foreground objects. This can also be observed in figure. Such a blending of intensities might lead to erroneous estimates. Figure 6 gives the initial exerimental results with real images in figure 5. Further imrovements in the disarity estimates can be obtained using a dynamic rogramming aroach and rectangular windows. Alication of a coarse-to-fine reconstruction strategy together with coarseto-fine dynamic rogramming [] to minimize the comutation time. 5 Conclusions The aer has resented a method based on discrete Chebyshev moments using which the disarity values in a stereo image air could be reresented in the moment sace. Such a reresentation allows ixel disarities to be equivalently denoted by offsets in the argument of continuous olynomial functions. Consequently, several analytical as well as numerical characterizations of the disarity values could be ossible, given a large set of equations involving moments (of various orders) of the left and right images. The ossibility of using this aroach in stereosis has been exlored, and some exerimental results were resented. Palmerston orth, ovember

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