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International Journal of Innovations in Engineering and Management, Vol. 2; No. 2: ISSN: 2319-3344 (July-Dec. 2013)

Performance Comparison of Gaussian and Butterworth High Pass
Filters

Ayush Dogra, Dr. Manjeet Singh Patterh

Punjabi University, Patiala (Punjab), India
Email: [email protected], [email protected]

Abstract: In this paper, Gaussian and Butterworth high pass filters are analyzed for image sharpening qualitatively
and quantitatively. The filters are implemented in frequency domain using MatLab software. Experiments are
performed on number of CT and MR images. To compare the performance of both the filters, processed images are
presented. The results demonstrate that Butterworth filter yields better results.

Keywords: Gaussian, Butterworth, Image Sharpening, Frequency Domain, High Pass Filter, Sharpenig

Accepted On: 11.09.2013

1. Introduction here is an indication of amount of change in
original & sharpened images.

The objective of image filtering is to process the
image so that the result is more suitable then the 2. Gaussian and Butterworth High

original image for a specific applications. A large Pass Filter
variety of image processing task can be

implemented using various filters. A filter that
Low pass filters and high pass filters are used to
attenuates high frequencies while passing low
show some details in the image while hiding other
frequencies is called low pass filter. Low pass
details. Low pass filters blur the image which
filter are usually used for smoothing while high
leads to noise reduction while high pass filter
pass filter is use for sharpening.
sharpen some image details, such as edges[6]. In
In literature, Zwirn and Akselrod [1] proposed an
the case of Gaussian filtering, the frequency
enhancement method for echocardiography
coefficients are not cut abruptly, but smoother cut
images. Echocardiography is such images which
of process is used instead. Thus also take
suffer from noise and difficulty in interpretation.
advantage of the fact that the DFT of a Gaussian
Fanga D. and his associated workers [2] proposed
function is a also a Gaussian function. The
an image enhancement method for smoothing
Gaussian high pass filters attenuates frequency
details and preserving edges of an image based on
components that are near to image centre. The
nonlinear diffusion equation. Kim B., Kim H. and
Gaussian high pass is given by [7]
Park D [3] proposed enhancement method for
fingerprint images. Haddad and Akansu [4] (cid:1)(cid:2)(cid:3),(cid:5)(cid:6)= −(cid:9)(cid:10)(cid:11)(cid:12)(cid:2)(cid:13),(cid:14)(cid:6)/(cid:16) (cid:11)(cid:18)(cid:12) (1)

employed binomial filters to achieve fast near- Another version of smoothing /sharpening filters is
Gaussian filtering. Clustering filter, an approach Butterworth filter. An advantage with Butterworth
proposed by Wong [5] which covered the studies filter is that we can control the sharpness of the
reveal the use of frequency domain filters. Yusuf, filter with the order. A Butterworth high pass filter
Nijad, Sara Tedmory [6] proposed and exploited keeps frequencies outside radius D and discards
0
hybrid method for enhancing digital x-ray images. value inside. It has gradual transition from 0 to 1
In this paper, we performed image sharpening on to reduce ringing artifacts. A Butterworth filter
CT and MRI images by Gaussian and Butterworth (BHPF) of order n and cut of frequency D is
0
high pass filter and evaluate their performance defined as [7]
qualitatively (visual inspection) and quantatively
by calculating mean square error (MSE). MSE (cid:1)(cid:2)(cid:3),(cid:5)(cid:6)= (cid:19) (2)
(cid:19)(cid:20)[ (cid:11)(cid:2)(cid:13),(cid:14)(cid:6)/(cid:11)(cid:18)](cid:12)(cid:23)
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International Journal of Innovations in Engineering and Management, Vol. 2; No. 2: ISSN: 2319-3344 (July-Dec. 2013)

3. Mean Square Error (MSE)

The Mean Square Error (MSE) is the error metric
used to compare image quality. The MSE
represents the cumulative squared error between
the compressed and the original image, Mean
Square Error (MSE) is given by [8, 9]

(cid:24)(cid:25)(cid:26) = ∑ , (3)
"∗$

M and N are the number of rows and columns in
the input images.

4. Methodology

Fig. 2 (a) Original CT image (b) FFT of CT image (c)
Image filtering refers to a process that removes the 4th order BHPF of FFT image (d) GHPF of FFT image

noise, improves the digital image for varied

application. The basic steps in frequency domain
filtering are shown in Fig. 1. Fourier transform
will reflect the frequencies of periodic parts of the
image. By applying the inverse Fourier transform
the undesired or unwanted frequencies can be
removed and this is called masking or filtering. A
filter is a matrix, and components of the filters
usually vary from 0 to 1. If the component is 1,
then the frequency is allowed to pass, if the
component is 0 the frequency is tossed out.

Fig. 3 (a) CT image (b) FFT of CT image (c) 4th order
BHPF of FFT image (d) GHPF of FFT image

Fig.1 Basic steps for filtering in frequency domain
5. Result and Discussion

Filtering in frequency domain consist of following

steps [6]
High pass filter give emphasis to the higher
1. Pre processing - multiplying the input image
frequencies in the image. The high pass image is
by (-1) to centre the transform.
x+y then added to the original image so as to obtain a
2. Computing F(u,v), using any of the
sharper image. It may be interesting to experiment
transformation methods.
with width and frequency threshold of the
3. Multiplying F(u,v) by a filter function H(u,v).
Butterworth or the Gaussian high pass filters and
4. Computing the inverse transformation of the
it will really interesting to compare the sharpening
result.
which is done in frequency domain and can
5. Post processing – multiplying the result by (-
compare it with the sharpening done in spatial
1) .
x+y domain. We will only demonstrate the image
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International Journal of Innovations in Engineering and Management, Vol. 2; No. 2: ISSN: 2319-3344 (July-Dec. 2013)

sharpening using Gaussian and Butterworth high
pass filter taking D =100,n=4(where D is cutoff
o o
frequency, n is the order of the Butterworth filter).
Fig. 2 (a) is the CT image, (b) depicts the FFT of
the image, and (c) shows the Butterworth high
pass filter of FFT image (d) shows the Gaussian
high pass filter of FFT image. Similar examples
are shown with other CT images as depicted in
Fig. (3, 4, 5, 6). Equivalent examples can be
repeated with MRI datasets shown in Fig.
(7, 8, 9, 10, 11) Now the resultant sharpened
images of CT and MRI datasets are shown in Fig.
(12, 13, 14, 15). Now these sharpened images can
be used in various image processing tasks, like
edge detection and ridge detection. For
quantitative analysis, computed the MSE between
original input images and reconstructed images
using equation 3. The following calculated values
for MSE between input images and reconstructed

images are shown in Table 1(a & b) Fig. 5 (a) CT image (b) FFT of CT image (c) 4th order
BHPF of FFT image (d) GHPF of FFT image

Fig. 4 (a) CT image (b) FFT of CT image (c) 4th order
BHPF of FFT image (d) GHPF of FFT image Fig. 6 (a) CT image (b) FFT of CT image (c) 4th order
BHPF of FFT image (d) GHPF of FFT image

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International Journal of Innovations in Engineering and Management, Vol. 2; No. 2: ISSN: 2319-3344 (July-Dec. 2013)

Fig. 7 (a) Original MRI image (b) FFT of MRI image
(c) 4th order BHPF of FFT MRI Image (d) GHPF of Fig. 9 (a) Original MRI image (b) FFT of MRI image
FFT MRI image (c) 4th order BHPF of FFT MRI Image (d) GHPF of
FFT MRI image

Fig. 8 (a) Original MRI image (b) FFT of MRI image Fig. 10 (a) Original MRI image (b) FFT of MRI image
(c) 4th order BHPF of FFT MRI Image (d) GHPF of (c) 4th order BHPF of FFT MRI Image (d) GHPF of
FFT MRI image FFT MRI image

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International Journal of Innovations in Engineering and Management, Vol. 2; No. 2: ISSN: 2319-3344 (July-Dec. 2013)

Fig 11 (a) Original MRI image (b) FFT of MRI image
(c) 4th order BHPF of FFT MRI Image (d) GHPF of
FFT MRI image

Fig.13 Sharpened CT images by 4th order BHPF

` Fig. 12 Sharpened CT images by GHPF

Fig.14 Sharpened MR images by GHPF

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International Journal of Innovations in Engineering and Management, Vol. 2; No. 2: ISSN: 2319-3344 (July-Dec. 2013)

emphasis on the high frequencies in the image.
The difference between Butterworth and Gaussian
filters is that the former provides much sharper
image than latter. The resultant images by BHPF
is much sharper than GHPF ,while analysis the
FFT of CT and MRI images, one sharp spike is
concentrated in the middle. By applying BHPF &
GHPF on the images, it is found that the color
intensities of high pass filter of FFT images
drastically change. The split edges are elongated
and wider in BHPF case than GHPF. But in GHPF
case the split edges are sharper. These elongated
split edges in BHPF leads to sharper image than
GHPF. Computing MSE between input images &
reconstructed images depicts that reconstructed
images by BHPF possess better image quality than
GHPF as MSE here indicates change and this
change is greater in BHPF as compare to GHPF.

References:

[1] Zwirn G. and Akselrod S., “ A Histogram-
Based Technique for Echocardiographic

Fig.15 Sharpened MR images by 4th order BHPF Image Enhancement,” IEEE Journal of
Computers in Cardiology, vol. 31, pp. 81-84,
Table 1 (a) MSE for CT images & (b) MSE for MR 2004.
Images [2] Fanga D., Nanninga Z., and Jianrua X.,
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Original CT Nonlinear Diffusion Equation,” Signal
& Sharpened GHPF BHPF
Processing Journal, vol. 88, no. 11, pp. 2850-
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1 52.6933 77.0619
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3 44.4186 58.8527 Enhancement Algorithm Based on Local
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Original MR [4] R.A. Haddad and A.N. Akansu, "A Class of
& Sharpened GHPF BHPF
Fast Gaussian Binomial Filters for Speech and

MR images
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3 27.5597 45.9106 39, pp 723-727, March 1991.
4 18.3468 29.5037 [5] Wong Y., “Image Enhancement by Edge-
5 60.6688 88.8303 Preserving Filtering,” in Proceedings of the 1st
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6. Conclusion [6] Yusuf, Nijad, Sara Tedmory “Exploiting

hybrid methods for enhancing digital X-ray
Gaussian and 4th order Butterworth high pass filter images”.The International Arab Journal of
with D =100 is used. High pass filter give Information Technology, Vol. 10, No.1,
o
January 2013
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International Journal of Innovations in Engineering and Management, Vol. 2; No. 2: ISSN: 2319-3344 (July-Dec. 2013)

[7] R.C.Gonzales, R.E.Woods, Digital Image
Processing, 2-nd Edition, Prentice Hall, 2002
[8] Mathworks,accelerating the pace of
engineering and sciencehttp:
//www.mathworks.in/ help/vision /ref/
psnr.html
[9] B. Girod, “What’s wrong with mean-squared
error,” in Digital Images and Human Vision
(A. B. Watson, ed.), pp. 207–220, the MIT
press, 1993.

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