a Novel Image Processing Filter Designed Using Discrete Fourier Invariant

A Novel Image Processing Filter Designed Using Discrete Fourier Invariant Signal

Roshni Ravi Dept. of Electronics and Communication Engineering

Rajiv Gandhi Institute of Technology, Kottayam Kerala, India

[email protected]

Josemartin M.J. Dept. of Electronics and Communication Engineering

Rajiv Gandhi Institute of Technology, Kottayam Kerala, India

[email protected]

AbstractIn this paper, a new image processing filter is proposed. In order to construct this image smoothing filter, one dimensional discrete Fourier invariant signal generated by an iterative design principle based on gradient descent method is used. The filter shape and values in spatial and frequency domain is almost the same. The proposed filter can be used as a kernel matrix in image processing to perform blurring as well as high frequency noise suppression. Also it can be used as an optimal two dimensional window for spatial-frequency spectral analysis of images.

KeywordsDiscrete Fourier invariant signal, 2D kernel matrix, image smoothing, spatial-frequency spectral analysis

I. INTRODUCTION All Fourier invariant signals have very interesting

property of shape invariance in time as well as in frequency domain, for example Gaussian signal, Dirac delta function, and hyperbolic secant functions. Two practical methods for the design of one dimensional (1D) discrete Fourier invariant signals are proposed in [1]. The direct design method involves splitting the signal into independent and dependent parts and calculation of the dependent for any independent part by using an obtained connection matrix. This method has accuracy problem for long signals. The iterative design method overcomes the accuracy problem and it is based on a successive approach by using any symmetrical discrete signal as the input. In this paper we use iterative method for the generation of 1D discrete Fourier invariant signals and interpolation techniques are then used to construct a two dimensional (2D) signal. Unlike the 2D Gaussian signal, the 2D signal obtained from this method is not only Fourier invariant but also have almost the same size in spatial domain and frequency domain.

A. Mathematical Background The Fourier transform encompasses a vast area of each

and every field of engineering. Fourier transform takes a function from time/spatial domain and maps it onto frequency domain [2]. For a continuous time domain signal, its Fourier transform is given by

And in a similar way the signal can be mapped back to

time/spatial domain from the frequency domain using inverse Fourier transform

In order to perform spectral analysis of discrete signals,

we use Discrete Fourier Transform (DFT) and the Centered Discrete Fourier Transform (CDFT).

For an N point discrete signal, its DFT is given by

n=0, 1, 2, . N, is the frequency sample. The CDFT is defined as

! " ! "

#

! " ! "

$

where and are the spectral and signal samples in frequency and time/spatial domain respectively. The convenience of using CDFT is that it makes time/spatial domain and frequency samples directly comparable, since the time/spatial domain samples at {0,1,2,..N} are mapped around the zero index {% ,% & ', } and the corresponding frequency samples at {0,,, } are centered around zero frequency {% ,% & ,, }.

The time/spatial and frequency points are normalized and are given by

%($ ) * + % , - , ) ($.

%($ ) /0 0 % , - , ) ($1

for * =1,2,..,N and 0 =1,2..,N If the N samples of the discrete signals are given by the

vector x and N spectral coefficients by vector X, then the matrix form of the CDFT operation can be represented as

X=Fx and x=F*X

2014 International Conference on Electronic Systems, Signal Processing and Computing Technologies

978-1-4799-2102-7/14 $31.00 2014 IEEEDOI 10.1109/ICESC.2014.88

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where F is a CDFT transform matrix given by

2' ! " ! " 3

for n=1,,N and k=1,,N.

B. Fourier Invariant Signals In iterative design method, discrete Fourier invariant

signals can be generated by minimizing the difference between the signal and its CDFT spectrum based on gradient descent method in successive iterations. After certain number of iterations the difference converges to zero and we obtain Fourier invariant discrete signal. Any symmetrical discrete signal of length N can be used as an initial signal at the start. 4 !4 546 '' 7 8' 8 9 - :;

In the of the signal ?> and the difference vector > between the signal and its spectrum are calculated by => @?>A> => % ?>(

@ BC' 7 C'D7 7 7CD' 7 CD'DE Matrix @is a real valued Fourier transform matrix of size KXK and given as

2' FGH IJ 6 % - " * % - " K

for n = 1,2..,K and k = 0,1,2..,N-1

2'L FGH IJ 6 % - " 8 % - " K

for 0 = 1,2..,K M Z[S> \

Z5> ] S> 2S'6 ^ 6O S> [2S' % \6 6O$

Now a new signal with a smaller maximum difference is given by ?>! ?> % _ `>. where is the gain constant of the iteration loop. For fast convergence should be as large as possible but stability condition of the iteration limits its value. The iteration method puts no constraint over the initial input signal except that the signal should be symmetric.

Flexibility of iterative method lies at its choice for initial input signal. When ramp signal is the input signal we obtain a Fourier invariant signal as shown in Fig. 1 and values depicted in Table I. Similarly when Gaussian signal

is given as the input signal we obtain a Fourier invariant signal as shown in Fig. 2 with values shown in table III. Table II shows the bandwidth and time-width comparison between the discrete Fourier invariant signal and the discrete Gaussian signal.

TABLE I. SIMULATION VALUES FOR ITERATIVE METHOD WITH INPUT AS RAMP SIGNAL

Time domain values

Corresponding frequency

domain values

Error difference

between time and frequency domain values

Corresponding time and frequency

values after 71 iterations

1 0.0898 0.9102 0.4067

2 -0.0745 2.8912 -0.0745

3 0.1088 4.0601 0.0989

4 -0.0601 4.8649 -0.0708

5 0.1351 6.0435 0.1351

6 -0.0435 6.8249 -0.0435

7 0.1751 8.0189 0.1751

8 -0.0189 8.7575 0.0256

9 0.2425 9.9726 0.2330

10 0.0274 10.6268 0.1393

11 0.3732 2.0745 0.3732

12 0.1402 11.8598 0.4580

13 0.6931 12.3069 0.9595

14 0.5352 13.4648 1.5672

15 1.9669 13.0331 2.5983

16 3.8288 12.1712 3.8287

17 42.0167 -25.0167 5.1250

Figure 1. Iterative design of Fourier invariant signal for N=34,

achieved after 71 iterations when input is a ramp signal.

TABLE II. TIME-WIDTH AND BANDWIDTH COMPARISON

Signal Time-width (T) Bandwidth

(B) Absolute

Difference (T-B)

Discrete Fourier Invariant Signal 0.0584 0.0654 0.0070

Discrete Gaussian Signal 0.1369 0.0175 0.1193

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TABLE III. SIMULATION VALUES FOR ITERATIVE METHOD WITH GAUSSIAN SIGNAL AS THE INPUT

Time domain values

Corresponding frequency

domain values

Error difference

between time and frequency domain values

Corresponding time and frequency

domain values after 71 iterations

0.0439 0.0061 -0.0378 0.0287

0.0634 -0.0062 -0.0696 -0.0062

0.0895 0.0063 -0.0832 -0.0069

0.1234 -0.0064 -0.1299 0.0066

0.1664 0.0066 -0.1597 -0.0069

0.2191 -0.0069 -0.2261 0.0073

0.2821 0.0073 -0.2748 -0.0040

0.3549 -0.0078 -0.3627 0.0083

0.4363 0.0084 -0.4280 -0.0025

0.5243 -0.0091 -0.5334 0.0101

0.6157 0.0101 -0.6056 0.0252

0.7066 -0.0113 -0.7179 0.0981

0.7926 0.0128 -0.7798 0.2088

0.8688 -0.0140 -0.8828 0.2088

0.9308 0.0423 -0.8885 0.3661

0.9745 0.5193 -0.4552 0.5193

0.9971 2.3611 1.3640 0.6226

Figure 2. Iterative design of Fourier invariant signal for N=34,

achieved after 85 iterations when input is a Gaussian signal.

II. A NOVEL 2D KERNEL MATRIX FROM THE 1D FOURIER INVARIANT SIGNAL

The kernel matrix which can be used as a novel image processing filter is constructed by using the 1D Fourier invariant signal. The generated 2D matrix almost looks alike in both spatial and frequency domains.

The interpolation algorithm used to generate the 2D matrix is presented here. For any 1D discrete Fourier invariant signal, x(n) of length N, the N N kernel matrix generated is denoted by X. Here N is assumed to be odd. Even if the generated 1D Fourier invariant signal is of even length, we can make it an odd length signal by removing one of the two equal samples values at the centre of the time axis. Let

Figure 3. 2D Gaussian matrix and its corresponding 2D-DFT matrix

Figure 4. 2D matrix constructed from 1D Fourier invariant signal by iterative method with ramp as the initial signal

Figure 5. Fourier Invariant (FI) 2D matrix from 1D Fourier invariant signal by iterative method with Gaussian as the initial signal

.

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III. APPLICATIONS

A. 2D Fourier Invariant Kernel Matrix Used as a Smoothing Filter

In image processing, while choosing a smoothing filter, there are two criteria to be fulfilled as described in [4]. The filter should be smooth and roughly band limited in the frequency domain to reduce the possible number of frequencies at which function changes can take place. Also the filter should be spatially localized. These two criteria are conflicting, and the Gaussian signal is a sub - optimum filter, since its distribution optimizes the two criteria.

The 2D Fourier invariant kernel matrix generated by iterative method can be used as image smoothing filter for noise removal in images, since it satisfies the two above mentioned criteria to a certain extent.

We have performed a comparison on the basis of performance evaluation of the 2D Fourier invariant matrix and the Gaussian kernel with standard deviation () 2.0, 2.5, 3.0, 3.5 and 4.0 on filtering a test image corrupted with Gaussian noise of mean 0 and variance 0.01 as shown in Fig. 6. To make the comparison genuine, the variance of the Gaussian kernel is selected in an appropriate range which is comparable to the variance of the 1D Fourier invariant signal. Root Mean Square Error (RMSE), Image Quality Index (IQI) and Peak Signal to Noise Ratio (PSNR) of the filtered images are calculated and tabulated in table IV, V, and VI and the filtered images are shown in Fig. 7.

Original image

Image with Gaussian noise

Figure 6. Test image and the image after adding Gaussian noise

TABLE IV. ROOT MEAN SQUARE ERROR (RMSE)

Kernel size

Fourier Invariant

2D Kernel Matrix

Gaussian Matrix

(=2.0) (=2.5) (=3.0) (=3.5) (=4.0)

5x5 12.4551 14.6221 15.0164 15.2414 15.3803 15.4717 7x7 12.5063 15.9307 16.7581 17.2503 17.5600 17.7657 9X9 12.6149 16.4176 17.7144 18.5480 19.0931 19.4626

11X11 13.2154 16.7788 18.4144 19.5697 20.3672 20.9247 13X13 13.7201 16.8136 18.6440 20.0614 21.1069 21.8690 15X15 14.3134 16.8223 18.7366 20.3279 21.5812 22.5389 17X17 14.6584 16.8199 18.7619 20.4457 21.8466 22.9698

TABLE V. IMAGE QUALITY INDEX(IQI)

Kernel Size

FI 2D Matrix

Gaussian Matrix (=2.0) (=2.5) (=3.0) (=3.5) (=4.0)

5x5 0.5779 0.5703 0.5604 0.5542 0.5502 0.5475 7x7 0.5855 0.5550 0.5348 0.5211 0.5119 0.5056 9X9 0.6000 0.5486 0.5138 0.4884 0.4709 0.4587

11X11 0.5917 0.5364 0.4935 0.4594 0.4345 0.4166 13X13 0.5948 0.5383 0.4869 0.4419 0.4075 0.3823 15X15 0.5849 0.5356 0.4826 0.4333 0.3933 0.3630 17X17 0.5797 0.5368 0.4845 0.4323 0.3872 0.3514

a)

b)

c)

d)

e)

f)

Figure 7. Filtered image using a) 5X5 Fourier invariant kernel b) 5X5 Gaussian kernel with standard deviation 2.0 c) 5X5 Gaussian kernel with standard deviation 2.5 d) 5X5 Gaussian kernel with standard deviation 3.0 e) 5X5 Gaussian kernel with standard deviation 3.5

f) 5X5 Gaussian kernel with standard deviation 4.0

From Table IV, V and VI it is clear that the 2D Fourier

invariant kernel behaves as a smoothing filter which removes noise in images by blurring.

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TABLE VI. PEAK SIGNAL TO NOISE RATIO (PSNR)

Kernel Size

FI 2D Matrix

Gaussian Matrix (=2.0) (=2.5) (=3.0) (=3.5) (=4.0)

5x5 26.2578 24.8646 24.6335 24.5043 24.4255 24.3740 7x7 26.2223 24.1201 23.6803 23.4289 23.2743 23.1731 9X9 26.1471 23.8586 23.1983 22.7989 22.5473 22.3808

11X11 25.7432 23.6696 22.8616 22.3331 21.9862 21.7516 13X13 25.4176 23.6516 22.7540 22.1176 21.6763 21.3682 15X15 25.0500 23.6471 22.7110 22.0029 21.4833 21.1061 17X17 24.8430 23.6483 22.6992 21.9528 21.3771 20.9416

B. 2D Fourier Invariant Matrix Used as an Optimal Spatial- Frequency Window

To perform time/frequency spectral analysis for 1D signal Short Time Fourier Transform (STFT) is used. In a similar way to perform spectral analysis of images, this 2D Fourier invariant kernel matrix can be used. Since 2D Fourier invariant matrix has optimum localization in spatial and frequency domain, it can be used as a 2D window for spatial-frequency spectral analysis.

For 1D signal spectral analysis, a window is multiplied with the signal in time domain and then the products STFT is calculated. Similarly for 2D case we have first multiplied our 33X33 Fourier invariant 2D matrix X with the test image A, then determined 2D DFT of the product and noted the centre value of the resulting 33X33 matrix. We again perform the same process by sliding the 2D window over test image with a shift of 33 as illustrated in Fig. 8.

2D Fourier invariant

kernel matrix X

Test image A Figure 8. Illustration of 2D windowing on a test image A by using the

Fourier invariant kernel matrix X

TABLE VII. SPATIAL- FREQUENCY ANALYSIS USING 2D FOURIER INVARIANT MATRIX

3.0303 2.7576 2.7273 2.6364 2.4545 2.6364 1.8485

1.8485 1.6970 1.6667 1.7273 1.8485 1.0606 1.0606

2.6364 2.1515 2.6061 1.8485 1.2727 2.3939 2.1515

2.4545 3.0909 2.8788 1.1515 2.3333 1.5455 1.7576

2.6061 4.3030 3.1212 3.1818 2.8182 2.8182 3.0303

3.0303 2.4242 3.7273 1.9091 1.1818 1.1818 1.4242

4.4242 1.2424 0.6970 3.0000 3.0303 2.9394 2.6667

Each value of the matrix/image shown in Table VII

indicates the frequency domain central sample values of the result obtained by the above multiplication, sliding and 2D DFT operations. Instead of taking only the centre values, we can also select any other sample values of the 2D DFT in this type of image spectral analysis.

IV. CONCLUSION The 2D Fourier invariant kernel matrix constructed using

the discrete Fourier invariant signal generated by the iterative method can be used as a novel image smoothing filter which provides optimum spatial localization and high frequency noise suppression. It can also be used as an optimal 2D window for spatial-frequency spectral analysis of images.

REFERENCES [1] Maja Temerinac-Ott, and Miodrag Temerinac, Discrete Fourier

invariant signals: design and application, IEEE transactions on Signal processing, vol. 60, no. 3, pp. 1108-1120, March 2012.

[2] A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems. Englewood Cliffs, NJ: Prentice-Hall, 1997.

[3] Rafel C. Gonzalez, Richard E. Woods, Digital Image Processing. Third edition, Prentice-Hall, 2008.

[4] D. Marr; E. Hildreth, Theory of edge detection Proceedings of the Royal Society of London. Series B, Biological Sciences, Vol. 207, No. 1167, Feb. 29, 1980), pp. 187-217.

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