full details about Spatial and Intensity Resolution , optical and digital zoom concepts and the common three interpolation algorithms for implementing zoom in image processing

1. 1
Image Interpolation Techniques
with
Optical and Digital Zoom Concepts
Musaab Mohammed Jasim.
Yildiz Technical University
Faculty of Electrical & Electronic
Computer Engineering Department
Seminar course
mus.albiaty85@gmail.com
ID :14501063 .
Abstract — Actually the digital images correspond to some
physical response in real 2-D space i.e. the optical intensity
received at the image plane of a camera or the ultrasound
intensity at a transceiver. So, it can be considered as a discrete
representation of data possessing both spatial (layout) and
intensity (color) information .by processing these data we can get
different results that mean obtain various image statuses and one
of these processes is Zooming [1].
Zooming includes : enlargement and shrinking processes where
these processes require two steps: the creation of new pixel
locations, and the assignment of gray (or color) levels to those
new locations [2]. in this papers we will address the interpolation
techniques to achieve the zooming by using three types of
algorithms then we try to discover how it be executed ,its effects
on the image and its results.
Keywords-Image processing, Digital Zooming , Interpolation
techniques .
I. INTRODUCTION
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, as well as others. Of the many image
interpolation techniques available, "nearest neighbor, bilinear and
bicubic" are the common non-adaptive methods [3]. In this papers we
will discuss all details relative with these techniques to achieve the
"image enlargement and shrinking" .
But to understand these topics , at the first we need understand
and address with the other topics such as Image Resolution (Spatial
and intensity resolution) , The differences between optical and digital
zooming , Linear interpolation .
II. IMAGE RESOLUTION CONCEPTS
Resolution is the capability of the sensor to observe or measure
the smallest object clearly with distinct boundaries while the Pixel is
actually a unit of the digital image . so the Resolution is the
measurement unit of the clarity in digital image field .There are
different types of the Resolution that is used to determine the digital
image clarity and one of them is "Pixel Resolution" which depends
upon the size of the pixel. So when the pixels are counted this will be
referred to as pixel resolution. the convention is to describe the pixel
resolution with the set of two positive integer numbers, where the
first number is the number of pixel columns (width) and the second is
the number of pixel rows (height), Below in Figure(1) is an
illustration of how the same image might appear at different pixel
resolutions, if the pixels were poorly rendered as sharp squares
(normally, a smooth image reconstruction from pixels would be
preferred, but for illustration of pixels, the sharp squares make the
point better).
Figure (1)
So the Pixel Resolution determine the number of pixels in the
image , but unfortunately, the count of pixels isn't a real measure of
the image clarity as most people think , there another concepts
determine the clarity such Spatial and Intensity Resolution for gray
images and Spectral Resolution for colored image [4].
Where the Spatial resolution can be defined as the number of
independent pixel values per unit length (depended on the Sampling
process of sensor). So it depend in the number of pixels and the area
in which these pixels are resolved (spreaded). Since the spatial
resolution refers to clarity , so for different devices , different
measure has been made to measure it.
‐ Dots per inch is usually used in monitors.
‐ Lines per inch or LPI is usually used in laser printers.
‐ Pixels per inch is measure for different devices such as tablets ,
Mobile phones e.t.c.
While Intensity Resolution is the bit depth or the colors range of
pixels in image , it is determined based on the number of bits which
has been assigned for each pixel .(depended on the quantization
process). Since , differences of spectrum or wavelength is needed to

2. 2
reproduce color. So the Spectral Resolution is used here where is
defined as "the ability to resolve spectral features and bands into
their separate components". The spectral resolution required by the
analyst or researcher depends upon the application involved.
III.OPTICAL ZOOM VS. DIGITAL ZOOM
Optical zoom means moving the zoom lens so that it increases the
magnification of light before it even reaches the digital sensor
(Before sampling process and quantization process). So, the optical
zoomed image occupies the full area of the sensor and is simply a
magnified the real life image [5].
A digital zoom is not really zoom, in the strictest definition of the
term. It degrades quality by simply interpolating the image after it
has been acquired at the sensor(After sampling and quantization
process) . because it is implemented after determination the Spatial
Resolution , so it mean resampling the image by creating new pixel
locations and assigning gray-level values (or color) to these locations.
There are two types of digital zoom. The most common form of
digital zoom involves image "interpolation" which is introduced and
discussed and is discussed here . The second type is called "smart
zoom".
Based on the above definitions we can understand that with
digital zoom the detail is clearly far less than with optical zoom. as
shown in the image below.
The original image
10X Optical Zoom 10X Digital Zoom
IV.IMAGE INTERPOLATIONS
Image interpolation occurs when we resize or distort our image
from one pixel grid to another. Image resizing is necessary when we
need to increase or decrease the total number of pixels, whereas
remapping can occur for distortion or rotating an image as shown in
Figure(2). Zooming refers to increase the quantity of pixels, so that
when we zoom an image we will able to see more detail. (Pixilation
Process).
Figure (2)
Common interpolation algorithms can be grouped into two
categories: adaptive and non-adaptive. Adaptive methods change
depending on what they are interpolating, whereas non-adaptive
methods treat all pixels equally. Non-adaptive algorithms include:
nearest neighbor, bilinear, bicubic, spline, sinc, and others, while
Adaptive algorithms include many proprietary algorithms in licensed
software such as: Qimage, PhotoZoom Pro and Genuine Fractals
[5][6]. In this papers we will address with three of Non-adaptive
algorithms for resizing purposes ,and try to understand its algorithm
and the difference between these method results , then we try to
implement these algorithms in MATLAB to see its result on the gray
and color images. These three methods are :
1‐ Nearest Neighbor interpolation.
2‐ Bilinear interpolation .
3‐ Bicubic interpolation .
Figure(3) show the meant of the resizing concept , and the roles
of interpolation to achieve it.
Figure(3) Resizing by using Interpolation
Nearest Neighbor Interpolation
Nearest Neighbor Interpolation, the simplest method, determines
the grey level value(or color) from the closest pixel to the specified
input coordinates, and assigns that value to the output coordinates. It
should be noted that this method does not really interpolate values, it
just copies existing values. Since it does not alter values, it is
preferred if subtle variations in the grey level values need to be
retained [7].
For one-dimension Nearest Neighbor Interpolation, the number of
grid points needed to evaluate the interpolation function is two. For
two-dimension Nearest Neighbor Interpolation, the number of grid
points needed to evaluate the interpolation function is four.

3. 3
 Nearest Neighbor algorithm
Figure(4) and (5) below show the two states of using Nearest
Neighbor Interpolation methods [8].
Figure(4) Enlargement
Figure(5) Reducing
Bilinear Interpolation
Bilinear Interpolation determines the grey level value (or color)
from the weighted average of the four closest pixels to the specified
input coordinates, and assigns that value to the output coordinates.
Bilinear interpolation considers the closest 2x2 neighborhood of
known pixel values surrounding the unknown pixel's computed
location. It then takes a weighted average of these 4 pixels to arrive at
its final, interpolated value. The weight on each of the 4 pixel values
is based on the computed pixel's distance (in 2D plane) from each of
the known points (linear interpolations).
But what is the (Linear Interpolation) and the (Weighted
Average) which are used to implementation the Bilinear method ??
 Linear interpolation between two known points
If the two known points are given by the coordinates (x0,y0) and
(x1,y1) , the linear interpolation is the straight line between these
points. For a value x in the interval (x0,x1) , the value y along the
straight line is given from the equation (1) . [9]
(1)
Which can be derived geometrically from the Figure(6) below
where it is a special case of polynomial interpolation with n = 1.
Figure(6)
Solving this equation for y, which is the unknown value at x,
gives the equation (2)
∗ 2
Which is the formula for linear interpolation in the interval (x0,x1)
Outside this interval, the formula is identical to linear extrapolation.
 weighted average
This formula can also be understood as a weighted average. The
weights are inversely related to the distance from the end points to
the unknown point; the closer point has more influence than the
farther point. Thus, the weights are and which are
normalized distances between the unknown point and each of the end
points. Because these sum to 1, as shown in the equation (3)
1 1 1 (3)

4. 4
Which yields the formula for linear interpolation given above.
The Figure (7) with the color arrows show the idea of weighted
average .
Figure (7)
 Calculating a weighted Average for Image
Based on the concepts are mentioned above , the weighted
average of the attributes (color, alpha, etc.) of the four surrounding
pixels (as is shown in the Figure(8) is computed and applied to the
screen pixel. This process is repeated for each pixel forming the
object being textured.
Figure (8)
To understand how we can calculate the weighted average of a
digital image , we will discuss a simple example of a gray digital
image with a bit depth (gray level) for four pixels from it, then we
will calculate the gray level of the interpolated pixel based on the
rules which have already been discussed [10].
 Example
Based on the Figure(9) we observe that the intensity value at the
pixel computed to be at row 20.2, column 14.5 can be calculated by
first linearly interpolating between the values at column 14 and 15 on
each rows 20 and 21, giving
, .
15 14.5
15 14
∗ 91
14.5 14
15 14
∗ 210 150.5
, .
15 14.5
15 14
∗ 162
14.5 14
15 14
∗ 95 128.5
and then interpolating linearly between these values, giving
. , .
21 20.2
21 20
∗ 150.5
20.2 20
21 20
∗ 128.5 146.1
This algorithm reduces some of the visual distortion caused by
resizing an image to a non-integral zoom factor, as opposed to nearest
neighbor interpolation, which will make some pixels appear larger
than others in the resized image.
Figure (9)
And for the color digital image we will use the same rules but it
is implemented for each channel (Red, Green, Blue) of the image.
 Bilinear algorithm
After we comprehended the weighted average rule and how it is
calculated , we will be able to understand the Bilinear Algorithm
which be built based on the notations of the Figure(10) .
Figure (10)
The details of the algorithm implementation and variables
definitions and the relation between them as shown in the following
paragraph .[8]

5. 5
BiCubic Interpolation
BiCubic Interpolation method determines the gray level value (or
color) from the weighted average of the 16 closest pixels to the
specified input coordinates as shown in Figure(11) , 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.
Figure (11)
So, Bicubic goes one step beyond bilinear by considering the
closest 4x4 neighborhood of known pixels — for a total of 16 pixels.
Since these are at various distances from the unknown pixel, closer
pixels are given a higher weighting in the calculation.
V. CONCLUSION
The digital Image is a visual representation in form of a function
f(x,y) where f is related to the brightness (or color) at point (x,y) , the
value of each point is acquired based on the light that reflect from the
objects on the sensors on the Digital camera , the electrical responses
of these sensor will be aggregated then after the sampling and
quantization processes the image pixels will been created. A lot of
applications are appeared in this field , where these applications
contain many and many of the processing methods and algorithms to
apply it on the digital image and one of these processes is the
"interpolation process" .
Interpolation process include image magnification or reduction,
subpixel image registration, to correct spatial distortions, and image
decompression, as well as others. There are a number of techniques
that can be used to enlarge an image. The three most common were
presented here. The Nearest-Neighbor and Bilinear interpolation
methods are very practical and easy to apply, due to their simplicity.
However, their accuracy is limited while Bicubic gave the best results
in terms of image quality, but took the greatest amount of processing
time.
VI.REFERENCES
1. Chris Solomon ,Toby Breckon, "Fundamentals of Digital Image
Processing",Chichester, West Sussex, PO19 8SQ, UK , 2011,
sec on 1.1 , p 20‐25.
2. Rafael C. Gonzalez , Richard E. Woods , "Digital Image
Processing", Second Edition , Prentice Hall, Upper Saddle River,
New Jersey 07458 , sec on 2.4.5 , p 75‐81.
3. S.J.Lebonah ,D.Minola Davids, PhD. , "A Novel Coding using
Downsampling Technique in Video Intraframe",International
Journal of Computer Applications® (IJCA).
4. Richard Alan Peters II,EECECS 253 Image Processing course
,Vanderbilt university ,school of engineering , Fall , 2011.
5. Bax Smith , EN9821 Design Assignment , www.engr.mun.ca
/~baxter /Publications /ImageZooming.pdf.
6. A Learning Community for Photographers, DIGITAL IMAGE
INTERPOLATION, www.cambridgeincolour.com/tutorials/image‐
interpolation.htm
7. University of Tartu , Digital Image processing ,Resizing Image ,
www.sisu.ut.ee/imageprocessing/book/3.
8. Image resolution , From Wikipedia , en.wikipedia.org /wiki
/Image_resolution .
9. Linear interpolation, From Wikipedia, en.wikipedia.org/wiki
/Linear_interpolation .
10. Bilinear interpolation, From Wikipedia , en.wikipedia.org/wiki
/Bilinear_interpolation.