### Er.Sudhir Sharma, IJRIT

85**IJRIT International Journal of Research in Information Technology, Volume 1, Issue 7, July, 2013, Pg. 85-90 **

International Journal of Research in Information Technology (IJRIT)

### www.ijrit.com

**ISSN 2001-5569**

## Review Paper on Interpolation of Digital Images using Modal Function

1

### Er.Sudhir Sharma,

^{2 }

### Er.Robin Walia

1M.tech Scholar

Department of Electronics & Communication Engineering, Maharishi Markandeshwar University, Mullana (Ambala), INDIA

2Asst.Prof.

Department of Electronics & Communication Engineering, Maharishi Markandeshwar University, Mullana (Ambala), INDIA

1 **er.sudhir466@gmail.com, **^{2 }**Robin1@gmail.com **

**Abstract **

Image interpolation techniques are referred in literature by many terminologies, such as image resizing, image resampling, digital zooming, image magnification or enhancement, etc. Basically, an image interpolation algorithm is used to convert an image from one resolution (dimension) to another resolution without losing the visual content in the picture. Image interpolation algorithms can be grouped in two categories, non-adaptive and adaptive. This paper, gives an overview of basic interpolation techniques and find that there is scope of improvement of quality in an image.

**Keywords: ***Interpolation, Resizing, Magnification, Zooming *
*.*

**1.**

**Introduction **

A digital image is buildup of elements called pixels. Mathematically, it can be defined as a two dimensional function f(x, y), where x and y are spatial (plane) co-ordinates. The value of ‘f’ at any pair of co-ordinate (x, y) is called the intensity ‘or’ gray level of the image at that point. The processing of digital images is called digital image processing[1].The most common digital image processing tasks include: resizing, zooming and colour enhancement.

This paper focused on zooming of digital images. Image zooming means changing the number of display pixels per image pixel only in appearance. For zoom level= 1, there is one display pixel per image pixel. However for zoom level= 2, there are 2 display pixels per image pixel in both x and y axis. This enlargement is usually quantified in terms of a number (greater than one) called magnification. In terms of applications, the process of image zooming is usually applied in diverse areas ranging from computer graphics, rendering, editing, medical image reconstruction, to online image viewing. The zooming operation is performed using resampling and interpolation operations.

### Er.Sudhir Sharma, IJRIT

86 Traditional image zooming techniques use up-sampling by zero-insertion followed by one-dimensional filtering to interpolate the high resolution (HR) samples [1-3].The image quality highly depends on the used interpolation technique. Image interpolation works in two directions, and tries to achieve a best approximation of a pixel's colour and intensity based on the values at surrounding pixels. The following example illustrates how resizing / enlargement works.

Fig.1.Resizing/enlargement work [1].

**2. Interpolation Techniques **

Several approaches have been proposed to guess reasonable high resolution patterns from low resolution original images. The problem is quite complex as, in general, no hints on the real high resolution signal are available. The most sophisticated interpolation methods try to extract statistical information on the relationship between high and low resolution images from a training set of natural images (or images of interest). Interpolation (sometimes called resampling) is an imaging method to increase (or decrease) the number of pixels in a digital image [4]. It is applied in diverse areas ranging from computer graphics, rendering, editing, medical image

reconstruction, to online image viewing. The interpolation techniques are categorized into two parts.

**2.1 Non-Adaptive Interpolation **

Non-adaptive algorithms have been known as the traditional methods in image interpolation, and include nearest neighbor, linear, cubic, bilinear, bicubic and others. These algorithms depend on calculating the colour value from the adjacent pixels, and interpolating the weighted average colour pixels to magnify the image[5]. In other words, non-adaptive algorithms are only used to treat all the pixels equally. The non-adaptive technique are as following:-

**2.1.1Linear Interpolation: It is a simple form of interpolation. This algorithm considers the closest two neighbor **
pixels at both side of the interpolating pixel. The colour value of the interpolated pixel is the average of these two
values.

**2.1.2Cubic Interpolation: It is more accurate than linear interpolation, because the consideration of the neighbor **
known pixels is greater than linear interpolation. It considers the closest four neighbor pixels at both side of the
interpolating pixel. The colour value of the interpolated pixel is the average of these four values [5-9].

**2.1.3Bilinear Interpolation: It is an extension of linear interpolation. It considers the closest neighborhood of 2*2 **
known pixel values that surrounding the interpolating pixel.

### Er.Sudhir Sharma, IJRIT

87 Fig-2 Bilinear Interpolation.The value of interpolated pixel is the average of the four known pixels. The result of this algorithm is much better than linear interpolation.

**2.1.4 Bicubic Interpolation: Bicubic interpolation is an extension of cubic interpolation. It considers the closest **
4x4 neighborhood (for a total of 16 pixels) of known pixels [5-9]. The closer pixels are given a higher weighting in
the calculation. The image produced by Bicubic is better than bilinear interpolation, because of the larger amount of
neighbor pixels considered. The figure shows as below:-

Fig-3 Bicubic Interpolation.

**2.1.5 Nearest Neighbor Interpolation: It is the most fundamental interpolation. It only considers one pixel (the **
closest one to the interpolating pixel). This has the effect of simply making each pixel bigger.

**2.2 Adaptive Interpolation:- **

Adaptive algorithms more rely on detecting the presence of edge. Generally, the aim of these algorithms is to minimize the unsightly interpolation artifacts such as edge, blur and jaggies. Nowadays, there have many adaptive algorithms been introduced, most of the algorithm are more preferable to be applied in mathematic way to detect the edges. These kinds of algorithm rely on the mathematic equations to calculate the position of pixels along edges rather than studying the pattern of the edges in the original image to retrieve the edge information. The adaptive techniques are as following:-

**2.2.1 Least Directional Differences based on Linear and Cubic: This method is more sophisticated and produces **
smoother edges than both bilinear and bicubic interpolations. The difference between this algorithm and traditional
linear and cubic interpolation is the consideration of known pixels [10-12].

Fig-4 Least Directional Difference Method.

### Er.Sudhir Sharma, IJRIT

88**2.2.2 Progressive Refinement Approach: It is a method that doubles the size of images, and the sharpness of edges**is preserved without introducing distinct artifacts in the magnified images. The algorithm consists of two steps; first step is the generation of an initial magnified image, and the second step is the progressively refining the magnified image to produce a high quality image. The main contribution of this approach is the image refinement [10-12]. This refinement is performed progressively by transforming the value of each pixel, producing a new magnified image that owns the colors closer to its input image.

**2.2.3Image Sharpening Algorithm: It is an algorithm that is inspired by the usage of gradient profile prior, but the **
author utilizes it to sharpen an image directly instead of using it as regularization for reconstruction. The main idea
of this algorithm is to find out the gradient profile of each pixel and change the sharpness of the gradient profile
accordingly. Bicubic interpolation to compute the intensity of those pixels which locations are not aligned with the
image grid. The extracted gradient profile is segmented to obtain a single mode that contains the center pixels [12-
13]. The center pixels are the boundaries between two colour domains, which are also the edges of the image.

**2.2.4 Edge-Directed Interpolation: It is an adaptive method that also focuses on edges to improve image **
magnification. The algorithm consists of two phases: rendering and correction. The rendering phase is edge-directed
which is based on bilinear interpolation modified to prevent the interpolation across edges [11-13]. The correction
phase is to modify the mesh values on which the rendering is based to account for the disparity between the true low
resolutions data, and that predicted by a sensor model operating on the high resolution output of the rendering phase.

The overall process is repeated iteratively until the satisfactory image is obtained.

**3. Implementation Method **

In zooming operation, the re-sampling and interpolation operations are used. The principle of image zooming is shown below in Figure 5. In Figure 5(a), the image A is magnified to 2 times to image B. The black circles are the unknown pixel values. The image A (2x2) is magnified to B (4x4); however in reality, the original image (nxn) is magnified to (2n-1)x(2n-1). This process of zooming includes three stages as reported in Figure 5 and explained below.

**A. Stage I: The first step is to expand the original (nxn) image to (2n-1)x(2n-1). This can be done by **
extending the initial image borders. For example in Figure 5(b), the X(5x5) is original image and Y(9x9) is the
magnified image. X(i,j) is the pixel in original image; where i is the ith row and j is the jth column. In the same way
in Y(m,n), m is the mth row and n is the nth column. The black dots represent the pixels in the original image (X)
and these pixels can be mapped in image (Y) as X(i,j)=Y(2i-1,2j-1). The white dots represent the unknown pixels.

When an image is magnified 2 times, total number of pixels will be four times; thus, the new picture have now 1 known and 3 unknown pixels.

**B. Stage II: In this stage, the centre pixel X (Figure 5(c)) has to be finding out. The centre pixel X is **
deduced with the help of proposed algorithm in the next section. This procedure is repeated for the complete image
until all the unknown pixels of the boundaries and the centre pixels are found.

**C. Stage III: This stage started from the beginning of the image and will find the left over pixels as shown **
in Figure 5(d). In Figure 5(d), A & B are the pixels from original image, and X1 and X2 are the pixels derived from
stage I. All the pixels (A, B, X1, X2) are considered as in Stage I and computed in the same way. Finally, the value
of M is computed and put in the place of centre pixel.

### Er.Sudhir Sharma, IJRIT

89 Fig-5(a) Magnifying an ImageFig-5(b) Stage I

Fig-5(c) Stage II

Fig-5(d) Stage III

**4. Conclusion **

From past six decades, many researches and advancements have taken place in this area, many systems have been developed but still after years of research and development on the quality to improve images which are effected by some artifacts during zooming of image. Different interpolation techniques are used to improve the quality of images. In this review paper some brief overview of the interpolation techniques are studied. With the use of basic interpolation techniques an adaptive method is proposed to improve the quality of image and results are seen by calculating MSR and PSNR.

**5. References **

[1] M.Zamani,“An Applied Two-Dimensional B-spline Model for Interpolation of Data”,International Journal of Advanced Research in Engineering and Technology (IJARET), Volume 3, Number 2, July-December (2012).

[2] M. Nagaraju Naik and P. Rajesh Kumar,“Spectral Approach To Image Projection With Cubic B- Spline Interpolation”, (IJECET),Volume3, Issue3, pp. 153 - 161, 2012.

[3] Zamani, M. “A simple 2-D interpolation model for analysis of nonlinear data”, Natural Science,Vol. 2, No. 6, P.641-646, June 2010.

### Er.Sudhir Sharma, IJRIT

90 [4] Zhang Min, Wang Jiechao, LI Zhiwei ,’An Adaptive Image Zooming Method with Edge Enhancementʹ, 3rd International Conference on Advanced Computer Theory and Engineering (ICACTE), pg -608-611, 2010.[5] X. Zhang & X. Wu, “Image Interpolation by Adaptive 2-D Autoregressive Modeling and Soft-Decision Estimation,” IEEE Transactions on ImageProcessing, Vol. 17, No. 6, pp. 887-896, 2008.

[6] R. Lukac, K. Martin & K. N. Plataniotis “Digital Camera Zooming Based on Unified CFA Image Processing Steps,” Research at the CAIP Center, pp. 15-24, 2004.

[7] Karl S. Ni, S. Kumar, N. Vasconcelos, Truong Q. Nguyen, “Single Image Super Resolution Based on Support Vector Regression”, 2001.

[8] Y. Wu, K. Vij, “An Image Illumination Correction Algorithm based on Tone Mapping,” IEEE Trans. on Image Processing, pp. 245-248, 2010.

[9] C. Vazquez, E. Dubois & J. Konrad, “Reconstruction of No uniformly Sampled Images in Spline Spaces,”

IEEE Transactions on Image Processing, Vol. 14, No. 6, pp. 713-725, 2005.

[10] Y. F. Sun, J. W. Liu & L. Wang, “A Multi-scale TVQI-based Illumination Normalization Model,” Proceedings of the International Multi Conference of Engineers and Computer Scientists, Vol. I, IMECS 2011.

[11] Y. H. I. Gu, & T. Tjahjadi, “Efficient Planar Object Tracking and Parameter Estimation Using Compactly Represented Cubic B-spline Curves,”IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans,Vol. 29, No. 4, pp. 358-367, 1999.

[12] M. F. Hossain, O. Chae, “Global and Local Transformation Function Mixture for Image Contrast Enhancement,” IEEE International Conference on Consumer Electronics (ICCE), pp. 185–186, 2009.

[13] M. A. A. Wadud, M. H. Kabir, M. A. A. Dewan & O. Chae, “A Dynamic Histogram Equalization for Image Contrast Enhancement,” IEEE Transactions on Consumer Electronics, Vol. 53, No. 2, pp. 593–600, 2007.