ADVANCED IMAGE PROCESSING CELLULAR NEURAL NETWORKS

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1 Papers International Journal of Bifurcation and Chaos Vol. 17 No. 4 (2007) c World Scientific Publishing Company ADVANCED IMAGE PROCESSING CELLULAR NEURAL NETWORKS MAKOTO ITOH Department of Information and Communication Engineering Fukuoka Institute of Technology Fukuoka Japan LEON O. CHUA Department of Electrical Engineering and Computer Sciences University of California Berkeley Berkeley CA USA Received January ; Revised November Many useful and well-known image processing templates for cellular neural networks (CNN s) can be derived from neural field models thereby providing a neural basis for the CNN paradigm. The potential ability of multitasking image processing is investigated by using these templates. Many visual illusions are simulated via CNN image processing. The ability of the CNN to mimic such high-level brain functions suggests possible applications of the CNN in cognitive engineering. Furthermore two kinds of painting-like image processings namely texture generation and illustration style transformation are investigated. Keywords: CNN; cortical information processing; cognitive science; cognitive engineering; illusion image processing; painting-like image processing; multitasking image processing; illustration style transformation; texture generation. 1. Introduction Cellular Neural Network (CNN) is a dynamic nonlinear system defined by coupling only identical simple dynamical systems called cells located within a prescribed sphere of influence such as nearest neighbors [Chua 1998; Chua & Roska 2002]. Because of its simplicity and ease for chip (hardware) implementation CNN has found numerous applications in image and video signal processing in robotic and biological visions and in higher brain functions. The dynamics of a CNN is governed by a set of nonlinear differential equations with parameters which must be derived analytically or numerically from given functions. Recently a method for designing CNN parameters was proposed [Itoh & Chua 2003]. In this design one has to choose a candidate from a set of basic CNN templates and tune its parameters to achieve a desired task. Almost all templates in this set were derived from neural field equations advection equations and reaction diffusion equations by discretizing spatial integrals and derivatives [Itoh & Chua 2005]. However some useful and well-known templates were not included in this set. The CNN templates are usually designed to achieve only one task. However the CNN has the potential ability to execute two or more tasks simultaneously since the CNN has three kinds of templates: a control (feed-forward) template afeedback template and a threshold template. We may tune these templates to accomplish multitask image processing. Visual illusion is defined as any fallacious perception of reality. Research on visual illusion can provide fundamental insights into the general brain mechanisms of perception and cognition. Many well-known visual illusions have been simulated by CNN image processing [Chua 1998; Chua & Roska 2002]. 1109

2 1110 M. Itoh & L. O. Chua A painting-like rendering technique usually employs digital image processing. There are several image processing techniques which can transform an image into a painting or other artistic styles. Since the CNN has pattern formation and selforganizing properties it has the potential ability to produce images with human expressive characteristics or important details found in real paintings. Studying painting-like image processing can also provide insights into general brain mechanisms. In this paper we first derive several well-known templates from some neural field models without output connections. Then we explore the potential ability of the CNN to achieve multitasking image processing by using these templates. Next we simulate many visual illusions via a CNN by using a set of basic templates. Finally we study various painting-like CNN image processing techniques. That is we investigate templates for generating textures in given images by exploiting CNN pattern formation properties. Furthermore we investigate CNN templates for transforming images into various artistic illustration styles. 2. From Continuous to Discrete Array The neural field [Amari 1978] is a mathematical model of the cortex. The neural field equation represents highly dense cortical neurons as a spatially continuous field. The dynamics of one-dimensional neural fields is defined by u(x t) xmax ( ) = u(x t)+ a(x x )f u(x t) dx t x min xmax + b(x x )s(x )dx + z(x) (1) x min where the interval [x min x max ] denotes the domain of the neural field u(x t) is the average membrane potential of a neuron at position x at time t a(x x ) and b(x x ) are the connectivity functions representing the average intensity of connections from neurons at location x to neurons at location x f(u) is the output function which determines the firing rate of the neuron as a function of its membrane potential s(x ) is the input stimulus applied externally to neurons at position x andz(x) isthe firing threshold of a neuron located at x [Amari 1978; Kubota & Aihara 2004]. The discretized version of Eq. (1) can be obtained by using the method in [Kubota & Aihara 2004]. Divide the domain of field [x min x max ]inton intervals [x i 1 x i ](i =1 2...N)withlength x where and x i = x min + i x (i =1 2...N) (2) x = x max x min. (3) N Let u i be the average membrane potential of neurons in the interval [x i 1 x i ] a ij and b ij be the average intensity of connections from neurons in the interval [x j 1 x j ] to neurons in the interval [x i 1 x i ] s i be the input stimulus applied externally to neurons in the interval [x j 1 x j ] and z i be the firing threshold in the interval [x j 1 x j ]. If we substitute the approximations u(x) u i u(x ) u j f(u j ) p j a(x x ) a ij b(x x ) b ij s(x ) s j z(x) z i (4) for each term in Eq. (1) and replace the integral by a summation we would obtain a one-dimensional fully-connected CNN equation [Chua 1998] du i = u i + = u i + N j=1 N j=1 (a ij p j + b ij s j ) x + z i (a ij p j + b ij s j ) + z i (5) where b ij = b ij x a ij = a ij x. If each cell (neurons in the interval [x i 1 x i ]) C i is coupled locally only to those neighbor cells inside some neighborhood (sphere of influence) N i theneq.(6)canbe written as du i = u i + ) (a ij p j + b ij s j +z i (6) j N i where N i denotes the r-neighborhood of cell C i. Consider next the two-dimensional neural field equation u(wt) = u(wt) t xmax ymax ( ) + a(w w ) f u(w t) dx dy + x min y min xmax ymax x min y min b(w w )s(w )dx dy + z(w) (7)

3 where w =(x y) R 2 and w =(x y ) R 2.In this case the domain [y min y max ] is also divided into M intervals [y i 1 y i ](i =1 2...M)withlength y. Thus the domain bounded by [x min x max ]and [y min y max ] is divided into MN subdomains. Let u ij be the average of u(x t) in the region bounded by [x i 1 x i ]and[y j 1 y j ]. Then the following standard CNN equation [Chua 1998; Chua & Roska 2002] can be obtained from Eq. (7) du ij = u ij + kl N ij (a kl p kl + b kl s kl ) +z ij (i j) {1...M} {1...N} (8) where u ij denotes the state of cell C ij p kl and s kl denote the output and input respectively of cell C kl N ij denotes the r-neighborhood of cell C ij and a kl b kl andz ij denote the feedback control and threshold template parameters respectively. The matrices [a kl ]andb =[b kl ] are referred to as the feedback template A and the feedforward (input) template B respectively. The output p ij and the state u ij of each cell are usually related via the piecewise-linear saturation function p ij = f(u ij )= 1 ( ) u ij +1 u ij 1. (9) 2 If we restrict the neighborhood radius of every cell to r = 1 and assume that z ij isthesameforthe whole network the template {A B z} is fully specified by 19 parameters which are the elements of the 3 3 matrices A and B namely α 1 1 α 10 α 11 α 0 1 α 00 α 01 α 1 1 α 10 α 11 β 1 1 β 10 β 11 B = β 0 1 β 00 β 01 (10) β 1 1 β 10 β 11 and the real number z. Thus the CNN is characterized by the following templates: B = α 1 1 α 10 α 11 α 0 1 α 00 α 01 α 1 1 α 10 α 11 β 1 1 β 10 β 11 β 0 1 β 00 β 01 z = z β 1 1 β 10 β 11 (11) Advanced Image Processing Cellular Neural Networks Image Processing Filter-Based CNN In this section we derive many well-known image processing filters from several neural field models without output connections Neural field models without output connections Consider the neural field equations without output connections namely a(w w ) = 0. Then Eq. (7) has the following form u(wt) = u(wt) t + xmax ymax x min y min b(w w )s(w )dx dy + z(w). (12) The solution u(wt)convergesto u(w ) = as t. xmax ymax x min y min b(w w )s(w )dx dy + z(w) (13) Spatial Fourier Transform-Based CNN Consider Eq. (12) over an infinite space that is [x min x max ] [y min y max ] =[ ] [ ]. (14) If the intensity function b(w w )isgivenby b(w w )=b(w )=e i(ω 1x +ω 2 y ) } (15) z(w) =0 then we obtain the following integral-differential equation u(wt) = u(wt) t + e i(ω 1x +ω 2 y ) s(w )dx dy. (16) The solution u(wt)convergesto U(ω 1 ω 2 ) = u(w ) = e i(ω 1x +ω 2 y ) s(w )dx dy (17) as t. Equation (17) is the spatial Fourier transform of s(w). The inverse spatial Fourier transform of U(ω 1 ω 2 )isgivenby

4 1112 M. Itoh & L. O. Chua s(w ) = 1 4π 2 e i(ω 1x +ω 2 y ) U(ω 1 ω 2 )dω 1 dω 2. (18) Applying the Riemann sum (see Riemann sum Wikipedia: The Free Encylopedia) to Eq. (16) and replacing u(wt)byu(ω 1 ω 2 ) we obtain du(ω 1 ω 2 ) = u(ω 1 ω 2 ) + n= m= s nm e i(ω 1n x+ω 2 m y) x y (19) where x = n x y = m y. (20) Setting x = y =1weobtain du(ω 1 ω 2 ) = u(ω 1 ω 2 ) + s nm e i(ω 1n+ω 2 m). (21) n= m= The solution u(ω 1 ω 2 )convergesto u(ω 1 ω 2 )= s nm e i(ω 1n+ω 2 m) (22) n= m= as t. The function u(ω 1 ω 2 ) is the discrete spatial Fourier transform of s nm.itsinverse transform is given by s nm = 1 4π 2 π π π π u(ω 1 ω 2 )e i(ω 1n+ω 2 m) dω 1 dω 2. (23) In the case of a finite array namely n {0 1...N 1} m {0 1...M 1} we usually use the following discrete spatial Fourier transform u(k l) = N 1 M 1 n=0 m=0 s nm e 2πi( kn N + lm M ). (24) Its inverse discrete spatial Fourier transform is given by s nm = 1 MN N 1 k=0 M 1 l=0 kn v(k l)e 2πi( N + lm M ). (25) Therefore from Eqs. (21) and (24) we obtain a discrete spatial Fourier transform-based CNN equation [Crounse & Chua 1995] du kl N 1 M 1 = u kl + b nm s nm (26) n=0 m=0 (k l) {0 1...N 1} {0 1...M 1} where b nm = e 2πi((kn/N)+(lm/M)).Thesolution u kl (t) converges to u kl ( ) = N 1 M 1 n=0 m=0 b nm s nm (27) as t. Spatial Cosine Transform-Based CNN Define a discrete cosine transform-based CNN by modifying Eq. (26) where du kl N 1 = u kl + α k α l b nm =cos π(2n +1)k 2N M 1 n=0 m=0 cos b nm s nm (28) π(2m +1)l 2M (29) and 1 k =0 N α k = 2 N 1 k N 1 (30) 1 l =0 M α l = 2 M 1 l M 1 The solution u kl (t) convergesto N 1 ũ kl = ukl ( ) =α k α l M 1 n=0 m=0 b nm s nm (31) as t. This equation corresponds to the discrete cosine transform. Its inverse discrete cosine transform is given by s mn = N 1 k=0 M 1 l=0 α k α l b kl ũ kl. (32) Spatial Filter-Based CNN Consider Eq. (12) over an infinite space that is [x min x max ] [y min y max ] =[ ] [ ]. (33) If b(w w )=b(w w ) we obtain a spatial filter u(wt) = u(wt) t + b(w w ) s(w )dx dy + z(w). (34)

5 The solution u(wt) converges to u(w ) = b(w w )s(w )dx dy + z(w) (35) as t. This equation can be transformed into the form U(ω 1 ω 2 )=B(ω 1 ω 2 )S(ω 1 ω 2 )+Z(ω 1 ω 2 ) (36) where U(ω 1 ω 2 ) B(ω 1 ω 2 ) S(ω 1 ω 2 )andz(ω 1 ω 2 ) are the spatial Fourier transforms of u(w ) b(w) s(w) and z(w) respectively that is U(ω 1 ω 2 ) = S(ω 1 ω 2 ) = B(ω 1 ω 2 ) = Z(ω 1 ω 2 ) = e iω 1x e iω 2y u(w )dxdy e iω 1x e iω 2y s(w)dxdy e iω 1x e iω 2y b(w)dxdy e iω 1x e iω 2y z(w)dxdy. (37) Their inverse spatial Fourier transforms are given by u(wt) = s(w) = b(w) = z(w) = e iω 1x e iω 2y U(ω 1 ω 2 )dω 1 dω 2 e iω1x e iω2y S(ω 1 ω 2 )dω 1 dω 2 e iω 1x e iω 2y B(ω 1 ω 2 )dω 1 dω 2 e iω1x e iω2y Z(ω 1 ω 2 )dω 1 dω 2. (38) Advanced Image Processing Cellular Neural Networks 1113 Their discrete versions are also given by U(k l) =B(k l)s(k l)+z(k l) (39) where U(k l) = N 1 M 1 kn u nm e 2πi( N + lm M ) and S(k l) = B(k l) = Z(k l) = u nm = 1 NM n=0 N 1 n=0 N 1 n=0 N 1 n=0 N 1 k=0 N 1 1 s nm = NM k=0 N 1 1 b nm = NM k=0 N 1 1 z nm = NM k=0 m=0 M 1 m=0 M 1 m=0 M 1 m=0 kn s nm e 2πi( N + lm M ) kn b nm e 2πi( N + lm M ) z nm e 2πi( kn N + lm M ) M 1 l=0 M 1 kn U(k l)e 2πi( N + lm M ) S(k l)e l=0 M 1 2πi( kn N + lm M ) B(k l)e 2πi( kn N + lm M ) l=0 Z(k l)e 2πi( kn N + lm M ). M 1 l=0 (40) (41) Thus the spatial filter for a finite array is realized by the equation du ij N 1 = u ij + k=0 M 1 l=0 b kl s kl + z ij (i j) (k l) {0 1...N 1} {0 1...M 1}. (42) If each cell is coupled locally only to all cells within a neighborhood of radius equal to r we would obtain a spatial filter-based CNN du ij = u ij + b kl s kl + z ij (kl) N ij (43) (i j) {0 1...N 1} {0 1...M 1}. The solution u ij (t) convergesto u ij ( ) = b kl s kl + z ij (44) (kl) N ij as t.

6 1114 M. Itoh & L. O. Chua Gaussian Filter-Based CNN Consider Eq. (12) over an infinite space [x min x max ] [y min y max ] =[ ] [ ]. (45) If the intensity function b(w w )isgivenby b(w w )=b(w w )= 1 w w 2 2πσ 2 e 2σ 2 = 1 (x x ) 2 +(y y ) 2 2πσ 2 e 2σ 2 (46) then we obtain the following integral-partial differential equation u(wt) t 1 = u(wt)+ 2πσ 2 e (x x ) 2 +(y y ) 2 2σ 2 s(w )dx dy + z(w) (47) where σ is a constant. The solution u(wt) converges to the Gaussian filter u(w ) = 1 (x x ) 2 +(y y ) 2 2πσ 2 e 2σ 2 s(w )dx dy + z(w) (48) as t. The Gaussian filter can be used as a low-pass filter for blurring or smoothing an image. If we set x =1 y =1 x =1 y =1 x = i x = i y = j y = j x = n x = n y = m y = m (49) and apply the Gauss integral formula to Eq. (48) (that is replace the integral by a summation and replace u(wt) s(w ) and z(w) byu ij s nm and z ij respectively) we would obtain du ij = u ij + n= m= 1 2πσ 2 e (i n)2 +(j m) 2 2σ 2 s nm + z ij. (50) If each cell is coupled locally only to all cells within a neighborhood of radius equal to r we would obtain a Gaussian filter-based CNN equation du ij = u ij + b kl s kl + z ij (kl) N ij (51) (i j) {1...N} {1...M} where b kl = 1 (i k) 2 +(j l) 2 2πσ 2 e 2σ 2. (52) The solution u ij (t) convergesto u ij ( ) = b kl s kl + z ij (53) (kl) N ij as t. If the intensity of the threshold is uniform namely z ij = z then Eq. (51) with σ =1and r = 2 (the neighborhood of cell C ij ) can be approximated by the following CNN templates B = b z = z (54) where b =1/273. Mean Filter-Based CNN Consider Eq. (12) in a finite space. If the intensity function b(w w ) is spatially uniform that is b(w w )=b we would obtain a mean filter described by u(wt) = u(wt) t xmax ymax + b(w w )s(w )dx dy + z(w) x min = u(wt) y min ymax xmax + b x min y min s(w ) dx dy + z(w). (55) The solution u(wt)convergesto xmax ymax u(w ) =b s(w )dx dy + z(w) x min y min (56)

7 as t. The mean filter is normally used to reduce noise in an image. Applying the Gauss integral formula to Eq. (55) we obtain the equation du ij = u ij + b N = u ij + b k=1 N k=1 M s kl x y + z ij l=1 M s kl + z ij (57) l=1 (i j) (k l) {1...M} {1...N} where b = b x y. If each cell is coupled locally only to all cells within a neighborhood of radius equal to r =1andz ij is spatially uniform that is z ij = z we would obtain a mean filter-based CNN equation du ( ij = u ij + b s i 1j 1 + s i 1j + s i 1j+1 + s ij 1 + s ij + s ij+1 + s i+1j 1 + s i+1j + s i+1j+1 ) + z. (58) The solution u ij (t) convergesto ( u ij ( ) =b s i 1j 1 + s i 1j + s i 1j+1 + s ij 1 + s ij + s ij+1 + s i+1j 1 + s i+1j + s i+1j+1 ) + z (59) as t. Thus Eq. (58) is characterized by the following template B = b z = z (60) where we usually set b =1/9.Ifthereisnoconnection between the diagonal cells and the center cell we would obtain the following template B = b z = z (61) where b =1/5. These templates can be used for noise removal. Nonuniform Mean Filter-Based CNN Consider the integral of the neural field equation: Advanced Image Processing Cellular Neural Networks 1115 where u(wt) = u(wt) t + b(w w)s(w ) dx dy + z(w). (62) b(w w) = b ϕ(x x)ϕ(y y) { ϕ(x) = 1 x d 0 x >d (63) and d is a constant. If we apply the iterated trapezoidal approximation [Itoh & Chua 2005] to this equation and each cell is coupled locally only to all cells within a neighborhood of radius equal to r =1( x = y = d) we would obtain du ij = u ij + b 4 ( s i 1j 1 + s i 1j + s i 1j+1 + s ij 1 + s ij+1 + s i+1j 1 + s i+1j + s i+1j+1 +8s ij ) x y + z ( = u ij + b s i 1j 1 + s i 1j + s i 1j+1 + s ij 1 + s ij+1 + s i+1j 1 + s i+1j + s i+1j+1 +8s ij ) + z (64) where b =(b /4) x y. Thesolutionu ij (t) converges to ( u ij ( ) =b s i 1j 1 + s i 1j + s i 1j+1 + s ij 1 + s ij+1 + s i+1j 1 + s i+1j + s i+1j+1 +8s ij ) + z (65) as t. Equation (64) is characterized by the following CNN template B = b z = z (66) We can also obtain the following CNN template without diagonal elements B = b z = z (67) where b =(b /2) x y. These templates can also be used for noise removal.

8 1116 M. Itoh & L. O. Chua 3.2. Neural field models with derivative inputs Consider the neural field equation u(wt) t = u(wt)+ b(w)s(w)dxdy + z(w). (68) If we replace the integral over x by the partial derivative with respect to x we would obtain the following partial differential equation u(wt) t = u(wt)+ b(w) s(w) dy + z(w). x (69) The solution u(wt) converges to u(w ) = b(w) s(w) dy + z(w) (70) x as t. Sobel Filter-Based CNN Assume that the intensity function b(w) anhe threshold z(w) satisfy b(w) =b ϕ(y y) z(w) =z. (71) Furthermore assume that each cell is coupled locally only to all cells within a neighborhood of radius equal to r = 1 ( y = d). If we apply the central difference formula and Simpson s rule [Itoh & Chua 2005] to the spatial derivative and the spatial integral in Eq. (69) respectively we would obtain the Sobel filter-based CNN equation du ij = u ij + b (s i 1j+1 s i 1j 1 )+2(s ij+1 s ij 1 )+(s i+1j+1 s i+1j 1 ) y + z x { } = u ij + b (s i 1j 1 s i 1j+1 )+2(s ij 1 s ij+1 )+(s i+1j 1 s i+1j+1 ) + z (72) or equivalently du ij = u ij + b (s i 1j+1 +2s ij+1 + s i+1j+1 ) (s i 1j 1 +2s ij 1 + s i+1j 1 ) y + z x { } = u ij + b (s i 1j 1 +2s ij 1 + s i+1j 1 ) (s i 1j+1 +2s ij+1 + s i+1j+1 ) + z (73) where b = b ( y/ x). The solution u ij (t) convergesto { } u ij ( ) =b (s i 1j 1 +2s ij 1 + s i+1j 1 ) (s i 1j+1 +2s ij+1 + s i+1j+1 ) + z (74) as t. Equation (73) is characterized by B = b z = z. (75) The Sobel filter can be used to detect edges in an image (see image processing Wikipedia: The Free Encyclopedia). Prewitt Filter-Based CNN Assume that the intensity function b(w) and the threshold z(w) satisfy b(w) =b ϕ(y y) z(w) =z (76) respectively. Apply the central difference formula and the Gauss integral formula to the spatial derivative and the spatial integral in Eq. (69) respectively. If each cell is coupled locally only to all cells within the neighborhood of radius r =1( x = y = d) we would obtain the Prewitt filter-based CNN equation du ij = u ij + b (s i 1j+1 s i 1j 1 )+(s ij+1 s ij 1 )+(s i+1j+1 s i+1j 1 ) y + z x { } = u ij + b (s i 1j 1 s i 1j+1 )+(s ij 1 s ij+1 )+(s i+1j 1 s i+1j+1 ) + z (77)

9 or equivalently du ij Advanced Image Processing Cellular Neural Networks 1117 = u ij + b (s i 1j+1 + s ij+1 + s i+1j+1 ) (s i 1j 1 + s ij 1 + s i+1j 1 ) y + z x { } = u ij + b (s i 1j 1 + s ij 1 + s i+1j 1 ) (s i 1j+1 + s ij+1 + s i+1j+1 ) + z (78) where b = b ( y/ x). The solution u ij (t) convergesto { } u ij ( ) =b (s i 1j 1 s i 1j+1 )+(s ij 1 s ij+1 )+(s i+1j 1 s i+1j+1 ) + z (79) as t. Equation (78) is characterized by the following CNN template B = b z = z (80) The Prewitt filter can be used to emphasize edges in an image (see image processing Wikipedia: The Free Encyclopedia). Consider next the neural field model without an integral term u(wt) = u(wt)+b(w) s(w) + z(w). (81) t x In this case the solution u(wt) converges to u(w ) =b(w) s(w) + z(w) (82) x as t. Roberts Filter-Based CNN Assume that the intensity function b(w) and the threshold z(w) are spatially uniform that is b(w) = b and z(w) = z. Applying the difference formula to the spatial derivative in Eq. (81) we obtain the following CNN equation du ij = u ij + b (s ij+1 s ij ) y + z x = u ij + b(s ij s ij+1 )+z (83) where b = b( y/ x). The solution u ij (t) converges to u ij ( ) =b(s ij s ij+1 )+z (84) as t. Equation (83) is characterized by the following CNN template B = b z = z (85) If we replace the partial derivative of x in Eq. (81) with the diagonal derivative we would obtain the Roberts filter-based CNN template B = b z = z (86) The Roberts filter can be used as a high-pass filter to enhance edges in an image (see image processing Wikipedia: The Free Encyclopedia) Neural field models with second derivative inputs Consider the neural field model with a second derivative input u(wt) t = u(wt)+ b(w) 2 s(w) x 2 dy + z(w). (87) In this case u(wt) converges to u(w ) = b(w) 2 s(w) x 2 dy + z(w) (88) as t. Edge Detection Filter-Based CNN Assume that the intensity function b(w) and the threshold z(w) satisfy b(w) =b ϕ(y y) z(w) =z. (89) If we apply the second derivative approximation [Itoh & Chua 2005] and the Gauss integral formula to the spatial derivative and the spatial integral in Eq. (87) respectively we would obtain the

10 1118 M. Itoh & L. O. Chua edge detection filter-based CNN equation { du ij = u ij + b (si 1j 1 2s i 1j + s i 1j+1 ) x 2 y + (s ij 1 2s ij + s ij+1 ) x 2 y + (s } i+1j 1 2s i+1j s i+1j+1 ) x 2 y + z { = u ij + b ( s i 1j 1 +2s i 1j s i 1j+1 )+( s ij 1 +2s ij s ij+1 ) } +( s i+1j 1 +2s i+1j s i+1j+1 ) + z (90) where b = b ( y/ x 2 ). Here we assumed that each cell is coupled locally only to all cells within a neighborhood of radius r =1( x = y = d). The solution u ij (t) convergesto { u ij ( ) =b ( s i 1j 1 +2s i 1j s i 1j+1 ) +( s ij 1 +2s ij s ij+1 )+( s i+1j 1 } +2s i+1j s i+1j+1 ) + z (91) as t. Equation (90) is characterized by the following CNN template: B =2b z = z (92) Line Detection Filter-Based CNN If the intensity function b(w) in Eq. (87) is not uniform we can obtain the line detection filterbased CNN equation du ij = u ij + ˆb { ( s i 1j 1 +s i 1j s i 1j+1 ) +( s ij 1 + s ij s ij+1 )+( s i+1j 1 } + s i+1j s i+1j+1 ) +z (93) which is characterized by where ˆb is a constant. B = ˆb z = z (94) 3.4. Neural field models with Laplacian inputs Consider the neural field equation u(wt) = u(wt)+ b(w)s(w)dxdy t + z(w). (95) If we replace the integral over x and y by the twodimensional Laplacian operator we would obtain the following partial differential equation ( ) u(wt) = u(wt)+ 2 b(w)s(w) t x 2 + ) (b(w)s(w) 2 y 2 + z(w). (96) The solution u(wt) converges ) ) (b(w)s(w) 2 (b(w)s(w) 2 u(w ) = x 2 + y 2 + z(w) (97) as t. Laplacian Filter-Based CNN Assume that the intensity function b(w) and the threshold z(w) are spatially uniform that is b(w) = b and z(w) = z. If we apply the second derivative approximation to Eq. (96) and assume x = y we would obtain the Laplacian filterbased CNN equation du ij = u ij + b s i 1j + s ij 1 + s ij+1 + s i+1j 4s ij x 2 + z ( = u ij + b s i 1j + s ij 1 + s ij+1 + s i+1j 4s ij )+z (98)

11 Advanced Image Processing Cellular Neural Networks 1119 where b = (b/ x 2 ). The solution u ij (t) converges to ( u ij ( ) =b s i 1j + s ij 1 + s ij+1 + s i+1j 4s ij ) + z (99) as t. The Laplacian filter-based CNN equation can be characterized by the following CNN template: B = b z = z (100) which can be used to detect edges of binary images. We also obtain the following CNN templates if we incorporate additional connections between the diagonal cells and the center cell: B = b z = z (101) To illustrate some application we show two wellknown CNN templates. One is the edge detection CNN template: B = b z = z (102) which extracts edges of objects in a binary image [Chua 1998]. The other is the corner detection CNN template: B = b z = (103) which extracts convex corners in a binary image [Chua 1998]. Sharpening Filter-Based CNN If the intensity function b(w) is not uniform we can obtain the following CNN template from Eq. (96) B = b z = z (104) If we change the sign of b that is setting b = b we would obtain the sharpening filter-based CNN template: B = b z = z (105) which can be used to enhance the sharpness of an image. We also obtain the following CNN template if we incorporate additional connections between the diagonal cells and the center cell B = b z = z. (106) 4. Multitask Image Processing In this section we explore the potential ability of the CNN to execute multitasking image processing tasks Pre-processors and post-processors The dynamics of the two-dimensional neural field equation u(wt) t = u(wt) ( ) + a(w w )f u(w t) dx dy ( ) + b(w w )s(w )dx dy + z(w) (107)

12 1120 M. Itoh & L. O. Chua is divided into two processes: one is the pre-process defined by β(w) = b(w w )s(w )dx dy + z(w) (108) and the other is the post-process defined by u(wt) t = u(wt) ( ) + a(w w )f u(w t) dx dy + β(w). (109) If we look at the post-processor (109) from another point of view it can be considered to be a neural field equation with a nonuniform threshold: z(w) = β(w) and zero input connections. On the other hand if the pre-processor is defined by γ(w) = b(w w )s(w )dx dy (110) then the post-process is given by u(wt) t = u(wt) ( ) + a(w w )f u(w t) dx dy + γ(w)+z(w) (111) which can be considered to be a neural field equation with both input stimulus γ(w) anhreshold z(w). Let ξ(w) be a stable equilibrium state of (107). Then ξ(w) satisfies ( ) ξ(w) = a(w w )f ξ(w) dx dy ( + = ) b(w w )s(w )dx dy + z(w) ( ) a(w w )f ξ(w) dx dy ( ) + γ(w)+z(w) = ( ) a(w w )f ξ(w) dx dy + β(w). (112) Observe that the pre-processed data β(w) =γ(w)+z(w) = b(w w )s(w )dx dy + z(w) (113) is modified by the nonlinear term ( ) a(w w )f ξ(w) dx dy. (114) The dynamics of the CNN equation du ij = u ij + a kl p kl kl N ij + b kl s kl + z ij (115) kl N ij is also divided into two processes: one is the preprocess defined by β ij = b kl s kl + z ij (116) kl N ij and the other is the post-process defined by du ij = u ij + a kl p kl + β ij. (117) kl N ij The post-processor (117) can be considered as a CNN equation with a nonuniform thresholds: z ij = β ij and zero input connections. On the other hand if the pre-processor is defined by the term γ ij = b kl s kl (118) kl N ij then the post-processor is given by du ij = u ij + a kl p kl + γ ij + z ij (119) kl N ij which can be considered as a CNN equation with input γ ij (without connections) and threshold z ij. Let ξ ij be a stable equilibrium state of (115). Then ξ ij satisfies ξ ij = ) a kl p kl + (γ ij + z ij kl N ij = a kl f(ξ kl )+β ij (120) kl N ij where β ij = γ ij +z ij. Observe that the pre-processed data β ij = γ ij + z ij = b kl s kl + z ij (121) kl N ij is modified by the nonlinear term a kl f(ξ kl ). (122) kl N ij Thus Eq. (115) has the potential ability to execute two tasks simultaneously.

13 4.2. Multipurpose post-processor Consider the following post-processor equation u(wt) ( ) = u(wt)+a f u(w t) dx dy t + bs(w)+z. (123) If we apply the trapezoidal rule to Eq. (123) we would obtain du ij = u ij + a ( f(u ij 1 )+f(u ij+1 )+f(u i 1j ) 2 ) + f(u i+1j )+4f(u ij ) x y + bs ij + z ( = u ij + a f(u ij 1 )+f(u ij+1 )+f(u i 1j ) ) + f(u i+1j )+4f(u ij ) +bs ij + z (124) which is characterized by the multipurpose postprocessor CNN template a B = b z = z (125) where a =(a/2) x y. Many useful functions can be realized by using this CNN template (see [Chua 1998; Roska et al. 1999; Hänggi et al. 1999; Itoh & Chua 2005]): 1. Hole-Filling CNN The template of the hole-filling CNN is given by B =6 z = 0. (126) The hole-filling CNN fills the interior of all contours in a binary image. 2. Filled-Contour Extraction CNN The template of the filled-contour extraction CNN is given by B =2 z = 4. (127) The filled-contour extraction CNN extracts from a binary image fed to initial states those regions which contain the boundary and which Advanced Image Processing Cellular Neural Networks 1121 completely filled the interior of all closed curves of an input image. 3. Selected Objects-Extraction CNN The template of the selected objects-extraction CNN is given by B =7 z = 1. (128) The selected objects-extraction CNN can be used to extract marked objects from a binary input image. 4. Face-Vase-Illusion CNN The template of the face-vase-illusion CNN is given by B = 6 z = 0. (129) The face-vase-illusion CNN simulates the wellknown visual illusion where the input image is perceived either as two symmetric faces or as a vase depending on the initial thought or attention. 5. Solid Black Framed Area-Extraction CNN The template of the solid black framed areaextraction CNN is given by B = 0.4 z = 2. The solid black framed area-extraction CNN extracts binary images representing objects of the initial state which fit and fill in the closed curve of a given binary image. 6. Noise Removal CNN The template of the noise removal CNN is given by B =0 z = 0. (130) (131) The noise removal CNN can be used to remove noise from an input image.

14 1122 M. Itoh & L. O. Chua If we incorporate the connections between the diagonal cells and the center cell we would obtain the following CNN template from Eq. (123): a B = b z = z (132) which can achieve the following tasks [Roska et al. 1999; Roska et al. 1999; Itoh & Chua 2005]: 1. Figure-Reconstruction CNN The template of the figure-reconstruction CNN is given by B =8 z = 6.5. (133) The figure-reconstruction CNN reconstructs a given input image from a part of an input image. 2. Concave-Filling CNN The template of the concave-filling CNN is given by B = z = 9. (134) The concave-filling CNN can be used to fill concave areas of a given binary image Mean filter-based image processing The template of the nonuniform mean filter is given by or B = b z = 0 B = b z = (135) (136) The image processing task of this filter is illustrated in Fig. 1. In all figures 1 in Secs. 4 and 5 white pixels represent the value of 1 black pixels +1. A red cell implies that its state u ij is greater than 1 whereas a green cell implies u ij < 1. Values between 1 and 1 are coded in gray-scale. Except for Fig. 22 all other figures are in gray scale since the output p ij of the CNN cell satisfies 1 p ij 1 (see Eq. (9)). We next show that many multitasking CNN templates can be obtained if we replace the template B of a multipurpose post-processor CNN with the template B of a nonuniform mean filter CNN. Fig. 1. Nonuniform mean filter CNN. 1 All numerical simulations in Secs. 4 and 5 are performed by using the CNN Simulator (Institute for Signal and Information Processing Laboratory ETHZ) [Hänggi et al. 1999].

15 Advanced Image Processing Cellular Neural Networks Mean Filter-Based Figure Reconstruction CNN The template of the figure reconstruction CNN is given by B = z = (137) The image processing task of the figure reconstruction template is illustrated in Fig. 2. Replacing the template B of Eq. (137) with that of Eq. (135) and choosing b =1/2 we obtain the following CNN template which can achieve two tasks: B = 1 2 z = (138) In this CNN noise removal is performed by the pre-processor and figure reconstruction is performed by the post-processor. The noise ɛ ij in the range ɛ ij < 1 is removed by this template (Fig. 3). The following template can also achieve the same tasks B = z = 6.5. (139) 2. Mean Filter-Based Concave-Filling CNN Replacing the template B of Eq. (134) with that of Eq. (135) and choosing b =1/16 and z =8 we obtain the following CNN template: B = 1 16 z = Fig. 2. Figure reconstruction CNN. (140) Fig. 3. Mean filter-based figure reconstruction CNN.

16 1124 M. Itoh & L. O. Chua which achieves both noise removal and concavefilling. The noise ɛ ij in the range 1 <ɛ ij 0is removed by this template (Fig. 4). 3. Mean Filter-Based Hole-Filling CNN Replacing the template B of Eq. (126) with that of Eq. (136) and choosing b =5/8 andz = 1.5 we obtain the following CNN template: B = 5 8 z = 1.5 (141) which achieves both noise removal and holefilling. The noise ɛ ij in the range ɛ ij < 1 is removed by this template (Fig. 5). 4. Mean Filter-Based Solid Black Framed Area Extraction CNN Replacing the template B of Eq. (131) with that of Eq. (136) and choosing b =1/20 and z = 2 we obtain the following CNN template: B = 1 20 z = 2 (142) which achieves both noise removal and solid black framed area extraction. The noise ɛ ij in the range ɛ ij < 1 is removed by this template (Fig. 6). 5. Mean Filter-Based Filled Contour Extraction CNN Replacing the template B of Eq. (127) with that of Eq. (136) and choosing b =1/2 andz = 3 we obtain the following CNN template: B = 1 2 z = 3 Fig. 4. Mean filter-based concave-filling CNN. (143) Fig. 5. Mean filter-based hole-filling CNN.

17 Advanced Image Processing Cellular Neural Networks 1125 Fig. 6. Mean filter-based solid black frame area extraction CNN. which achieves both noise removal and filled contour extraction. The noise ɛ ij in the range ɛ ij < 1 is removed by this template (Fig. 7) Sharpening filter-based image processing The template of the sharpening filter is given by B = b z = (144) We next show that many multitasking CNN templates can be obtained if we replace the template B of the multipurpose post-processor CNN with the B template of the sharpening CNN. 1. Sharpening Filter-Based Horizontal Hole Detection CNN The template of the horizontal hole detection CNN [Chua 1998] is given by B = z = 0. (145) Replacing the template B of Eq. (145) with that of Eq. (144) and choosing b = 1/10 and z = 0 we obtain the following CNN template: B = z = 0 (146) which has the potential ability to achieve both image sharpening and horizontal hole detection (Fig. 8). Fig. 7. Mean filter-based filled contour extraction CNN.

18 1126 M. Itoh & L. O. Chua Fig Sharpening Filter-Based Horizontal and Vertical Line Detection CNN The template of the horizontal and vertical line detection CNN [Itoh & Chua 2005] is given by B = z = 0. Sharpening filter-based horizontal hole detection CNN. (147) Replacing the template B of Eq. (147) with that of Eq. (144) and choosing b =1/3 andz =0we obtain the following CNN template: B = 1 3 z = (148) which has the potential ability to achieve both image sharpening and horizontal and vertical line detection (Fig. 9). 3. Sharpening Filter-Based Filled Contour Extraction CNN Replacing the template B of Eq. (127) with that of Eq. (-4) and choosing b =1/2 andz = 4 we obtain the following CNN template: B = z = 4 (149) which has the potential ability to achieve both image sharpening and filled contour extraction (Fig. 10). 4. Sharpening Filter-Based Hole-Filling CNN Replacing the template B of Eq. (126) with that of Eq. (144) and choosing b =5/9 andz = 4 Fig. 9. Sharpening filter-based horizontal and vertical line detection CNN.

19 Advanced Image Processing Cellular Neural Networks 1127 Fig. 10. Sharpening filter-based filled contour extraction CNN. we obtain the following CNN template: B = z = 4 which has the potential ability to achieve both image sharpening and hole-filling (Fig. 11). 5. Sharpening Filter-Based Solid Black Framed Area Extraction CNN Replacing the template B of Eq. (131) with that of Eq. (144) and choosing b =1/2 andz = 4.5 we obtain the following CNN template: B = (150) z = 4.5 (151) which has the potential ability to achieve both image sharpening and solid black framed area extraction (Fig. 12) Laplacian filter-based image processing Laplacian Filter-Based Horizontal Line Detection CNN The template of the threshold CNN [Chua 1998] is given by B = z = 0. (152) Fig. 11. Sharpening filter-based hole-filling CNN.

20 1128 M. Itoh & L. O. Chua Fig. 12. Sharpening filter-based solid black frame area extraction CNN. Fig. 13. Replacing the template B of Eq. (152) with that of the Laplacian-filter CNN B = b Laplacian filter-based Muller Lyer illusion CNN z = 0 (153) and choosing b =1/10 and z = 2.1 we obtain the Muller Lyer illusion CNN template: B = 1 10 z = (154) which simulates the well-known Muller Lyer illusion and removes the noise. The noise ɛ ij in the range ɛ ij < 1 is removed by this template (Fig. 13). 5. Visual Illusion CNN Visual illusion is defined as the fallacious perception of some actually existing object. Research on visual illusion can provide fundamental insights into general brain mechanisms of perception and cognition. Some well-known visual illusions such as the Muller Lyer illusion and the Herring-grid illusion have been simulated by the CNN image processing [Chua 1998; Chua & Roska 2002]. Because of its rich image processing properties the CNN can simulate many more visual illusions using simpler templates. For more details of visual illusions simulated in this paper see optical illusion Wikipedia: The Free Encyclopedia.

21 Advanced Image Processing Cellular Neural Networks Muller Lyer Illusion CNN The threshold CNN template [Chua 1998] is given by B = b z = z (155) where z is a constant. Replacing the template B of this CNN with that of the Laplacian-filter CNN (153) and choosing z = 2.1 we obtain the Muller Lyer illusion CNN template [Chua 1998]: B = b z = 2.1 (156) where b =0.1. The Muller Lyer illusion CNN simulates the visual illusion where two line segments of identical length but terminated respectively by converging and diverging arrowheads are perceived as if one segment is shorter than the other (Fig. 14). The following horizontal line detection CNN template with a threshold parameter is given in [Itoh & Chua 2005]: B = z = z. (157) Replacing the template B of this CNN with that of the Laplacian-filter CNN we obtain another kind of Muller Lyer illusion CNN template (Fig. 15); namely B = b z = 2.1 (158) which deletes the arrowheads. 2. Ponzo Illusion CNN I By choosing z = 1 in Eq. (157) we obtain the Ponzo illusion CNN template: B = z = 1. (159) The Ponzo illusion CNN simulates the visual illusion where two line segments of identical length inside two slanted lines like an inverted V are perceived as if one segment is shorter than the other (Fig. 16). The slanted lines can be further deleted if we use the template of the horizontal line detection CNN with z =0 (Fig. 17): B = z = 0. (160) Fig. 14. Muller Lyer illusion CNN.

22 1130 M. Itoh & L. O. Chua Fig. 15. Another kind of Muller Lyer illusion CNN. Fig. 16. Ponzo illusion CNN I. Fig. 17. Another kind of Ponzo illusion CNN I.

23 Advanced Image Processing Cellular Neural Networks Ponzo Illusion CNN II By replacing the template B of the threshold CNN (152) with that of the Laplacian-filter CNN and by choosing z = 10 we obtain yet another kind of Ponzo illusion CNN template: B = z = (161) This Ponzo illusion type CNN simulates the visual illusion where line segments of identical length inside two slanted lines are perceived as if one segment is shorter than the others (Fig. 18). 4. Horizontal-Vertical Illusion CNN By replacing the template B of the threshold CNN with that of the nonuniform mean filter CNN and by choosing z = 5 we obtain the following Horizontal-Vertical illusion CNN template: B = b z = (162) The horizontal-vertical illusion CNN simulates the visual illusion where the vertical line segmentisperceiveobelongerthananequallength horizontal line segment (Fig. 19). In order to show the visual illusion effectively two pictures like an inverted T are given as initial input images. Observe the illusion is persistent even if the vertical lines have different thicknesses. 5. Herring-Grid Illusion CNN The template of the Laplacian post-processor CNN is given by a B = z = z. (163) By replacing the template B of this CNN with that of the Laplacian-filter CNN and by choosing z = 1 we obtain the Herring-grid illusion CNN template in [Chua 1998] B = z = 1 (164) where the boundary condition for the input is fixed to 0. The Herring-grid illusion CNN simulates the visual illusion where an input image of closely-spaced uniform black squares is perceived at a distance as if there were gray patches present at the intersections of the horizontal and vertical white grid lines separating the squares (Fig. 20). 6. Mach Effect Illusion CNN The template of the sharpening filter CNN can be used to obtain the following Mach effect Fig. 18. Ponzo illusion CNN II.

24 1132 M. Itoh & L. O. Chua (d) input image. (e) initial state. (f) output image. Fig. 19. Horizontal-vertical illusion CNN. (a) input image. (b) initial state. (c) output image at t =1.2. Fig. 20. Herring illusion CNN.

25 Advanced Image Processing Cellular Neural Networks 1133 illusion CNN template: B = z = 0 q q q q 1+8q q q q q (165) where q =0.2. The Mach effect illusion CNN simulates the visual perception of edge enhancement seen at boundaries between abrupt changes in luminance (Fig. 21). 7. Ebbinghaus Illusion CNN The template of the multipurpose postprocessor can be used to obtain the following Ebbinghaus illusion CNN template: B = z = 8. (166) The Ebbinghaus illusion CNN simulates the visual illusion where two central squares of the same size appear to be of different sizes (Fig. 22). 8. Poggendorff Illusion CNN The template of the multipurpose postprocessor can be used to obtain the following Poggendorff illusion CNN template: B = z = (167) Note that some elements of the template A are zero. The Poggendorff illusion CNN simulates the visual illusion where the two ends of a straight line segment passing behind an obscuring rectangle are perceived to have an offset (Fig. 23). 9. Ambiguous Illusion CNN Using the logic not operation template with Fig. 21. Mach effect illusion CNN. Fig. 22. Ebbinghaus illusion CNN.

26 1134 M. Itoh & L. O. Chua Fig. 23. a threshold parameter [Chua 1998] we obtain the ambiguous illusion CNN templates: B = z = c Poggendorff illusion CNN. (168) where c = 1 or 1. The ambiguous illusion CNN simulates the visual illusion where you either see the word LIFE or you see five white shapes. Once you are able to read the letters it is difficult not to see the word (Fig. 24). Observe that unlike the other images shown so far the images displayed in Figs. 24(c) and 24(f) are not gray-scale output images but are color images representing coded values of the cell state variables. By using the cell state image we can see the ambiguous illusion clearly. 10. Face-Vase Illusion CNN As stated in the previous section the face-vase illusion CNN template [Chua 1998; Itoh & Chua 2003] is given by: B = z = (169) The face-vase illusion CNN simulates the visual illusion where the image is perceived either as two symmetric faces or as a vase depending on the initial thought or attention (Fig. 25). 11. Stripe Illusion CNN The template of the one-dimensional nonuniform mean filter is given by B = z = z (170) Replacing the feedback template A of the stripe illusion CNN with that of the threshold CNN template and choosing z = 4 we obtain the following stripe illusion CNN template: B = z = (171) The stripe illusion CNN simulates the visual illusion where a square is perceived to be longer in the vertical direction when a square is drawn with horizontal lines (Fig. 26). 12. Baldwin Illusion CNN Replacing the template B of the threshold CNN with that of the uniform mean filter CNN we obtain the following Baldwin illusion CNN template: B = z = 2. (172) The Baldwin illusion CNN simulates the visual illusion where the distance between the two

27 Advanced Image Processing Cellular Neural Networks 1135 (a) input image. (b) initial state. (c) cell state image. (d) input image. (e) initial state. (f) cell state image. (g) input image. (h) initial state. (i) output image at t =0.4. (j) input image. (k) initial state. (l) output image at t = 20. Fig. 24. Ambiguous illusion CNN. (a) (f) c = 1. (g) (l) c = 1. Note that the figures (c) and (f) are the cell state images and the figures (i) and (l) are the output images for c = 1 att =0.4 an = 20 respectively. small boxes on the right appears to be longer than the distance between the two large boxes on the left. The distance between the two large boxes is the same as the distance between the two small boxes (Fig. 27). 13. Munsterberg Illusion CNN (Cafe-Wall Illusion CNN) Replacing the template B of the multipurpose post-processor with that of the nonuniform mean filter we obtain the following

28 1136 M. Itoh & L. O. Chua (a) input image. (b) initial state. (c) cell state image. (d) input image. (e) initial state. (f) cell state image. Fig. 25. Face-vase illusion CNN. Fig. 26. Stripe illusion CNN.

29 Advanced Image Processing Cellular Neural Networks 1137 Fig. 27. Munsterberg illusion CNN (Cafe-wall illusion CNN) template: B = z = Baldwin illusion CNN. (173) The Munsterberg illusion CNN simulates the visual illusion where the row of black rectangles is perceived as sloping alternately upward and downward (Fig. 28). 14. Size Illusion CNN The template of the multipurpose postprocessor can be used to obtain the following size illusion CNN: B = z = 8. (174) The size illusion CNN simulates the visual illusion where an expanding rectangle appears to be larger than a shrinking rectangle (Fig. 29). 15. Enhanced Darkness Illusion CNN The template of the nonuniform mean filter CNN can be used to obtain the following template of the enhanced darkness illusion CNN: B = z = 0. (175) The enhanced darkness illusion CNN simulates the visual illusion where all gray squares have the same brightness but the gray squares on black background appear to be darker than those on white background (Fig. 30). 16. Enhanced Brightness Illusion CNN The template of the sharpening filter CNN can be used to obtain the following template of the enhanced brightness illusion CNN: B = z = (176) The enhanced brightness illusion CNN simulates the visual illusion where two gray objects have the same brightness but the gray object on black background appears to be brighter than that on white background (Fig. 31). 17. Zöllner Illusion CNN Replacing the template B of the horizontal-line detection CNN with that of the Laplacian-filter CNN and choosing z = 5 we obtain Zöllner illusion CNN template: B = z = (177) The Zöllner illusion CNN simulates the visual illusion where two parallel lines appear to be tilted at an angle or to be expanded. In this computer simulation the output image indicates the expanding direction (Fig. 32).

30 1138 M. Itoh & L. O. Chua (a) input image. (b) initial state. (c) output image at t =0.33. (d) input image. (e) initial state. (f) output image at t =1.1. Fig. 28. Munsterberg illusion CNN. Fig. 29. Size illusion CNN.

31 Advanced Image Processing Cellular Neural Networks 1139 (a) input image. (b) initial state. (c) output image at t =0.3. Fig. 30. Enhanced darkness illusion CNN. Fig. 31. Enhanced brightness illusion CNN. Fig. 32. Zöllner illusion CNN.

32 1140 M. Itoh & L. O. Chua 18. Bourdon Illusion CNN Replacing the template B of the multipurpose post-processor CNN with that of the diagonal Laplacian-filter CNN and choosing z = 6 we obtain the following Bourdon illusion CNN template: 19. Oppel-Kun Illusion CNN Replacing the template B of the threshold CNN with that of the one-dimensional nonuniform mean-filter CNN and choosing z = 4 we obtain the following Oppel Kun illusion CNN template: B = B = (179) z = 6 (178) where the boundary condition for the input is fixed to 0. The Bourdon illusion CNN simulates the visual illusion where the right upper line of two triangles appears to tilt clockwise and the left counter-clockwise (Fig. 33). z = 4. The Oppel Kun illusion CNN simulates the visual illusion where a space containing nothing looks narrower than a space divided by some objects (Fig. 34). 20. Delboeuf Illusion CNN Replacing the template B of the threshold CNN with that of the mean-filter CNN and choosing (a) input image. (b) initial state. (c) output image at t =2.4. Fig. 33. Bourdon illusion CNN. Fig. 34. Oppel Kun illusion CNN.

33 Advanced Image Processing Cellular Neural Networks 1141 Fig. 35. Delboeuf illusion CNN. z = 2 we obtain the following Delboeuf illusion CNN template: B = z = 2. (180) The Delboeuf illusion CNN simulates the visual illusion where a square placed on a big square looks smaller than that placed on a small square (Fig. 35). 6. Painting-Like Image Processing CNN There are several image processing techniques which can transform an image into a painting or other artistic styles. The CNN can produce images with human expressive characteristics or important details found in real paintings since it has pattern formation and self-organizing properties. Research on painting-like image processing of the CNN may provide some insight into general brain mechanisms Texture generation image processing In this section we show that the CNN pattern formation property can generate textures in images Checkerboard Texture The template of checkerboard pattern formation CNN [Chua 1998] is given by B = b z = 0 (181) where b = 0. This template generates a checkerboard pattern under a random initial condition. If we set b = 2 then this template generates an interesting checkerboard texture in a given image as shown in Fig Striped Texture The template of the striped pattern formation CNN [Chua 1998] is given by B = b z = 0 (182) where b = 0. This template generates a striped pattern under a random initial condition. If we set b = 2 then this template generates an interesting striped texture in a given image as shown in Fig All numerical simulations in this section are performed by using the simulator CELL (Circuit Lab. Department of Electric Engineering University of Rome Tor Vergata ). In this simulator color image processing is possible by means of a three-layer network (R-G-B). We applied the same template to each color component.

34 1142 M. Itoh & L. O. Chua (a) input image. (b) initial state. (c) output image at t = 20. Fig. 36. Checker texture. (a) input image. (b) initial state. (c) output image at t = 10. Fig Cortex Texture The template of the cortex pattern formation CNN [Chua 1998] is given by B = b z = z (183) where b =0andz = 0. This template generates cortex patterns under random initial conditions. For b = this template generates an interesting cortex texture in a given image as shown in Figs Striped texture. 4. Squiggle Texture The template of the squiggle pattern formation CNN [Chua 1998] is given by B = b z = z (184) where b = 0. This template generates a cortex pattern under a random initial condition. If we set b =20andz = 0 then this template generates an interesting squiggle texture in a given image as shown in Fig. 41.

35 Advanced Image Processing Cellular Neural Networks 1143 (a) input image. (b) initial state. (c) output image at t = 10. Fig. 38. Cortex texture (b = 1.5). (a) input image. (b) initial state. (c) output image at t = 10. Fig. 39. Cortex texture (b = 2). (a) input image. (b) initial state. (c) output image at t = 10. Fig. 40. Cortex texture (b = 3).

36 1144 M. Itoh & L. O. Chua (a) input image. (b) initial state. (c) output image at t = Fingerprint Texture The template of the fingerprint enhancement CNN [Crounse & Chua 1995] is given by a B = Fig. 41. Squiggle texture (b = 20 z = 0). z = 0 (185) where a = 0. This template generates fingerprint patterns under random initial conditions. If a =1ora = 2 then it generates a fingerprint texture in given input images as shown in Figs. 42 and Maze Texture The templates of maze pattern formation CNNs are given by 1 8 and B = z = 0 (186) z = 0. B = (187) These templates generate maze (or striped) patterns under random initial conditions. They also (a) input image. (b) initial state. (c) output image at t = 10. Fig. 42. Fingerprint texture (α = 1).

37 Advanced Image Processing Cellular Neural Networks 1145 (a) input image. (b) initial state. (c) output image at t = 10. generate a maze (or striped) texture in given input images as shown in Figs. 44 and Miscellaneous Textures The following templates also generate interesting textures in given input images as shown in Figs B = z = 0. B = 1 2 z = 0. B = 1 2 z = 0. Fig. 43. Fingerprint texture (α = 2) (188) 6.2. Illustration-style image processing The CNN can transform given input images into various interesting artistic illustration styles because it is endowed with the pattern formation property. We next show that some well-known CNN templates produce illustration-like images from a mandrill image. Multipurpose post-processor CNN template is given by B = z = z (189) where z =5orz = 9. This template produces the illustration-like images in Figs. 49 and 50. Patch pattern formation CNN template [Chua 1998] is given by B = 0 b 0 z = 0 (190) where b =0orb = 0.5. This template produces the illustration-like images in Figs Cortex pattern formation CNN template is given by B = z = 1 (191) which produces the illustration-like image in Fig. 53.

38 1146 M. Itoh & L. O. Chua (a) input image. (b) initial state. (c) output image at t = 10. Fig. 44. Maze texture I. (a) input image. (b) initial state. (c) output image at t =3.5. Fig. 45. Maze texture II. (a) input image. (b) initial state. (c) output image at t = 10. Fig. 46. Miscellaneous texture I.

39 Advanced Image Processing Cellular Neural Networks 1147 (a) input image. (b) initial state. (c) output image at t = 10. Fig. 47. Miscellaneous texture II. (a) input image. (b) initial state. (c) output image at t = 10. Fig. 48. Miscellaneous texture III. (a) input image. (b) initial state. (c) output image at t =1.7. Fig. 49. Multipurpose post-processor CNN (z = 5).

40 1148 M. Itoh & L. O. Chua (a) input image. (b) initial state. (c) output image at t =0.06. Fig. 50. Multipurpose post-processor CNN (z = 9). (a) input image. (b) initial state. (c) output image at t = 10. Fig. 51. Patch pattern formation CNN (b = 0). (a) input image. (b) initial state. (c) output image at t = 20. Fig. 52. Patch pattern formation CNN (b = 0.5).