Spectral approach to image projection with cubic b spline interpolation

International Journal of Electronics and JOURNAL OF ELECTRONICS AND ISSN 0976 –
INTERNATIONAL Communication Engineering & Technology (IJECET),
6464(Print), ISSN 0976 – 6472(Online) Volume 3, Issue 3, October- December (2012), © IAEME
COMMUNICATION ENGINEERING & TECHNOLOGY (IJECET)
ISSN 0976 – 6464(Print)
ISSN 0976 – 6472(Online)
Volume 3, Issue 3, October- December (2012), pp. 153-161
IJECET
© IAEME: www.iaeme.com/ijecet.asp
Journal Impact Factor (2012): 3.5930 (Calculated by GISI) ©IAEME
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SPECTRAL APPROACH TO IMAGE PROJECTION WITH CUBIC
B-SPLINE INTERPOLATION

1. M. Nagaraju Naik, 2. P. Rajesh Kumar
1. Assoc. Prof., Mahaveer Institute of Science and Technology, Hyderabad, A P, India.
nagraju_naik@yahoo.co.in
2. Assoc. Prof, A.U. College of Engineering (Autonomous), Vizag, A P, India.
rajeshauce@gmail.com.

ABSTRACT

This paper proposes an energy spectrum interpolating method based on energy variations in an
image. As the size of an image is increased, so the pixels, which comprise the image, become
increasingly visible, making the image to appear soft. Super scalar representation of image
sequence is limited due to image information present in low dimensional image sequence. To
project an image frame sequence into high-resolution static or fractional scaling value, a scaling
approach is developed based on energy spectral interpolation by combining both Fast Fourier
transform and Bicubical interpolation.

Keywords: Bicubical interpolation, Super Resolution, Digital image, smoothness

I. INTRODUCTION

Image processing is a form of signal processing for which the input is an image, such as a
photograph or video frame; the output of image processing may be either an image or a set of
characteristics or parameters related to the image. Most image-processing techniques involve
treating the image as a two-dimensional signal and applying standard signal-processing
techniques to it. In the area of image processing there is a need to improve the resource
requirement for progressive image processing using resource optimization techniques. In earlier
approaches it is observed that image sequencing can be improved by optimizing usage of
available resources. The earlier proposed methods based on super resolution [1-4] were observed
to be developed keeping available resources and there constrains in mind. Today’s applications
demand is high-resolution representation of gray scale and color [8] of image data for real time
interfacing and communications. With the incorporation of developed optimization scheme as

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International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 –

outlined above can provide a significant improvement in coding but in current scenario these
methods may get constrained. As the available resources such as bandwidth, power, coding
techniques are limited to certain minimum values. To achieve high resolution representation
images are to be retained for good visual quality. As resource optimizations are constrained,
coding based on vector regression [6, 7, 9] techniques are stated to improve quality in image
processing. To achieve higher visual quality the stated interpolation approaches were carried out
in frequency representation [5, 10] using transformation techniques.
Though these interpolation methods are efficient to produce a HR image from a low LR image
they are not able to provide efficient visibility. So a new interpolating method is proposed in this
paper based on energy evaluation of an image using FFT and interpolating by Bicubical
interpolation and the paper is organized as follows. Section II gives brief information about
different types of interpolation methods, Section III gives describes the proposed method and
section IV gives the system architecture. The results and conclusions are drawn in Section V and
Section VI respectively.

II. INTERPOLATION APPROACH

Interpolation is the process of estimating the values of a continuous function from discrete
samples. Image processing applications of interpolation include image magnification or
reduction, sub pixel image registration, to correct spatial distortions, and image decompression.
There are so many interpolation techniques like linear interpolation, bilinear interpolation and
Cubical interpolations.

II.1 Bilinear interpolation

Bilinear Interpolation determines the grey level value from the weighted average of the four
closest pixels to the specified input coordinates, and assigns that value to the output coordinates.
First, two linear interpolations are performed in one direction and then one more linear
interpolation is performed in the perpendicular direction. For one-dimension Linear
Interpolation, the number of grid points needed to evaluate the interpolation function is two. For
Bilinear Interpolation (linear interpolation in two dimensions), the number of grid points needed
to evaluate the interpolation function is four.
For linear interpolation, the interpolation kernel is:
u(s) = {0 |s| > 1
{1 – |s| |s| < 1 …………….. (1)
Where‘s’ is the distance between the point to be interpolated and the grid point being considered.
The interpolation coefficients
ck = f (xk).

II.2 Bi-Cubic Interpolation

Cubic Convolution Interpolation determines the grey level value from the weighted average of
the 16 closest pixels to the specified input coordinates, and assigns that value to the output
coordinates. The image is slightly sharper than that produced by Bilinear Interpolation, and it
does not have the disjointed appearance produced by Nearest Neighbor Interpolation.
First, four one-dimension cubic convolutions are performed in one direction (horizontally in this
paper) and then one more one-dimension cubic convolution is performed in the perpendicular
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direction (vertically in this paper). This means that to implement a two-dimension cubic
convolution, a one-dimension cubic convolution is all that is needed.
For one-dimension Cubic Convolution Interpolation, the number of grid points needed to
evaluate the interpolation function is four, two grid points on either side of the point under
consideration. For Bicubic Interpolation (cubic convolution interpolation in two dimensions), the
number of grid points needed to evaluate the interpolation function is 16, two grid points on
either side of the point under consideration for both horizontal and vertical directions.
Though these interpolation methods are efficient to produce a HR image from a low LR image
they are not able to provide efficient visibility. So a new interpolating method is proposed in this
paper based on energy evaluation of an image using FFT and interpolating by Bicubical
interpolation, and is discussed in next section.

III. PROPOSED METHOD

This method is accomplished in two steps. First the image interpolation is done by Bicubical
interpolation and second the projection is done using Fast Fourier transform.

III.1 Image Coding

The following figure describes the image conversion from LR to HR. On the left hand side four
low-resolution images are shown Motion estimation is used to estimate the pixel positions of the
st
three images with respect to the 1 image. Once this information is calculated accurately, it is
possible to project this information on a desired high-resolution grid.

Figure 1 Conversion of low-resolution images to high-resolution images

Any Super-resolution algorithm is to estimate the motion between given LR frames. A good ME
is a hard prerequisite for SR. In this paper the motion is restricted to shifts and rotation, so a very
simple (though accurate) approach is enough for image registration. Rotate the individual images
at all the angles and correlate them with the first image. The angle that gives the maximum
correlation is the angle of rotation between them. The angle can be calculated as follows
.
Angle (i) = max index (correlation (I1(θ), Ii (θ))) ………(2)
Where I1(θ) is the pixel intensity of the reference pixel and Ii(θ) is the intensity of the ith pixel. It
turned out that though the first method is computationally expensive, but gives more precise
results, so it was used in this project.

III.2 Shift Calculation

Once rotation angle is known between different images, shift calculation can be performed.
Before calculating the shift, all the images are rotated with respect to the first image. For
determining the amount of shift in any pixel of an image,

155


Fi(uT) = ej2πu∆s. F1 (uT) ______(3)
This is obtained by applying Fourier Transform of a reference pixel matrix. The shift angle ∆s
from the above relation can be calculated as:
∆s = [angle (Fi(uT)/ F1(uT))]/2π____(4)
And in matrix form,
∆s= [∆x ∆y] T________ (5)
Where,
U(x, y) is the pixel coordinate, ∆x is the variation of current x-position from reference x-
position, ∆y is the variation of current y-position from reference y-position, Fi (uT) is the
transform of transposed ith pixel, F1 (uT) is the transform of transposed reference pixel, ∆s is the
shift angle respectively.
In the next step the projection of pixel values is going to be done and called as iterative back
projection.

III.3 Iterative Back Projection

The iterative back-projection (IBP) technique [6] can accomplish the HR image interpolation and
de-blurring simultaneously. Its underlying idea is that the reconstructed HR image from the
degraded LR image should produce the same observed LR image if passing it through the same
blurring and down sampling process. The iterative back-projection (IBP) technique can minimize
the reconstruction error by iteratively back projecting the reconstruction error into the
reconstructed image. Taking into account several considerations, a method that was fairly simple
and straightforward - Fourier algorithm (P-G Algorithm) is proposed.

III.3.1 Fourier Projection

This method assumes two things:
• Some of the pixel values in the high-resolution grid are known.
• The high frequency components in the high-resolution image are zero.
It works by projecting HR grid data on the two sets described above. The steps are:
• Form a high resolution grid. Set the known pixels values from the low-resolution images
(after converting their pixel position to the ref frame of first low-resolution image). The
position on the HR grid is calculated by rounding the magnified pixel positions to nearest
integer locations.
• Set the high-frequency components to zero in the frequency domain.
• Force the known pixel values in spatial domain. Iterate.

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Figure 2 Flow Chart for P-G Algorithm

By making the high-frequency equal to zero, this method tries to interpolate the unknown values
and so correct the aliasing for low-frequency components. Also, by forcing the known values, it
does predict some of the high-frequency values. The set of images walks through the actual
working of this algorithm. Initially, the HR grid is filled with known pixel values and makes the
unknown pixel values to be zero.
In the next step, the higher frequencies can be made zero in the frequency domain. This
effectively is low-pass filtering the image. The unknown pixels now have got some value, and
the known values have gone down in amplitude, due to low-pass filtering. The magnitude of
known pixels can be increased by forcing them to what they should be. This again creates some
high-frequency components by iteratively doing this again and again, correcting the low-
frequency values (by guessing the values for unknown pixels) and finding some the high-
frequency components by forcing the known values is achieved. By juggling between the two
data sets, i.e. forcing the high frequency to zero and forcing the known values, we have estimated
the value of unknown pixels.
Thus a super resolute image is produced and the model of the system related to this paper is
shown below.

IV. SYSTEM MODEL

The Architecture of Proposed Method is shown below, here frame generator takes low resolution
image sequences as input and converts it into static frames. These static frames converted into
grey level in the pre-processing step. Next these grey level frames converted into frequency
domain using FFT transformation to compare we are using Cubic-B-Spline method. The
transformed data to than interpolated (spectral projection/spectral resolution) using FFT and
Cubic-B-Spline. The projected data is aligned over a predefined grid format to obtain high
resolution image. This image sequence is compared with original data to extract Mean Error.

157


Figure 3 Architecture of proposed method

The concept of resolution projection of image stream is developed using spectral and frequency
interpolations and evaluated for computational time and retrieval accuracy.

IV.1 Operational Description

1) Input Interface: The developed system is processed over a very low image sequence
represented in low dimensional projection. To evaluate the performance of suggested scaling
system, a low dimensional, colored image streams are read and transformed into frame sequence
using input interface unit. The processed frame sequence is then passed to a pre-processing unit
for the equalization of input frame sequence for further processing.
1) Pre-Processing: This unit extracts the gray pixel intensity of the continuous frame sequence
and pass to the transformation unit for further processing. The gray pixel intensity are extracted
from the input information segregated colored information.
2) Transformation Unit: This unit transforms the given input information into power spectral
distribution using Fourier transformation. It is observed in conventional architectures that the
energy distribution of the original data could be used as interpolating information to represent
high quality images. But it is observed that spectral distributions need not be sufficient for
accurate interpolation, as the frequency resolution for spectrum energy coefficients may vary
distinctly. To achieve better representation Cubic-B-Spline method is incorporated for such
requirement.
3) Interpolation: Once the spectral resolutions were obtained, the pixel is to project on a
higher grid level depending upon the scale value. Scaling of the image is achieved by
interpolating the pixel information based on energy distribution of the given image sequence. To
achieve better interpolation rather than energy resolution, spectral resolution could provide high-
resolution accuracy developed using Cubic-B-Spline approach. The interpolated information is
then projected on a grid projection to represent the given low dimensional image sequence into
high-resolution image sequence. The results related to functional description of system
architecture are shown below.

V. RESULTS & OBSERVATIONS

For the evaluation of the suggested method a simulation implementation is carried out for a
sequence of video frames. A real time video sample

158


Figure 4 Original image sequence considered

The original frame sequence is taken for processing of the image coding system. The original
frame sequence is taken at a very low resolution with pixel representation of 150x250 size frame.
These 5 frame sequences are passed to the developed system for pre processing.

Figure 5 Scaled image sequences at 1:2.5 ratio using Fourier approach

The interpolation is carried out for the spectral distributed image coefficients obtained after
Fourier transformation. The interpolation is made for the spectral distributed data as shown
above.

Figure 6 Scaled image sequences at 1:2.5 ratio using cubical-b-spline approach

The observation clearly illustrates the accuracy in retrieval in terms of visual quality as compared
to the conventional Fourier based coding technique.
P ro c e s s in g t im e p lo t
100
F o u rie r in t e rp o la t io n
90 C u b ic -b -s p lin e in t e rp o la t io n

80

70
Computation time(Sec)

60

50

40

30

20

10

0
1 2
O b s e rva t io n

Figure 7 Computation time taken for the two methods

159


The system developed is also evaluated for the computation time taken for the computation and
projection of the frame sequence for interpolation. The total time taken for reading, processing
and projecting is considered for the processing system and the conclusions are drawn below.

VI. CONCLUSIONS

The energy spectral resolution projecting is carried out using Fourier transform techniques,
where a low dimensional image sequence is projected to a high grid based on energy distribution.
To improve resolution accuracy, a frequency based projection scheme is developed. To realize
the frequency spectral resolution Cubic-B-Spline method is used. It is observed that the
resolution accuracy with respect to visual quality, mean error and computational time is
comparatively improved compared to conventional Fourier based interpolation technique. For
the evaluation of the suggested approach, the system is tested over various low dimensions of
image sequence and scaled over fixed and fractional scaling value. Due to the higher visual
quality the system find applications in various real time applications such as Television
processing, Image conferencing, Internet image processing, Tele medicine etc.

VII. REFERENCES

[1] S. P. Kim, N. K. Bose, and H. M. Valenzuela, “Recursive reconstruction of high resolution
image from noisy undersampled multiframes”, IEEE Trans. Acoust., Speech, Signal Processing,
vol. 38, pp. 1013-1027, June 1990.
[2] S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super
resolution”, IEEE Trans. Image Processing, vol. 13, pp. 1327-1344, Oct. 2004.
[3] X. Li and M. T. Orchard, “New edge-directed interpolation”, IEEE Trans. Image Proc., vol.
10, pp. 1521-1527, Oct. 2001.
[4] H. A. Aly and E. Dubois, “Specification of the observation model for regularized image up-
sampling,” IEEE Trans. Image Processing, vol. 14, pp. 567-576, May 2005.
[5] R. S. Prendergast and T. Q. Nguyen, “Spectral modelling and Fourier domain recovery of
high-resolution images from jointly undersampled image sets”, under review for IEEE Trans.
Image Proc., submitted Dec. 18, 2006.
[6] K. S. Ni and T. Q. Nguyen,“Image superresolution using support vector regression”, IEEE
Trans. Image Proc., vol. 16, pp. 1596- 1610, June 2007.
[7] S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical
overview”, IEEE Signal Processing Mag., vol. 20, pp. 21-36, May 2003.
[8] B. Narayanan, R. C. Hardie, K. E. Barner, and M. Shao, “A computationally efficient super-
resolution algorithm for image processing using partition filters,” IEEE Trans. on Circ. Syst. For
Image Technology, vol. 17, no. 5, pp. 621-634, May 2007.
[9] S. Farsiu, M. Elad, and P. Milanfar, “Image-to-image dynamic superresolution for grayscale
and color sequences,” EURASIP Journal of Applied Signal Processing, Special Issue on
Superresolution Imaging, vol. 2006, pp. 1–15, 2006.
[10] R. C. Hardie, “A fast image super-resolution algorithm using an adaptive Wiener filter,”
IEEE Trans. Image Proc., vol. 16, no. 12, pp. 2953-2964, Dec. 2007.

160


M. Nagaraju Naik received his B.Tech from S.V.University, Thirupati, A. P. , India
in 1999. Masters Degree in Digital Systems and Computer Electronics from JNTU
Anantapur A.P., India in 2005. He is pursuing Ph.D in Andhra University College of
Engineering (Autonomous). His interest area Video Processing, Image Processing
and Signal Processing

Dr. P. Rajesh Kumar, Associate Professor Department of ECE, Andhra University
Vizag. A.P., India. Received his Ph.D degree in 2007 on’Radar Signal
Processing’.He is presently engaed in research in Image Processing. His interest area
Signal Processing, Radar Signaling, Image Processing.

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Spectral approach to image projection with cubic b spline interpolation

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