A method for defining analog circuits for the minimization of discrete functionals: An image processing application

CIRCUITS SYSTEMS SIGNAL PROCESS

VoL 18, No. 5, 1999, PP. 457-477

A METHOD FOR DEFINING ANALOG CIRCUITS FOR THE MINIMIZATION OF DISCRETE FUNCTIONALS: AN IMAGE PROCESSING APPLICATION*

Marco Storace, 1 Mauro Parodi, 1 Diego Pastorino, and Veronica Tripodoro

Abstract. The solutions of many physical-mathematical problems can be obtained by minimizing proper functionals. In the literature, some methods for the synthesis of analog circuits (mainly cellular neural networks) are presented that find the solution of some of these problems by implementing the discretized Euler-Lagrange equations associated with the pertinent functionals.

In this paper, we propose a method for defining analog circuits that directly minimize (in a parallel way) a class of discretized functionals in the frequently occurring case where the solution depends on two spatial variables. The method is a generalization of the one presented in Parodi et al., Internat. J. Circuit TheoryAppl., 26, 477--498, 1998. The analog circuits consist of both a (nonlinear) resistive part and a set of linear capacitors, whose steady-state voltages represent the discrete solution to the problem. The method is based on the potential (co-content) functions associated with voltage-controlled resistive elements. As an example, we describe an application in the field of image processing: the restoration of color images corrupted by additive noise. Key words: Nonlinear circuit theory, co-content, functional minimization, image processing.

1. Introduction

Analog circuits for parallel information processing are being increasingly used to solve a large variety of problems, thus avoiding the bottleneck of serial processing. Classical examples concern cellular neural networks (CNNs), which are exploited, for instance, to model population dynamics [11], for image processing, and, more generally, to solve many kinds of partial differential equations (such as reaction- diffusion) [7].

* Received February 9, 1998; revised April 23, 1999. This work was supported by the M.U.R.S.T. research project, "Neural and non-linear circuits for one- and multi-dimensional signal processing applications" and by the University of Genoa.

1Biophysical and Electronic Engineering Department, University of Genoa, Via Opera Pia 1 l a, 1-16145 Genoa, Italy. E-mail: [email protected].

458 STORACE, PARODI, PASTORINO, AND TRIPODORO

Some of these circuit architectures refer to the CNN elementary cell defined by Chua in [9]. In other cases, the circuit structure is derived from the mathematical formulation of the problem considered. The definition of circuits able to minimize functionals in a parallel way is certainly one of the most interesting goals of circuit theorists and designers, considering the wide field of application, in particular for image processing. From a general point of view, the definition of an image processing circuit requires a discretized image, i.e., an image composed of a finite number of state variables. The continuous representation f (x , y) of the image then must be approximated by considering finite sets {J~k} of variables (e.g., grey levels or RGB brightness levels). This is the basic prerequisite for processing the image through a lumped circuit. The nature and the characteristics of the circuit elements depend on the kind of processing to be performed.

The most common early vision problems require the extraction of some information (e.g., edges) from the image of a scene. Generally speaking, such an image is degraded by discretization, noise, sensor nonlinearity, or other unwelcome dis- tortions. In order to solve some ill-posed [22] early vision problems, the class of admissible solutions must be restricted by using suitable a priori knowledge, either in the form of variational constraints on the possible solutions or in terms of statistical properties of the solution space, as outlined in [22].

In variational regularization, the solution space is restricted by a functional consisting of a sum of terms that constrain the solution to be "regular." The solution can be obtained by minimizing the functional [28] or by solving the Euler equations associated with it [18], [30]. Some examples of lumped circuits for image processing (in particular, CNNs) can be found in both cases (e.g., see [15], [171, [211, [23], [24], [27]).

In the simplest case, the image-discretization process can be accomplished by considering a finite set of variables {J)k} at the nodes of a regular grid (e.g., a square-mesh grid) and by approximating spatial operators (e.g., the gradient) of the functional by finite-difference techniques [11. More refined approaches subdivide the image region into subdomains of finite elements and approximate the image values in each element in terms of a limited number of parameters [1O]. In probabilistic reguIarization, the functional is obtained directly in discrete form by defining the problem as a Bayesian estimation of proper probability functions [121, [13].

In both cases, the problem can be formulated in terms of the minimization (with respect to the variables representing the image) of a functional H that is a weighted sum of two terms: H = H1 + ~H2 [13]. For discretized images, the regularization term H~ is a double sum of local potential functions q~ and restricts the class of admissible solutions by imposing a smoothness constraint: H1 = ~ ~ rb(fjk, fpq) [13], [22]. The first sum is extended to all pixels of the image. The second sum, in the simplest case, ranges over the set Njk of all the horizontal and vertical nearest neighbor pixels at (p, q). The data term H2 is a single sum of local potential functions qJ that constrains the solutions to remain close to observed data:/42 = ~ qJ(3)~; )~k) [13], [22]. The sum extends over all

ANALOG CIRCUITS MINIMIZING DISCRETE FUNCTIONALS 459

pixels at (j, k) of the image, and the set offixed brightness values {3~k } represents the (degraded) image to be processed. The constant term )~ is a weighting coefficient that balances the influence between the two terms of H (note that X multiplies the data term, hence it is just the inverse of the "regularization parameter").

The functional H for discretized images is the starting point in this paper. The problem of the discretization of continuous functionals, either by finite-element methods (see [24]) or by finite-difference methods (see [26]), is not discussed here. For definiteness, the local functions ~ and ~ considered are taken according to the guidelines proposed in [13], but the key concepts of the method proposed in this paper are more general and can also be applied to local functions found on the basis of other criteria.

A circuit that finds a minimum of the discretized functional H can be obtained by taking advantage of the properties of the potential functions (e.g., the co-content function) of a proper resistive multiterminal structure. The circuit elements are defined by using rather a general method (whose guidelines were given in [19]) that relates the 2D array corresponding to the discretized image to a structure consisting of both a resistive grid and a set of dynamic elements. The nonlinear characteristics of the resistive elements of the circuit are derived from the functional H and the co-content concept. The circuit structure and its properties are presented and discussed with reference to some general theorems of circuit theory, thus setting the basis for any concrete realization. Starting from the case of grey-level images and introducing proper generalizations, the method is finally applied to define an analog circuit for the restoration of color images corrupted by Gaussian noise.

The essentials of the method are summarized in Section 2. In particular, Section 2.2 discusses the definition of �9 according to the constraints defined in [13] 1 for grey-level images. In Section 3, the method is applied to the restoration of RGB color images corrupted by Gaussian noise. Section 4 provides some image processing examples.

2. Outline of the method

As stated in Section 1, the specific problem that we consider in this paper is the definition (from a circuit theory point of view) of a circuit to restore color images corrupted by null-mean Gaussian white noise. This goal is attained through the minimization of a functional H for a discretized image.

The basic idea of the method is to associate the functional H with a potential (co- content) function ~ of a proper voltage-controlled resistive multiterminal element. This multiterminal resistor, connected to a set of linear capacitors, provides a

1 In the Markov model, dp is the potential function of the Gibbs energy, and minimizing H corresponds to a statistical maximum a posteriori criterion (MAP) [13].


(dynamic) circuit whose equilibrium state corresponds to a stationary point (usually a minimum) of the co-content function of its resistive part [5]. We assume that the variables (grey levels or RGB brightness levels) {3~k} representing the discretized image are taken at the nodes (j, k) of a square-mesh grid on the image plane~ Denoting by f the set of variables {3~k} of the discretized image, the considered functional H ( f ) (hence the corresponding ~) can be written as a sum of local functions, each depending on a limited number of variables fjk. Then, because of the additive property of the co-content [5], [ 16] in the corresponding expression for ~, each local function can be interpreted as the co-content of a voltage-controlled resistive element of the circuit that represents an elementary constituent, or tessera, of the resulting multiterminal element.

An application of this method to the restoration of grey-level noisy images was proposed in [19], [25]. As an introduction to the more general case of a color image, we now summarize some elements of the method for the case of grey-level images.

2.t. Basic definitions.

The co-content concept is strictly static, as it concerns resistive circuits only. The co-content G (v) of a voltage-controlled two-terminal nonlinear resistor described by the constitutive relation i = g(v) can be expressed as [16]

G(v) = g(~) d~. (1)

The lower limit of integration does not need to be specified until the function is actually evaluated. Usually, it is set at the point v = 0.

According to a well-known theorem [5], [ 16], the total co-content G of a resistive multiterminal structure consisting of two-terminal resistors can be obtained as the sum of the co-contents of all the structure branches. As a general statement, two potential (co-content) functions ~1 and ~2 can be associated with the terms Hi (f_f) and ~.H2(_.f) of the functional pertinent to the problem under consideration. In the case of grey-level images, for instance, any dimensionless grey level fj'k over the range [0,255] is related to a voltage value vjk over the range [Vmin, Vmax] via the linear transformation

vjk = Vafj~ + Vmin, (2)

Vmax--Vmi~ ThUS, the voltages vjk represent the grey levels of the where VA = 255 �9 discretized image and are the variables of the potential functions ~1 and G2. The functions ~1 and G2 are interpreted as the co-content functions of resistive multiterminal structures (9~1 and 9~2, respectively) that can be synthesized by using two-terminal resistors only.

A generalization of this method to the case of RGB color images will be proposed in Section 3.


0

A

Figure 1. Typical idealized example of a local potential function qb (j~ - fpq).

2.2. Definition o f g~ 1 (for grey-level images).

Following [13], the generic structure of the term H1 ( f ) to be considered is

H l ( f ) ~ j ~ k ~ d~ ( '~k- fpq) -- pqENjk S "

(3)

As shown in [13], in the case of image restoration and edge detection, d~(x) is even, increasing for x ~ 9~ +, and l i m x ~ q~(x) = 0. A typical idealized example of local potential function qb is shown in Figure 1, together with the "threshold" parameter A. The analytical expression of this function is inessential on an introductory level; hence, for conciseness, we do not report it here. The term A is a positive constant (e.g., see [13] for details), and Njk = {fpq:l (J, k) - (p, q)12 < r} denotes the neighborhood set of order r (integer number) of the pixel at (j, k). In the following discussion, only the first-order neighborhood set Njk = {(j - 1, k), ( j + 1, k), (j, k - 1), (j, k + 1)} will be considered.

The shape of q~ in Figure 1 can be intuitively derived by making reference to a single potential term of the sum (3). If IY~k - fpq[ < A, in the dynamic process towards equilibrium the difference 3~k - fpq tends to approach the value 0, corresponding to the minimum value of ~; by contrast, if [J~k - fpq[ > A (horizontal arms of ~), the difference tends to remain constant in the minimization process. The former case corresponds to the absence of edges between the pixels at (j, k) and (p, q), whereas the latter case corresponds to the presence of an edge.


. . . . . . . . . . . . . .

~A

- V T 0 V T Vjk__Vp q

Figure 2. Typical idealized example of a local potential function +(vjk - Vpq).

The local potential function �9 can be associated with the co-content + of a nonlinear resistor Rjkpq connecting the generic pixels at (j, k) and (p, q). Co- content �9 (see Figure 2) must have the same shape as ~, but

(i) a constant term can be added to it, and

(ii) it can be scaled by a constant coefficient, provided that the same scaling operation is performed on the qJ function in Ha,

because, in either case, the positions of the minima of the global functional H are unchanged.

Without loss of generality, the independent variable of + is the difference I)jk -- Vpq (corresponding to fjk -- fpq according to equation (2)). The voltage parameter VT corresponds to A, and the relation between + and �9 is

+(1)jk -- ~Jpq) = di.~M [1-~-rJi.~( OJk ~?pq ) t , (4)

where the constant term c~m has been introduced in accordance with (i) and (ii). Because of the square-mesh grid distribution of the pixels, the term Hi ( f )

corresponds to the total co-content G1 (v) = ~ j k ~pq~Njk ~(1)J k -- Upq) of agrid of identical (under the assumption of first-order neighborhood set) voltage-controlled nonlinear resistors R1. These resistors connect each pixel at (j, k) to its neighbors in Njk, as shown in Figure 3. The original set of variables _f is represented by the vector v of the node voltages vjk with respect to the ground.


jk ~ pq / . / " / I ,I~ R, ~V j-l,k-l./" ~ j-l,k /"" r'rT'rr-~ j-l,k+l ..... " "j~ L...."~ ............. . . , ~ _ _ - ~ ..............

...... .j.',k:.! J, . j,k+ ..............

.J..+.!,_k.zL_/ ~ j + ~ , k / ~ J+I,~+J.Z_ ........

., ..e !

Figure 3. The resistive grid ~)~1'

The current ijgpq that flows through the resistor RI results from the tocal co-

. d$ The obtained characteristic can be easily content function ~ as tjkpq ~--- d(vjk_Opq).

included in the class of nonlinear functions for anisotropic diffusion methods [3], [20]. These methods are strictly related to the variational or probabilistic methods that tackle image processing problems by minimizing a functional [4], [20], [30].

2.3. Definition of fit2 (for grey-level images).

The generic structure of the second addend of H is ).H2(_.f) = ~ j k )~k~ (3~k; J~k),

where the minimum value of the local potential function q~(~k; ~k) is obtained when J~k is equal to the constant term J~k (i.e., the grey level of the pixel at (j, k) in the image to be processed). Following [13], here ~(J~k; J~k) = (J~k -- J~k) 2. Techniques for determining X are provided by standard regularization theory [29]. In the case of images corrupted by null-mean Gaussian white noise with variance cr 2, the maximum a posteriori criterion [13] leads us to assign the value ~ to the regularization parameter ~. (see the Appendix).

In turn, the local potential function )~ku(j)k; ~k) can be associated with the co-content ~ (vjk; f)jk) of a nonlinear resistor R2jk connecting the node j k (at the voltage vjk) to the ground. The constant term vjk corresponds to 3~k and must be interpreted as a parameter of the R2jk characteristic. According to equation (2),


R2jk R

Vjk ,'Sk /

/ I /

/

Figure 4. The nonlinear resistor R2jk corresponding to the local co-content function ^ X~M, klJ(Vjk; ~)jk) = V2a tUjk -- ~jk) 2.

the relation between ~ and qJ can be expressed, without loss of generality, as

' (5)

Then, the term X H2 ( f ) corresponds to the total co-content G2 (v) = Y~j k ~ (vj k; f~jk) of a set of voltage-controlled nonlinear resistive elements R2jk connecting each node j k to the ground. The characteristic of R2jk can be easily derived from the

�9 d~, Unlike the resistors Rt of 9~1, each local co-content function ~ as tjk = doik. element R2jk is different from the others, as shown in the example in Figure 4, because its characteristic depends on the constant value vjk (a voltage source),

whereas the series resistors R = ~ are identical and linear. 2Zq~M

2.4. Definition of the complete circuit.

The complete nonlinear circuit is obtained by connecting 9ql to 9%2 and by adding linear capacitors C between each node-pixel at (j, k) and the ground. A detail of the complete circuit for grey-level images is shown in Figure 5. In this circuit, each voltage vjk is the state variable of a capacitor C. The temporal evolution of the circuit is governed by the following expressions:

_cdVjk = 09 (6) dt Ovjk '


t t ,,JY RSj--i ,k

/l~J'k-1 R, i j'-i+'l-

.-"i J+l'

Figure 5. A detail of the complete circuit.

Z dvj ___ c(dvJ ) 2 dt . Ovj~ dt jk \ dt ] - (7)

Then, from the general property defined in expression (7), the initial values vjk(O) representing the image to be processed are modified by the circuit so as to reach a minimum of the co-content function g. As a consequence, the final processing image corresponds to a minimum of the functional H.

As stated in Section 1, higher-order neighborhood sets correspond to more accurate approximations of the spatial operators (i.e., approximations using a larger number of points [1]). This refinement requires more complex circuit structures, where the number of resistors (no longer identical) connected to the node j k increases [25].

3. A circuit for the restoration of RGB color images

An RGB color image can be described in terms of three sets of variables fR, fG,

f__B corresponding to the brightness levels of the R,G,B channels, respectively. In

this case, the pixel at (j, k) is associated with three node voltages v~ homologous

with the components fff (C = R,G,B) via relation (2). As a generalization of the

grey-level image case, here the local functions ~ defining the functional HI depend


on the voltage variables viC and vCq, whereas the functions + of / /2 depend on

v~ (C = R,G,B).

The interpretations of ~ and ~ in terms of resistive elements are based on the definition of co-content for a complete voltage-controlled, reciprocal m-port element. Denoting by i and v the port current vector and the port voltage vector, respectively, and by i = g(v) the constitutive relation, we have [5]:

f G(v) = g(~). d~, (8)

and the current component i~ of i (n = 1 . . . . . m) is

OG in = - - . (9)

OVn

3. I. Definition o f 9~1.

Following [13] and [19], it can be shown that, in the case of a grey-level image, the local potential function +(x) can be chosen as the piecewise quadratic C 1 function:

+(x) =

+--g--x 2 0 < Ix! < V0 VoVr - -

[ (~-vr) 2 ] ~PM [ vr(vo:vr) + 1 Vo <_ jxj < VT �9

~PM iX I ~ VT

(lo)

This function has a minimum value at x = 0, whereas the inflection points (with discontinuous second derivatives) of the welI are :k(Vo, ~-o), where +o = +M

and the terms Vr and +M have the usual meanings. We assume that this function can also be used as the basis for color images. The elements considered for this generalization are as follows:

(i) In the case of an RGB color image, a pair of pixels (at (j, k), (p, q)) is associated with the three-component voltage vector

vB 4 (ii) In equation (10), we take x as the Euclidean norm v of the voltage vector v,

and we interpret the function

as the co-content of a (reciprocal) voltage-controlled, resistive three-port (see Figure 6).


i R >

pq (R)

R1 --I iGl

V G

i B <

w pq (B) V B

Figure 6. The three-port R1 corresponding to the local co-content function + (v).

With this generalization, the constitutive equations of the corresponding three-port are reasonably simple as the port currents i c are given by

2 ~ ~ c 0 < v < go ic O~ VoVr

-- OvC -- Vr(Vo-Vr) 2~'t4 vc v-vrv ' Vo _< v _< VT, C = R,G,B " (12)

0 v>_Vr

The resistive grid 9li is then built up by properly connecting each set of the three nodes j k to its homologous neighbors through the obtained three-port RI, as shown in Figure 7.

3.2. Definition of 912.

In the case of additive null-mean Gaussian noise with standard deviation cr, the local potential function tP(fjk) is taken as a sum of quadratic terms that constrain the variables fie to remain close to the observed fixed values j~c:

*(~, ~, ~; I~,jk ~, ~-~/= Z ; ( ~ - ~/2. (13) c

Moreover, the weighting coefficient )~ is equal to 1/2cr 2 (see the Appendix). Then, the corresponding co-content is (from equation (5))

+(v~,v~,v}3k;-R -G ~B C~M 1 ~ ( v ~ - ' c 2 '~sk, "J~, "J~) = 2-~ vX c ~,j~), (14)

468 STOP, ACE, PARODI, PASTORINO, AND TRIPODORO

j - l , k

Figure 7. A detail of the resistive grid 9~.

. . . . . . . . . . . . . . . . . . I |

R ' 2iki O

/~VJ k "' I ~ R i

\ \ jk %+-c 1 V', i

/ / /

Figure 8. The internal structure of the four-terminal resistive element R2j k,

where the fixed voltages fi~ (C = R,G,B) are derived from the corresponding terms

)~c via equation (2).

For each of the vCk components, the corresponding current is

C = R,G,B, (15)


, i - l . k

Figure 9. A detail of the complete circuit.

and can be interpreted as the current flowing in a two-terminal element of the type displayed in Figure 4. Then, the resulting co-content function + can be associated with a four-terminal voltage-controlled resistive element R2jk consisting of three resistors RCjk connected to the ground, as shown in Figure 8.

3.3. Definition of the complete circuit.

The complete circuit can be obtained by connecting each node jk of H1 and 9~2 to the ground through a linear capacitor C. A detail is presented in Figure 9. The temporal evolution can be treated in the same way as for grey-level images (see Section 2.4).

4. Examples

The behavior of the resulting ideal circuit was numerically evaluated by using the programming tool MATLAB. To this end, the parameters N, M, Vr, Vo, ~M, C, Vmin, and Vmax are fixed. Preliminarily, we choose V0 = -~ in order to

keep constant in the potential well the absolute value g = 4".,-~2 of the second v i


derivative of the local function $ . The simulations described in this section share the following parameter values: M = N; V0 = -~; g = 10 -3 s ~M = V~g; C = 1 nF; Vmin = 0 V; Vmax = 10 V.

The original image shown in Figure 10a was corrupted by null-mean Gaussian noise with standard deviation a = 40, thus obtaining the input picture presented in Figure 10b. In the numerical simulation, this image was processed by the circuit defined by N = 256 and Vr = 3.5 V. The steady state of the circuit, reached after a few microseconds, corresponds to the image in Figure 10c.

Another e• is given in Figure 11. In this case, two circuits, defined by N = 256 and two different values of Vr, were simulated to process the input picture in Figure 1 lb, characterized by null-mean Gaussian noise with standard deviation cr = 30. The final results are shown in Figures 11c (for Vr = 2.2 V) and 1 ld (for Vr = 2.8 V). A comparison of these figures demonstrates the rote played by the parameter Vr in determining the quality of the image processing.

An intuitive interpretation of VT can be given in the simple case of grey-levei images [19]; the larger the value of Vr, the larger the gradient needed to detect an edge between two pixels. In the case of RGB images, Vr plays the same role but with respect to the quadratic norm of v. This fact is evident in Figure I lc and 1 ld; in Figure 1 ld a greater value of ~ allowed the circuit to generate an image smoother than the one in Figure 1 lc.

Original image (Figure 1 lb)

Circuit simulation

(after 6/zs)

MCD

CPF

Median filter

PSNR

13.670

Vr = 2.2 V 16.913

Vr = 2.5 V 19.341

Vr = 2.8 V 18.594

after 50 iterations 16.522



after 400 iterations 14~359

after 500 iterations 14.377 l


radius 1 pixel 19.791

radius 2 pixels 20.583

radius 3 pixels 19.880

Table 1.


In Table I, a comparison is shown between the best results in terms of pixel signal-to-noise ratios (PSNRs) obtained by our simulation program and three other methods. Two different anisotropic diffusion algorithms [2] and a median filter were applied to the Lena test image. Each of the two algorithms, mean curvature diffusion (MCD) and corner-preserving filter (CPF), was separately run over the R, G, and B channels of the original image (Figure 11 b). In both cases, the three intensity levels related to the pixel at (j, k) (i.e., fiR, fj~k' and fj~) in the resulting images were used as RGB components of the pixel at (j, k) in the processed image (the best results in terms of PSNR are shown in Figure 12b and 12c). Note that, on the contrary, our processing method (see Figure 12a) maintains a sort of coupling that combines the information provided by the three channels. The image shown in Figure 12a, where noise is still present, corresponds to the best results in terms of PSNR. For higher values of the parameter VT, noise tends to disappear, but PSNR increases. The best result obtained by using the median filter (of radius equal to 2 pixels) is shown in Figure 12d. In the MCD case, the image structure was preserved, and the noise was reduced. However, corners and other features characterized by higher-order structures (e.g., edge intersections) were rounded. In the CPF case and in our case, the comer structures were preserved much better, also as compared with the image processed by the median filter.

5. Concluding remarks

The solution f(x, y) (in general, a vector) of many problems corresponds to the minimization of a functional. After a square-mesh grid discretization, the functional can often be approximated by a sum of locally defined identical functions. "Locally" means that each function depends on the values of the solution f at a limited set of points. Typical examples of local functions considered in this paper are the functions q~ and �9 used for the restoration of both grey-level and color images. The interpretation of these local functions as potential (co-content) functions of resistive elements allows one to conceive the discretized solution as a set of equilibrium values (voltages) of the circuit obtained by connecting the resistive elements to a set of linear capacitors. This results from a well-known theorem of circuit theory [5], [16].

Generally speaking, the resistive elements are nonlinear and either one-ports (if f is a scalar function, as in the case of grey-level images) or m-ports (in the more general case when f is a vector, for instance, of the R, G, and B components in color images).

In [19], it was shown that the obtained circuit (for grey-level images) can be interpreted in terms of a CNN architecture. Following the same guidelines, the circuit obtained in this paper also could be represented as a CNN (see, e.g., Figure 13). However, the representation of the circuit described in Section 3 is the most "natural," as it results directly from associating the discretized functional with the co-content of a lumped circuit.


a) b)

c)

Figure 10. (a) The original peppers image, (b) the corrupted image, and (c) the result of the numerical simulation with VT = 3.5 V.


a) b)

c) d)

Figure 11. (a) the original Lena image, (b) the corrupted image, and (c,d) the results of the numerical simulations with Vr = 2.2 V and Vr = 2.8 V.

We think that the proposed method for the definition of analog circuits to minimize discrete functionals has an application field much wider than that of image processing only, as it can be applied to any problem solvable by resorting to a discrete variational formulation. Its main advantages are the parallel computation and the possibility of defining the characteristics of the resistive elements in a very simple way. The synthesis of such elements can be obtained by exploiting the PWL techniques proposed by Chua [6], [8], [14].


a) b)

c) d)

Figure 12. (a) Result of the numerical simulation of the circuit with Vr = 2,5 V; (b,c) images obtained after 100 iterations of the MCD algorithm and after 500 iterations of the CPF algorithm; (d) result of the application of a median filter of radius equal to 2 pixelso All the results were obtained by processing the image shown in Figure 1 lb and correspond

to the best PSNR result in Table 1.

Acknowledgments

The authors are grateful to Dr. A. E1-Faltah and Dr. R. Estes for their complete C source code for the two diffusion algorithms.


~ 0

,jk (It)

" - v v . J

i ~= ff(lv~};{v~)) for any pq ~Njk C = R,G,B

ljk (o)

for any pq ~ N j k

C = R,G,B

TJk (B)

Figure 13. Possible CNN representation of the proposed circuit. The functions h •, h G, and h B directly follow from equation (12).

Appendix

Each considered image is thought of as a Markov random field. The value of the constant coefficient )~ can then be derived from Bayesian estimation and the MAP criterion according to the Hammersley-Clifford theorem [ 13]. In order to obtain the brightness values ~3~ } of the restored image (described by the field F), starting from the observed data (i.e., the elements {gj~} of the field G), it is necessary to maximize the conditional probability P(F I G):

P(F) . P(G [ F) max P(F [ G) = max = max{P(F) �9 P(G [ F)},

P(G)

where (see, e.g., [12]) P(F) = �89 -U(F) is the Gibbs distribution of F (Z is the

partition function [12]), and P(G I F) = 1 e -~(F-a)z is the probability

density of a null-mean Gaussian noise with variance a2. Then

max P ( F I G ) = m a x Z q r ~ r ~ e

As

max P(F ] G) = rain U(F) + ~ 2 ( F - G) 2 = rain[HI(F) +)~H2(F)],

it is straightforward to define L = 2@"


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A method for defining analog circuits for the minimization of discrete functionals: An image processing application

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