Backprojection of Some Image Symmetries Based on a Local Orientation Description

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1 Backprojection of Some Image Symmetries Based on a Local Orientation Description Björn Johansson Computer Vision Laboratory Department of Electrical Engineering Linköping University, SE Linköping, Sweden bjorn@isy.liu.se LiTH-ISY-R ISSN

2 Backprojection of Some Image Symmetries Based on a Local Orientation Description Björn Johansson Computer Vision Laboratory Department of Electrical Engineering Linköping University, SE Linköping, Sweden bjorn@isy.liu.se October 30, 2000 Abstract Some image patterns, e.g. circles, hyperbolic curves, star patterns etc., can be described in a compact way using local orientation. The features mentioned above is part of a family of patterns called rotational symmetries. This theory can be used to detect image patterns from the local orientation in double angle representation of an images. Some of the rotational symmetries are described originally from the local orientation without being designed to detect a certain feature. The question is then: given a description in double angle representation, what kind of image features does this description correspond to? This inverse, or backprojection, is not unambiguous - many patterns has the same local orientation description. This report answers this question for the case of rotational symmetries and also for some other descriptions. 1

3 Contents 1 Introduction 3 2 Local orientation in double angle representation, z 3 3 Some symmetries on local orientation z General assumption Rotational symmetries, z = e i2β(ϕ) Theory Experiments Polar symmetries, z = e i(n1 ln r+n2ϕ+α) Theory Experiments Cartesian symmetries, z = e i(a1x+a2y+α) Theory Experiments Acknowledgment 13 2

4 1 Introduction Human vision seems to work in a hierarchical way in that we first extract low level features such as local orientation and color and then higher level features. No one knows for sure what these high level features are but there are some indications that curvature, circles, spiral- and star patters are among them [7], [13]. Perceptual experiments also indicate that corners and curvature are very important features in the process of recognition and one can often recognize an object from its curvature alone [1], [3]. One way to detect the features mentioned above is to use the theory of rotational symmetries. They have been described in the literature a number of times, see e.g. [6], [2], [5], [11], [12], [4], [9], [10]. Hyperbolic-, circular-, star-, and other patterns can be described in a compact way using local orientation in double angle representation. They can therefore be detected using simple correlations on the local orientation image. These descriptions can be generalized to include a larger class of image patterns called rotational symmetries. This larger class is described on the local orientation and therefore we do not know directly what image patterns they describe. The most useful rotational symmetries has been proven to describe the patterns mentioned above, but there are still unexplored symmetries. The question is basically: given a description on local orientation, what class of image patterns does this correspond to? This report answers this question for the case of rotational symmetries and also for some other descriptions. The double angle representation is non-linear and the answer is therefore not trivial. This report contains a lot of tedious mathematical calculations, but the idea is fairly easy: Start with a local orientation description in double angle representation. In each point, decode this representation into an orientation angle and assume that the image gradient has the same angle. When we know the image gradient we can solve a differential equation to get the underlying image pattern (the solution is not unique). 2 Local orientation in double angle representation, z There exists a number of ways to detect local orientation in images. We will not deal with these methods in this report but rather concentrate on orientation representations. The classical representation of local orientation is simply a 2D-vector pointing in the dominant direction with a magnitude that reflects the orientation dominance (e.g. the energy in the dominant direction). An example of this is the image gradient. A better representation is the double angle representation, where we have a vector, or a complex number z, pointing in the double angle direction. This means that if the orientation has the direction θ we represent it with a vector pointing in the 2θ-direction, i.e. z = e i2θ. Figure 1 illustrates the idea. This representation has at least two advantages: We avoid ambiguities in the representation: It does not matter if we choose to say that the orientation has the direction θ or, equivalently, θ + π. In the double 3

5 Im z Re z Figure 1: Double angle representation of local orientation. angle representation both choices get the same descriptor e i2θ. Averaging the double angle orientation description field makes sense. One can argue that two orthogonal orientations should have maximally different representations, e.g. vectors that point in opposite directions. This is for instance useful in color images and vector fields when we want to fuse descriptors from several channels into one description. This descriptor will in this paper be denoted by a complex number z. The argument arg z points out the dominant orientation and the magnitude z usually represent the certainty or energy of the orientation. 3 Some symmetries on local orientation z Lets repeat the question we are trying to answer: given a description on local orientation, what class of image patterns does this correspond to? This section start with the orientation image z and go backwards to the original image, f, from which the orientation image was calculated, i.e. Double angle representation z Grayscale image f This inverse is of course not unambiguous, but we get a hint by making the assumption that the image gradient is parallel to the dominant orientation. There are two assumptions made in this report. The first one, mentioned above, is not very restrictive and says that the image gradient is parallel to the local orientation. The second one assumes that the image pattern can be described as a separable function in some suitably chosen coordinate system which depend on the selected symmetry. 4

6 3.1 General assumption The assumption that the image gradient f is parallel to the local orientation β = β(x, y) can be written mathematically as ( ) ( ) fx cos β f = //, where β = 1 arg z (1) f y sin β 2 i.e. ( ) ( fx cos β = ± f f x 2 + fy 2 y sin β ) (2) We have to divide the phase (arg z) by two to get rid of the double angle representation. The price is the direction unambiguity (±). Equation 2 squared gives fx 2 = ( ) fx 2 + fy 2 cos 2 β fy 2 = ( ) fx 2 + fy 2 sin 2 β f x sin β = ±f y cos β (3) From equation 2 we see that the - -solution is false and we arrive at the final equation: Backprojection equation: f x sin β = f y cos β (4) Every pattern description based on the local orientation is described by a function β(x, y). This function can be put into equation 4 which can be solved to get the corresponding gray-image pattern f(x, y). The remaining subsections solve the backprojection equation for some special cases of β(x, y). The first subsection deals with the rotational symmetry class. The other two subsections deals with polar and Cartesian symmetries, which may not be as useful as the rotational symmetries, but are included more as curiosity and art. 3.2 Rotational symmetries, z = e i2β(ϕ) Theory The name rotational symmetries is not entirely logical but it is an established name and should not be changed. They are actually a special case of polar symmetries described in section 3.3. They can be described using the local orientation description as z = e i2β(ϕ) (5) The orientation β only depends on ϕ and is constant along the r-dimension. The most useful ones are the zeroth, first, and second order rotational symmetries: z = e i(nϕ+α), n =0, 1, 2 (6) 5

7 They are proven to describe patterns like corners, curvature, circles, and stars (see e.g. [8]). The other rotational symmetries has not been thoroughly examined before. We shall now solve the backprojection equation 4 assuming β = β(ϕ). In this case it is easier to switch to polar coordinates: x = r cos ϕ (7) y = r sin ϕ The derivatives f x and f y can be written in polar coordinates using the chain-rule: fx = f x = f r r f y = f y = f r x + f ϕ ϕ r y + f ϕ sin ϕ x = f r cos ϕ f ϕ r ϕ cos ϕ y = f r sin ϕ + f ϕ r (8) If this is inserted in equation 4 we get ( sin ϕ f r cos ϕ f ϕ r ) ( sin β = f r sin ϕ + f ϕ cos ϕ r After some re-shuffling we get the polar version of equation 4: ) cos β (9) Polar backprojection equation: f r r sin(β ϕ) =f ϕ cos(β ϕ) (10) This equation is still quite difficult to solve, but if we make the assumption that f is polar separable, i.e. f(r, ϕ) =R(r)Φ(ϕ), we get R (r)φ(ϕ)r sin(β ϕ) =R(r)Φ (ϕ)cos(β ϕ) R (r) R(r) r = Φ (ϕ) coth(β ϕ) (11) Φ(ϕ) Since β only depends on ϕ we know that the left side only depends on r and the right side only depends on ϕ. Therefore both sides have to be constant: Andweget R (r) R(r) r = K Φ (ϕ) Φ(ϕ) coth(β ϕ) =K Rotational symmetries: R(r) =C 1 r K Φ(ϕ) =C 2 e K tan(β ϕ)dϕ (12) f(r, ϕ) =C(re tan(β(ϕ) ϕ)dϕ ) K (13) 6

8 Provided a function β(ϕ), this function can be solved numerically to get the final solution. There are some cases where we can solve equation 13 analytically. Suppose z is the n:th order rotational symmetry e i(nϕ+α), i.e β(ϕ) = 1 2 nϕ + 1 2α. Then we get Φ(ϕ) = C 2 e K tan(( n 2 1)ϕ+ α 2 )dϕ = / / = = C 2 e K n 1 1 ln cos(( n 2 1)ϕ+ α 2 ) 2 = (14) = C 2 cos(( n 2 1)ϕ + α K 2 ) n 1 2 / / There is one exception to the solution above: If n =2we get Φ(ϕ) = C 2 e K tan( α 2 )dϕ = = C 2 e K tan( α 2 )ϕ (15) If we choose choose K =1 n 2 and skip the. in the case n 2and K =1in the case n =2we get the final solution: n:th order rotational symmetry: f(r, ϕ) =Cr (1 n 2 ) cos(( n 2 1)ϕ + α 2 ) n 2 (16) f(r, ϕ) =Cre tan( α 2 )ϕ n =2 What does the patterns in equation 16 look like? One way to visualize them is to plot trajectories or isobars (inspired by [5]). To get the trajectories we can for instance plot 1+cos(ωf(r, ϕ)) g(r, ϕ) = (17) 2 in the case n 2. ω determines the frequency of the repetitive pattern. For the case n =2is turns out that g(r, ϕ) = 1 2 (1 + cos(ω cos( α 2 )lnf(r, ϕ))) = (18) = 1 2 (1 + cos(ω(cos( α 2 )lnr +sin(α 2 )ϕ))) is a better, well behaved, choice. It is easy to show that if f(r, ϕ) is a solution to the backprojection equation 4 then every function g(r, ϕ) =h(f(r, ϕ)) is also a solution. We can thus generate a bigger class of functions than polar separable functions that solves the symmetry equation. Figure 2 shows some examples of functions from equation 16. Another choice of g could be a log-norm function: g(r, ϕ) =e ω ln2 (100 f(r,ϕ)/ω) (19) This will give a non-repetitive pattern (a trajectory of f). Different ω gives different patterns, e.g different ω when plotting the first order symmetry (n = 1) will give various degree of curvature, see figure 3. 7

9 3.2.2 Experiments Figure 2 and 3 contains some examples of the functions described in equation 16. n = 4 n = 3 n = 2 n = 1 n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 α = 3π/2 α = π α = π/2 α = 0 Figure 2: Some examples of rotational symmetries z = e i(nϕ+α). The backprojection is found in equation 16 and they are plotted using trajectory functions 17 and 18. ω =5 ω =10 ω =20 ω =40 ω =80 Figure 3: Some examples from equation 16 using function 19. There are a lot of other patterns that can be described as a rotational symmetry. The figure shows some examples. In these cases we have to use equation 13 and approximate the integral. z = e 2iϕ + e 2iϕ z = e 1iϕ + e 2iϕ z = e 4iϕ +1.5e 2iϕ z = 1+e 2iϕ +0.5e 4iϕ z = e iϕ + e iϕ Figure 4: Some examples from equation 13 using trajectory function 17. 8

10 3.3 Polar symmetries, z = e i(n 1 ln r+n 2 ϕ+α) Just out of curiosity it could be interesting to see what happens if we also let the symmetry description depend on the radius. Therefore we try to examine the description z = e i(n1 ln r+n2ϕ+α) (20) (We select ln r instead of just r because that gives easier equations below.) Theory As before we start from the polar backprojection equation 10: In this case we have f r r sin(β ϕ) =f ϕ cos(β ϕ) β = m 1 ln r + m 2 ϕ + v where m 1 = n 1 2,m 2 = n 2 2,v= α (21) 2 Instead of switching to polar coordinates as we did in the previous case we make the following substitution: s = m 1 ln r +(m 2 1)ϕ (22) t = (1 m 2 )lnr + m 1 ϕ This gives m f r = f 1 s r + f t 1 m2 r (23) f ϕ = f s (m 2 1) + f t m 1 If this is inserted in the polar backprojection equation we get (m 1 f s +(1 m 2 )f t )sin(s + v) =((m 2 1)f s + m 1 f t )cos(s + v) f s (m 1 sin(s + v)+(1 m 2 )cos(s + v)) = f t (m 1 cos(s + v) (1 m 2 )sin(s + v)) f s cos(s + v arctan m1 1 m 2 )= f t sin(s + v arctan m1 1 m 2 ) This finally gives where ϑ = s + v arctan m1 1 m 2 f s cos ϑ = f t sin ϑ (24) Now assume the function f is separable in the s, t variables, that is f(s, t) =S(s)T (t) (25) 9

11 If this is inserted in equation 24 we get S T cos ϑ = ST sin ϑ S cos ϑ S sin ϑ = T = const = K (26) T The solution becomes: Which give the final formula: S(s) = C 1 e K ln cos ϑ = C 1 cos ϑ K T (t) = C 2 e Kt (27) f(r, ϕ) =Cr K(1 m2) e Km1ϕ cos(m 1 lnr+(1 m 2 )ϕ+v arctan Experiments Some functions from equation 28 is plotted in figure 5. m1 1 m 2 ) K (28) n1 = 3 n2 = 3 n2 = 2 n2 = 1 n2 = 0 n2 = 1 n2 = 2 n2 = 3 n1 = 3 n1 = 2 n1 = 1 n1 = 0 n1 = 1 n1 = 2 Figure 5: Some examples of polar symmetries z = e i(n1 ln r+n2ϕ). The backprojection is found in equation 28 and they are plotted using trajectory functions g =(1+cos(ω log f))/2. 10

12 3.4 Cartesian symmetries, z = e i(a 1x+a 2 y+α) Theory Start from the backprojection equation 4: f x sin β = f y cos β Make the substitution u = e γ cos β v = e γ sin β which gives where γ is defined by γ x = β y γ y = β x (29) f x = f x = f u u x + f v v x = = f u e γ (γ x cos β β x sin β)+f v e γ (γ x sin β + β x cos β) = = f u ( β y u β x v)+f v ( β y v + β x u) f y = f y = f u u y + f v v y = = f u e γ (γ y cos β β y sin β)+f v e γ (γ y sin β + β y cos β) = = f u (β x u β y v)+f v (β x v + β y u) (30) If this is inserted in the backprojection equation we get ( fu (β x u β y v)+f v (β x v + β y u) ) u = ( f u ( β y u β x v)f v ( β y v + β x u) ) v f u (β x u 2 β y uv + β y uv + β x v 2 )=f v ( β y v 2 + β x uv β x uv β y u 2 ) This finally gives In this case we have Equation 31 then becomes f u β x = f v β y (31) β = 1 2 (a 1x + a 2 y + α) γ = 1 2 ( a 2x + a 1 y) (32) a 1 f u = a 2 f v (a 1 u + a 2 )f =0 (33) v If we make the substitution u = a 1 U + a 2 V (34) v = a 2 U a 1 V 11

13 we have U = a 1 u + a 2 v and the differential equation becomes f =0 (35) U which has the solution f(u, V ) = h 1 (V )=h 1 ( a2u a1v )=h a 2 2 (a 2 u a 1 v)= 1 +a2 2 = h 2 (a 2 e γ cos β a 1 e γ sin β) = = h 2 (e γ a a2 a1 2 cos(β + arctan a 2 )) And the final formula becomes (36) f(x, y) =e 1 2 ( a2x+a1y) cos( a1 2 x + a2 2 y + α a1 2 + arctan a 2 ) (37) Experiments Some functions from equation 37 is plotted in figure 6. a1 = 3 a2 = 3 a2 = 2 a2 = 1 a2 = 0 a2 = 1 a2 = 2 a2 = 3 a1 = 3 a1 = 2 a1 = 1 a1 = 0 a1 = 1 a1 = 2 Figure 6: Some examples of polar symmetries z = e i(a1x+a2y). The backprojection is found in equation 37 and they are plotted using trajectory functions g = (1 + cos(ω log f))/2. 12

14 4 Acknowledgment This work was supported by the Swedish Foundation for Strategic Research, project VISIT - VIsual Information Technology. References [1] F. Attneave. Some informational aspects of visual perception. Psychological Review, 61, [2] H. Bårman and G. H. Granlund. Corner detection using local symmetry. In Proceedings from SSAB Symposium on Picture Processing, Lund University, Sweden, March SSAB. Report LiTH ISY I 0935, Computer Vision Laboratory, Linköping University, Sweden, [3] Irving Biederman. Recognition-by-components: A theory of human image understanding. Psychological Review, 94(2): , [4] J. Bigün. Optimal orientation detection of circular symmetry. Report LiTH ISY I 0871, Computer Vision Laboratory, Linköping University, Sweden, [5] J. Bigün. Local Symmetry Features in Image Processing. PhD thesis, Linköping University, Sweden, Dissertation No 179, ISBN [6] Josef Bigün. Pattern recognition in images by symmetries and coordinate transformations. Computer Vision and Image Understanding, 68(3): , [7] Jack L. Gallant, Jochen Braun, and David C. Van Essen. Selectivity for polar, hyperbolic, and cartesian gratings in macaque visual cortex. Science, 259: , January [8] G. H. Granlund and H. Knutsson. Signal Processing for Computer Vision. Kluwer Academic Publishers, ISBN [9] Björn Johansson and Gösta Granlund. Fast Selective Detection of Rotational Symmetries using Normalized Inhibition. In Proceedings of the 6th European Conference on Computer Vision, volume I, pages , Dublin, Ireland, June [10] Björn Johansson, Hans Knutsson, and Gösta Granlund. Detecting Rotational Symmetries using Normalized Convolution. In Proceedings of the 15th International Conference on Pattern Recognition, volume 3, pages , Barcelona, Spain, September IAPR. [11] H. Knutsson and G. H. Granlund. Apparatus for determining the degree of variation of a feature in a region of an image that is divided into discrete picture elements. US-Patent , 1988, (Swedish patent 1986). 13

15 [12] H. Knutsson, M. Hedlund, and G. H. Granlund. Apparatus for determining the degree of consistency of a feature in a region of an image that is divided into discrete picture elements. US-Patent , 1988), (Swedish patent 1986). [13] Gerald Oster. Phosphenes. Scientific American, 222(2):82 87,

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