2-D discrete signal interpolation and its image resampling application using fuzzy rule-based inference

2-D discrete signal interpolation and its image resampling

application using fuzzy rule-based inference

Jia-Lin Chen, Jyh-Yeong Chang∗, Kun-Li Shieh

Department of Electrical and Control Engineering, National Chiao Tung University, Taiwan, ROC Received August 1997; received in revised form March 1998

Abstract

This paper describes an interpolation algorithm for two-dimensional (2-D) discrete signals using fuzzy rule-based in-ference. The original signal is estimated by the main-surface function in the interpolation region, and four sub-plane functions surrounding the interpolation region. The main-surface is a bilinearly interpolated function passing through four signal samples in the interpolation region and the four sub-planes re ect the tendencies of pixels from the left, right, up, and down of the interpolation region. Drawing fuzzy inferences about signals from these ve functions, we can estimate original signals very well even when the signals are buried in noise. We veried the method by computer simulations of some assumed 2-D signals and by resampling of the actual image data. c 2000 Elsevier Science B.V. All rights

reserved.

Keywords: 2-D discrete signal interpolation; Fuzzy inference; Membership function; Image resampling

1. Introduction

Interpolation algorithms for 2-D signals have been proven useful in a wide range of applications, especially in image processing systems [11]. All interpolation algorithms generally start by determining a continuous interpolation function from a set of discrete signal samples, and then resample this interpolated function according to the number of points specied. The simplest algorithms are the nearest neighbor and bilinear interpolations [16]. The more rened cubic B-spline interpolation algorithm, proposed by Hou and Andrews [7], is, however, computationally complex. Several alternative cubic convolution techniques [3,15,20] have been proposed to reduce its computational complexity. It must be noted that all these algorithms were proposed to estimate original signals in noise-free environments. However, during the courses of transmission, storage, and retrieval, noise is generally embedded in signals from various sources, i.e., during coding and decoding, errors are induced in signals and transmitted signals are usually corrupted by noise. Because the interpolation curves obtained by using these algorithms are so calculated that they pass exactly through the given noisy

∗_{Corresponding author.}

0165-0114/00/$ - see front matter c 2000 Elsevier Science B.V. All rights reserved.

points, these algorithms are obviously not good at estimating original signals from noisy signals. To improve this situation, we propose a new 2-D noisy signal interpolation algorithm, based mainly on the fuzzy inference rule.

Fuzzy logic has been successfully applied to various kinds of engineering problems [1,2,9,10,18]. Fuzzy theory allows the inclusion of fuzzy if–then rules concerning fuzzy concepts. These rules may come from human experts or be generated automatically by matching input–output pairs through training routines [4,6,12]. In the eld of image processing, successful image enhancement results have been reported after the fuzzy indices of input images were reduced [13,14], and [17] used fuzzy inference rules derived from pixels of in-terest and its neighboring pixels. In particular, a fuzzy rule-based interpolation algorithm for one-dimensional discrete signals has recently been proposed [19]. This algorithm interpolates between two noise-free points in Euclidean space by aggregating the fuzzy sets dened on the surrounding points. An advanced algo-rithm for the noisy points is also proposed; this interpolation is done by classifying the neighboring noisy points into groups. In this direction, this paper investigates a fuzzy rule-based interpolation algorithm for 2-D signals.

This paper is organized into four sections. The architectural framework of the entire system is discussed in detail in Section 2. Derivation of the main surface and the four sub-planes are then presented. This is followed by construction of the membership functions and fuzzy rules for inferring 2-D signals. Computer simulations of assumed 2-D signals and image resampling examples are presented in Section 3. Comparison results are also given. Section 4 contains some concluding remarks.

2. 2-D signal interpolation by fuzzy rule-based inference

During the courses of transmission, storage, and retrieval, 2-D signals are usually corrupted by noise. Since the signals are inevitably buried in noise, schemes that can remove the noise during the course of interpolation are required. In the spatial domain, a successful and frequently used method for removing noise from images is neighborhood averaging [5, Section 4.3], a method mainly concerned with masks. If the center of a mask is located at pixel (x; y) in an image, the gray level of (x; y) is replaced by the average gray level of all pixels in the area of the mask. In a similar manner, our proposed 2-D signal interpolation scheme uses fuzzy inference, a more sophisticated neighborhood averaging technique involving application of fuzzy logic to sam-pled noisy data. That is, we propose a 2-D fuzzy rule-based interpolation scheme in which fuzzy inference is used to derive results similar to those obtained by using neighborhood averaging for interpolation tasks. First, we nd the main-surface function of the interpolation region and then nd four surrounding plane functions, called sub-planes, of the main-surface. The main-surface is a bilinearly interpolated function passing through four signal samples, and the sub-planes re ect the tendencies of pixel uctuations from four directions, left, right, up, and down, of the interpolation region. Next, some fuzzy if–then rules for these ve functions are presented to carry out the interpolation task. Because of the inference rules employed for interpolation, orig-inal signals can be correctly interpolated and resampled despite the origorig-inal signals having been corrupted by noise.

Let ˆf(x; y) be the estimated function and Is(i; j); i = 0; 1; : : : ; M − 1; j = 0; 1; : : : ; N − 1, be the M × N sampled noisy data as shown in Fig. 1(a). In Fig. 1(b), we dene an interpolation region Ri; j as follows:

Ri; j= {(x; y) | xi6x6xi+1; yj6y6yj+1}; i = 0; 1; : : : ; M − 2; j = 0; 1; : : : ; N − 2: (1) From (1) we can similarly represent the eight regions, Ri−2; j; Ri−1; j; Ri+1; j; Ri+2; j; Ri; j−2; Ri; j−1; Ri; j+1, and Ri; j+2, around the region of interest. In our interpolation algorithm, we use ve functions, one surface function generated from Ri; j and the other four plane functions generated from Ri−1; j; Ri+1; j; Ri; j−1, and Ri; j+1, respectively, to determine the estimation function ˆf(x; y). Function Fc(x; y), i.e., the main-surface, passes through the four signal samples, Is(i; j); Is(i + 1; j); Is(i; j + 1), and Is(i + 1; j + 1); of the interpolation

Fig. 1. The interpolation region and its contiguous regions.

region Ri; j. Let fl(x; y); fr(x; y); fu(x; y), and fd(x; y) be the four sub-planes that pass through the local signal averages of the four contiguous regions, Ri−1; j; Ri+1; j; Ri; j−1, and Ri; j+1, and re ect pixel uctuation tendencies. Subscripts l; r; u and d denote left, right, up, and down, respectively. To interpolate a point (x; y) in the region Ri; j; ˆf(x; y) is obtained by taking into account the fuzzy aggregation of the function Fc(x; y) in the interpolation region and the four planes, fl(x; y); fr(x; y); fu(x; y), and fd(x; y) next to the interpolation region.

Let the original M × N 2-D signal Is(i; j) be interpolated during resampling periods Nx and Ny in the x- and y-directions, respectively. In region Ri; j we can then calculate the new coordinates (xi; yj) of the given sample points and the new coordinate (x; y) of the interpolated point as follows (see Fig. 1): xi= iNx, i = 0; 1; : : : ; M − 1, yj= jNy, j = 0; 1; : : : ; N − 1; x = xi+ u; u = 0; 1; : : : ; Nx− 1, y = yj+ v; v = 0; 1; : : : ; Ny− 1. In what follows, the procedures for deriving the main-surface, sub-planes, and for constructing the fuzzy rules are described.

2.1. Deriving the interpolation main-surface

Function Fc(x; y) is bilinearly interpolated from the four corner signals of region Ri; j and we call function Fc(x; y) the main-surface. Suppose Fc(x; y) is represented by the following:

In (2), the four unknown coecients, a; b; c; and d, can easily be determined from the following four equations using these four signal samples. I.e.,

Is(i; j) = axi+ byj+ cxiyj+ d; Is(i; j + 1) = axi+ byj+1+ cxiyj+1+ d; Is(i + 1; j) = axi+1+ byj+ cxi+1yj+ d; Is(i + 1; j + 1) = axi+1+ byj+1+ cxi+1yj+1+ d: (3) Thus, we have       a b c d      =       xi yj xiyj 1 xi yj+1 xiyj+1 1 xi+1 yj xi+1yj 1 xi+1 yj+1 xi+1yj+1 1       −1      Is(i; j) Is(i; j + 1) Is(i + 1; j) Is(i + 1; j + 1)      : (4)

After mathematical manipulations (see Appendix A), (2) takes the following form:

Fc(x; y) = (x; y) Is(i; j) + (x; y) Is(i; j + 1) + (x; y) Is(i + 1; j) + (x; y) Is(i + 1; j + 1); (5) where (x; y) = (1 − (x − xi)=Nx)(1 − (y − yj)=Ny); (x; y) = (1 − (x − xi)=Nx)((y − yj)=Ny); (x; y) = ((x − xi)=Nx)(1 − (y − yj)=Ny); (6) and

(x; y) = ((x − xi)=Nx)((y − yj)=Ny); xi6x6xi+1; yj6y6yj+1;

i = 0; 1; : : : ; M − 2; j = 0; 1; : : : ; N − 2: (7)

2.2. Deriving the four sub-planes

To derive the sub-plane, using the left-hand surrounding plane fl as an illustrative example, we need to know the relationship between the local signal average of the interpolation region and the local signal averages of the two consecutively contiguous regions on the left-hand side of the interpolation region. These local signal averages are calculated as follows:

Gi; j=1₄{Is(i; j) + Is(i; j + 1) + Is(i + 1; j) + Is(i + 1; j + 1)};

i = 0; 1; : : : ; M − 2; j = 0; 1; : : : ; N − 2: (8)

In (8), Gi; j is the local signal average of the interpolation region Ri; j. The local signal averages, Gi−2; j and Gi−1; j, of these two contiguous regions, Ri−2; j and Ri−1; j can be calculated in a similar manner.

Sub-plane fl(x; y) may be inclined or at. If it is inclined, it is parallel to the inclined plane included in the cuboid Cubl of the contiguous region Ri−1; j. Cuboid Cubl is bounded by the maximum and minimum of the sampled noisy data, Is(i − 1; j); Is(i − 1; j + 1); Is(i; j), and Is(i; j + 1), in the region of interest Ri−1; j,

Fig. 2. The cuboid, Cubl, in Ri−1; j.

as shown in Fig. 2; hence, the cuboid will enclose these sampled data. To this end, the maximum Maxi−1; j and minimum Mini−1; j of Ri−1; j are dened as

Maxi−1; j= max{Is(i − 1; j); Is(i − 1; j + 1); Is(i; j); Is(i; j + 1)}; Mini−1; j= min{Is(i − 1; j); Is(i − 1; j + 1); Is(i; j); Is(i; j + 1)};

i = 1; 2; : : : ; M − 1; j = 0; 1; : : : ; N − 2: (9)

Sub-plane fl(x; y) is determined according to the geometric perspective among the following three contiguous regions, {Ri−2; j; Ri−1; j; and Ri; j}. To derive fl(x; y), the following parameters, G1; G2; G3; (xc; yc); max; and min, must be dened. G1; G2, and G3 are, respectively, the local signal averages of the regions Ri−2; j; Ri−1; j; and Ri; j. (xc; yc) denotes the coordinate of the center of Ri−1; j; whereas max and min represent the maximum, Maxi−1; j, and the minimum, Mini−1; j, of Ri−1; j, respectively. In summary, to derive fl(x; y), these parameters are dened as follows (see Fig. 3(a)): G1= Gi−2; j, G2= Gi−1; j, G3= Gi; j; xc= (xi−1+ xi)=2; yc= (yj + yj+1)=2; max = Maxi−1; j; min = Mini−1; j. Next we use these parameters to derive the sub-plane fl(x; y). When deriving fl(x; y), we must distinguish among three cases.

Case 1: G1¡G2¡G3. In this case, pixel values tend to increase from the left-hand side of Ri; j, and sub-plane fl(x; y) is introduced to re ect this tendency. Sub-plane fl(x; y) is an inclined plane that passes through the point ((xc; yc); G2) and parallels plane Plh in the cuboid Cubl, where the subscript lh indicates the tendency from lower gray levels toward higher ones as the pixel moves from the left-hand side of Ri; j. Plane Plh and its normal vector Vlh are shown in Fig. 4(a). After the coordinates of the vertices of Cubl have been taken, the equation of the normal vector Vlh is given by

Vlh= (v1; v2; v3) = (Nx; 0; max − min) × (0; Ny; 0) =_N0 max − min y 0   ;max − min Nx 0 0  ;Nx 0 0 Ny  ; (10)

Fig. 3. The positional relationships among the parameters G1, G2, and G3, referred to as the three-region sets, {Ri−2; j; Ri−1; j; Ri; j}, {Ri; j; Ri+1; j; Ri+2; j}, {Ri; j; Ri; j+1; Ri; j+2},

and {Ri; j−2; Ri; j−1; Ri; j}; of the three regions.

Fig. 4. When fl is derived, (a) case 1 is satised; (b) case 2 is

satised.

where × is the vector cross product. Let fl(x; y) = z be represented implicitly as

v1x + v2y + v3z + d = 0: (11)

Since ((xc; yc); G2) is on the plane fl(x; y), it follows that ((xc; yc); G2) satises (11). Hence,

d = −v1xc− v2yc− v3G2: (12)

With vector Vlh and d given above, the left-hand interpolation sub-plane fl(x; y) is given by

Case 2: G1¿G2¿G3. In this case, pixel values tend to decrease from the left-hand side of Ri; j, and sub-plane fl(x; y) is introduced to re ect this tendency. Sub-plane fl(x; y) is again an inclined plane that passes through point ((xc; yc); G2) and parallels plane Phl in the cuboid Cubl, where the subscript hl indicates a tendency from higher gray levels toward lower ones as the pixel moves from the left-hand side of Rij. Fig. 4(b) shows plane Phl and its normal vector Vhl. Normal vector Vhl is given by

Vhl= (v1; v2; v3) = (Nx; 0; min − max) × (0; Ny; 0) =_N0 min − max y 0   ;min − max Nx 0 0  ;Nx 0 0 Ny  : (14)

Case 3: Neither case 1 nor case 2 condition holds. No pixel tendency can be inferred in this case. Plane fl(x; y) is a horizontal plane that passes through point ((xc; yc); G2). It is represented as follows:

fl(x; y) = G2: (15)

Referring to Figs. 3(b)–(d), we can derive fr(x; y); fu(x; y), and fd(x; y) in a manner similar to that used to derive fl(x; y).

2.3. Constructing the fuzzy sets and inference rules for interpolation

Since the 2-D signal is interpolated via fuzzy inference on the main-surface and sub-planes, the membership functions for the main-surface and sub-planes are dened rst. As shown in Fig. 5, Wc(x; y) and Wi; j(x; y) are the respective membership functions assigned to the main-surface and sub-planes. Wc(x; y) is a cubic of height “1”. Wi; j(x; y) is a pyramid of height “1” and is given by (see Fig. 6)

Wi; j(x; y) = 1 − max{2|x − xc|=Lengthx; 2|y − yc|=Lengthy};

i = 0; 1; : : : ; M − 2; j = 0; 1; : : : ; N − 2; (16)

where (xc; yc) = ((xi+xi+1)=2; (yj+yj+1)=2) is the center coordinate of each region and Lengthx and Lengthy represent the lengths of the pyramid in the x- and y-directions, respectively. The values of Length_x and Length_y can be chosen freely and they determine relative weighting of sub-planes on computing ˆf(x; y). In this paper, we set Length_x= 3Nx and Lengthy= 3Ny. We use Wi−1; j(x; y); Wi+1; j(x; y); Wi; j−1(x; y), and Wi; j+1(x; y) to represent the membership functions assigned to fl(x; y); fr(x; y); fu(x; y), and fd(x; y), respectively, of which only Wi−1; j(x; y) and Wi+1; j(x; y) are shown in Fig. 5 for simplicity.

Finally, the interpolation of ˆf(x; y) in the region Ri; j is carried out according to the following rules: Rule 1: If the interpolation membership function is Wc;

then the corresponding interpolation surface is Fc(x; y): Rule 2: If the interpolation membership function is Wi−1; j;

then the corresponding interpolation plane is fl(x; y): Rule 3: If the interpolation membership function is Wi+1; j;

then the corresponding interpolation plane is fr(x; y): Rule 4: If the interpolation membership function is Wi; j−1;

then the corresponding interpolation plane is fu(x; y): Rule 5: If the interpolation membership function is Wi; j+1;

then the corresponding interpolation plane is fd(x; y):

Fig. 5. Membership functions assigned to the main-surface and the sub-planes, and the fuzzy rules for interpolation in region Ri; j.

Fig. 6. The pyramid membership function Wi; j(x; y).

Thus the interpolated value, ˆf(x; y), in the region Ri; j is given as follows: ˆ

f(x; y)

=Wi−1; j(x; y)fl(x; y) + Wi+1; j(x; y)fr(x; y) + 4Fc(x; y) + Wi; j−1(x; y)fu(x; y) + Wi; j+1(x; y)fd(x; y) Wi−1; j(x; y) + Wi+1; j(x; y) + 4 + Wi; j−1(x; y) + Wi; j+1(x; y) ; xi6x6xi+1; yj6y6yj+1; i = 0; 1; : : : ; M − 2; j = 0; 1; : : : ; N − 2: (18) In (18), the weighting of the main-surface is four times greater than those of the sub-planes. This is to balance the eect of the main-surface with the eects of the remaining four sub-planes. Note that, at the edges of the images, we may eliminate some items from (18) in case G1; G2; or G3 cannot be dened. For example, sub-plane fl(x; y) and Wi−1; j(x; y) must be eliminated from interpolation equation (18), when interpolating the left edge of the image; sub-planes fl(x; y); fu(x; y), and their correspond-ing weightcorrespond-ing membership functions have to be ignored when interpolatcorrespond-ing the left-upper corner of the image.

3. Simulations and results

In order to conrm the validity of the proposed 2-D fuzzy rule-based inference interpolation algorithm, we conducted computer simulations on synthetic signals, as shown in the following examples. We used ve simulations to test the proposed algorithm. In these simulations, performance in terms of reconstruc-tion accuracy of the present algorithm was compared with bilinear interpolareconstruc-tion and the cubic convolureconstruc-tion interpolation proposed by Keys [8]. With respect to reconstruction accuracy, we used the mean square er-ror (MSE) and the mean absolute erer-ror (MAE) as the comparison indices. These two erer-ror indices are given by MSE =_MN1  M−1X i=0 N−1 X j=0 [f(i; j) − ˆf(i; j)]2  ; MAE = 1_MN  M−1X i=0 N−1_X j=0 |f(i; j) − ˆf(i; j)|  ; (19)

where f(i; j) and ˆf(i; j) represent the original and the resampled signals, respectively, and both of them consist of M × N pixel arrays.

Then, we applied the proposed algorithm to image resampling examples and compared the results with the other two interpolation algorithms.

3.1. Synthetic signal interpolation 3.1.1. Simulation 1

In this synthetic signal interpolation simulation, the original signal was given as follows:

S1(x; y) = 20x exp(−x2− y2); −26x62; −26y62: (20)

A 45 × 45 point array S1(xk; yl), from which k = 0; 1; : : : ; 44; l = 0; 1; : : : ; 44; was created by uniformly sam-pling S1(x; y) in the dened region. The noisy point array S10(xk; yl), was created by adding Gaussian noise of zero mean and unit variance to S1(xk; yl), and its MSE and MAE were 1.105 and 0.8, respec-tively. The 12 × 12 point sampled noisy data array Is(i; j) was obtained from Is(i; j) = S10(xk; yl), where i = 0; 1; : : : ; 11; j = 0; 1; : : : ; 11; k = 4i; l = 4j. We then put Is(i; j) with Nx= 4 and Ny= 4 into the pro-posed algorithm, to derive the resampling function ˆf(x; y). And, putting Is(i; j) into the bilinear interpolation

Table 1 Table 2

The MSEs for the resampling signals ˆf(x; y) from the three

in-terpolation algorithms applied in ve simulations The MAEs for the resampling signals ˆterpolation algorithms applied in ve simulationsf(x; y) from the three in-Noise Fuzzy Bilinear Cubic

Simulation 1 1.015 0.355 0.549 0.696 Simulation 2 8.429 3.669 6.021 7.199 Simulation 3 31.596 6.420 7.825 9.987 Simulation 4 1.684 0.876 1.095 1.313 Simulation 5 0.966 0.394 0.504 0.656

Noise Fuzzy Bilinear Cubic Simulation 1 0.801 0.479 0.591 0.668 Simulation 2 2.534 1.686 2.174 2.43 Simulation 3 5.515 2.311 2.489 2.764 Simulation 4 0.915 0.662 0.751 0.838 Simulation 5 0.789 0.500 0.565 0.649 Note: Simulations 2 and 3 were calculated based on the loss

region only, and the others on the whole signal region, −26x62,

−26y62.

Note: Simulations 2 and 3 were calculated based on the loss region only, and the others on the whole signal region, −26x62;

and cubic convolution interpolation algorithms, we respectively obtained the corresponding resampling signals ˆ

f(x; y). To compare reconstruction accuracy, the MSEs of the resampling signals ˆf(x; y) obtained from the bilinear, cubic spline, and the proposed algorithm were 0.549, 0.696, and 0.355, respectively. And the MAEs were, respectively, 0.591, 0.668, and 0.479. The results of these algorithms are shown in Tables 1 and 2. From these MSEs and MAEs, we can see the reconstruction accuracy of the fuzzy rule-based in-terpolation algorithm was better than those of the other two inin-terpolation algorithms. The proposed al-gorithm not only generated reconstructed signals from the region of interest but also kept track of pixel variation trends from the four surrounding sub-planes, and hence yielded better noise removal results.

3.1.2. Simulation 2

We assumed that the noisy signal S0

1(x; y) was due to a complete loss of the original signal S1(x; y) in (20) from the following region, {(x; y) | −0:26x60:25; −0:296y60:16}. MSEs and MAEs for ˆf(x; y) were calculated only in the lost 6 × 6 region, by the three interpolation algorithms, these results are also shown in Tables 1 and 2. From these two tables, it is evident that the proposed method demonstrated the best signal recovery ability.

3.1.3. Simulation 3

In this simulation, the noisy signal S0

1(x; y) was also assumed to be due to the loss of signal S1(x; y) in the 6 × 6 region, {(x; y) | 0:616x61:06; −0:836y6 − 0:38}. MSEs and MAEs for the resampled signal

ˆ

f(x; y) calculated only in the 6 × 6 lost region, are also shown in the two tables mentioned above. The fuzzy inference approach led to the smallest error.

3.1.4. Simulation 4

In Simulation 4, noisy signal S0

1(x; y) suered from losses in regions of Simulation 2 and 3, i.e., {(x; y) | −0:2 6x60:25; −0:296y60:16} and {(x; y) | 0:616x61:06; −0:836y6−0:38}. The MSEs and MAEs for the

Fig. 7. Lena’s eye images. The primary data was 64 × 64, with display dimensions of 256 × 256 using zero-order hold. (a) The original image; (b) The noisy image.

45 × 45 point array ˆf(x; y) at −26x62 and −26y62 are also shown in Tables 1 and 2, respectively. The smallest error was still recorded by the proposed approach.

3.1.5. Simulation 5

In the last synthetic signal simulation, we considered another signal S2, which was rougher than S1, given by S2(x; y) = 3(1 − x)2exp[−x2− (y + 1)2] − 10(x=5 − x3− y5) exp[−x2− y2]

−(1=3) exp[−(x + 1)2_{− y}2_{]; −26x62; −26y62:}

Fig. 8. Resampling using three dierent methods. (a) Fuzzy rule-based interpolation algorithm; (b) bilinear interpolation algorithm; (c) cubic convolution interpolation algorithm.

A 49 × 49 point array S2(xk; yl), where k = 0; 1; : : : ; 48; l = 0; 1; : : : ; 48, was also created by uniform sampling S2(x; y). And a 49 × 49 point noisy array S20(xk; yl), was created similar to that in Simulation 1. The noisy sig-nal’s MSE and MAE are also shown in Tables 1 and 2. The 13 × 13 point sampled noisy data array Is(i; j) was sampled from the noisy point array S0

2(xk; yl); i.e., Is(i; j) = S20(xk; yl), where i = 0; 1; : : : ; 12; j = 0; 1; : : : ; 12; k = 4i; l = 4j. The sampled noisy data was resampled to a 49 × 49 point array ˆf(x; y), and the MSEs and MAEs for ˆf(x; y) are also shown in Tables 1 and 2. As in the other simulations, the fuzzy inference approach produced the smallest errors.

In these simulations and their error measures, the proposed fuzzy rule-based interpolation algorithm demon-strated the best noise removal and signal recovery abilities.

3.2. Image resampling

2-D signal interpolation algorithms have a wide range of application areas in many image processing systems. For example, 2-D interpolation algorithms are used for image resampling to change the sizes of digital images interactively. To demonstrate the power of our algorithm on image resampling, we applied the 2-D interpolation scheme on a 256 × 256 image of Lena’s eye. The result was compared with results obtained from the bilinear and cubic convolution interpolation algorithms. The original image, as shown in Fig. 7(a), had primary data of 64 × 64, with display dimensions of 256 × 256 obtained from zero-order hold. Fig. 7(b) shows the noisy image from zero-order hold, which was created by adding zero-mean Gaussian noise to the primary image at a signal-to-noise ratio (SNR) of 11.7 dB. Next, the 64 × 64 noisy image was put into the three algorithms and resampled to 256 × 256 images. The resampled images from the fuzzy rule-based interpolation, bilinear, and cubic convolution interpolation algorithms are shown in Figs. 8(a)–(c). Comparing these three gures, we can see that our algorithm removed noise better than the other two algorithms and outperformed the bilinear interpolation algorithm in reducing blocky eect. Moreover, the interpolated image of the proposed algorithm is comparable to that of the cubic convolution interpolation algorithm, in spite of the much simpler scheme used by our method, as compared to the third-order scheme of the cubic convolution algorithm.

4. Conclusion

An interpolation algorithm for 2-D signal using fuzzy rule-based inference has been proposed in this pa-per. The proposed interpolation scheme rst nds a bilinearly interpolated main-surface that passes through four signal samples in the region to be interpolated, and nds four sub-planes, surrounding the interpolation region, that re ect the pixel tendencies of the interpolation region. Some fuzzy if–then rules for these ve functions were presented for carrying out 2-D signal interpolation. Because of the inference rules employed for interpolation, the original signal can be interpolated and resampled very well despite the original signal having been corrupted by noise. When we applied the proposed algorithm to image resampling, the result was also very successful.

Appendix A. The deriving of Fc(x; y)

In this appendix, the equivalence of (2) and (5) is proved. From (4),     a b c d     =     xi yj xiyj 1 xi yj+1 xiyj+1 1 xi+1 yj xi+1yj 1 xi+1 yj+1 xi+1yj+1 1     −1    Is(xi; yj) Is(xi; yj+1) Is(xi+1; yj) Is(xi+1; yj+1)    : (A.1)

Since Nx= xi+1− xi and Ny= yj+1− yj, the inverse of the matrix in (A.1) can be derived by     xi yj xiyj 1 xi yj+1 xiyj+1 1 xi+1 yj xi+1yj 1 xi+1 yj+1 xi+1yj+1 1     −1 =       −yj+1=NxNy yj=NxNy yj+1=NxNy −yj=NxNy −xi+1=NxNy xi+1=NxNy xi=NxNy −xi=NxNy 1=NxNy −1=NxNy −1=NxNy 1=NxNy xi+1yj+1=NxNy −xi+1yj=NxNy −xiyj+1=NxNy xiyj=NxNy      : (A.2) By substituting (A.2) into (A.1), we have

a = −_Nyj+1 xNyIs(i; j) + yj NxNyIs(i; j + 1) − yj+1 NxNyIs(i + 1; j) − yj NxNyIs(i + 1; j + 1); b = −_Nxi+1 xNyIs(i; j) + xi+1 NxNyIs(i; j + 1) + xi NxNyIs(i + 1; j) − xi NxNyIs(i + 1; j + 1); c = 1 NxNyIs(i; j) − 1 NxNyIs(i; j + 1) − 1 NxNyIs(i + 1; j) + 1 NxNyIs(i + 1; j + 1); d = xi+1_Nyj+1 xNy Is(i; j) − xi+1yj NxNyIs(i; j + 1) − xiyj+1 NxNyIs(i + 1; j) + xiyj NxNyIs(i + 1; j + 1): (A.3)

By substituting the above four coecients into Fc(x; y) = ax + by + cxy + d and after a few mathematical manipulations, we have Fc(x; y) =(xi+1− x)(y_N j+1− y) xNy Is(i; j) + (xi+1− x)(y − yj) NxNy Is(i; j + 1) + (x − xi_N)(yj+1− y) xNy Is(i + 1; j) + (x − xi)(y − yj) NxNy Is(i + 1; j + 1): (A.4)

Since xi+1− x = Nx− (x − xi) and yj+1− y = Ny− (y − yj), it follows that

Fc(x; y) =[Nx− (x − x_N i)] x [Ny− (y − yj)] Ny Is(i; j) + [Nx− (x − xi)] Nx (y − yj) Ny Is(i; j + 1) + (x − x_N i) x [Ny− (y − yj)] Ny Is(i + 1; j) + (x − xi) Nx (y − yj) Ny Is(i + 1; j + 1); (A.5) which is exactly (5).

References

[1] K. Arakawa, Y. Arakawa, A nonlinear digital lter using fuzzy clustering, Proc. IEEE Internat. Conf. Acoust. Speech, Signal Process. IV, San Francisco, March 1992, pp. 309–312.

[2] J.C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum Press, New York, 1981.

[3] T.C. Chen, R.J.P. deFigueiredo, Two-dimensional interpolation by generalized spline lters based on partial dierential equation image models, IEEE Trans. Acoust. Speech, Signal Process. 33 (1985) 631–642.

[4] I. Enbustsu, K. Baba, N. Hara, Fuzzy rule extraction from a multilayered neural network, Proc. Internat. Joint Conf. Neural Networks, Seattle, July 1991, pp. 461–465.

[5] R.C. Gonzalez, R.E. Woods, Digital Image Processing, Addison-Wesley, Reading, MA, 1992.

[6] I. Hayashi, H. Nomura, N. Wakami, Construction of fuzzy inference rules by NDF and NDFL, Internat. J. Approx. Reason. 6 (1992) 241–266.

[7] H.S. Hou, H.C. Andrews, Cubic splines for image interpolation and digital ltering, IEEE Trans. Acoust. Speech, Signal Process. 26 (1978) 508–517.

[8] R.G. Keys, Cubic convolution interpolation for digital image processing, IEEE Trans. Acoust. Speech, Signal Process. 29 (1981) 1153–1160.

[9] R. Krishnapuram, J.M. Keller, Fuzzy set theoretic approach to computer vision: an overview, Proc. IEEE Internat. Conf. Fuzzy Systems, San Diego, March 1992, pp. 135–142.

[10] C.C. Lee, Fuzzy logic in control systems: fuzzy logic controller – Part I and Part II, IEEE Trans. Systems Man, Cybernet. 20 (1990) 404–435.

[11] J.S. Lim, Two-Dimensional Signal and Image Processing, Prentice-Hall, Englewood Clis, NJ, 1990.

[12] C.T. Lin, C.S.G. Lee, Real-time supervised structure=parameter learning for fuzzy neural network, Proc. IEEE Internat. Conf. Fuzzy Systems, San Diego, March 1992, pp. 1283–1291.

[13] S.K. Pal, R.A. King, Image enhancement using smoothing with fuzzy sets, IEEE Trans. Systems Man, Cybernet. 11 (1981) 494–501. [14] S.K. Pal, S. Mitra, Multilayer perceptron, fuzzy sets, and classication, IEEE Trans. Neural Networks 3 (1992) 683–697. [15] S.K. Park, R.A. Showengerdt, Image reconstruction by parametric convolution, Comput. Vision, Graphics, Image Process. 23 (1983)

258–272.

[16] W.K. Pratt, Digital Image Processing, Wiley, New York, 1991.

[17] F. Russo, G. Ramponi, Combined FIRE lters for image enhancement, Proc. IEEE Conf. Fuzzy Systems I, Orlando, June 1994, pp. 260–264.

[18] M. Sugeno, An introductory survey of fuzzy control, Inform. Sci. 36 (1985) 59–83.

[19] E. Uchino, T. Yamakawa, T. Miki, S. Nakamura, Fuzzy rule-based simple interpolation algorithm for discrete signal, Fuzzy Sets and Systems 59 (1993) 259–270.

[20] M. Unser, A. Aldroub, M. Eden, Fast B-spline transforms for continuous image representation and interpolation, IEEE Trans. Pattern Anal. Mach. Intell. 13 (1991) 277–285.