n k =0 l=0

j= ’1

where

0 ¤ h, i ¤ n ’ 1

and

Low-pass and high-pass filtering shown in the previous section is achieved by zeroing out

certain terms of the transform. Eliminating a particular term removes the corresponding

frequency level throughout the image. Therefore, filtering can be done by performing a

Fourier transform, zeroing out the desired frequency level and calculating the inverse

transform to return the altered image back into the spatial domain.

One weakness of the Fourier transform is that it decomposes an image into its frequency

components but its representation provides no information about where in the image these

frequencies occur. Unfortunately, it is impossible to obtain the spatial frequency in an

image at a particular location due to Heisenberg™s uncertainty principle. Wavelet

transforms attempt to overcome this limitation by expressing an image as a linear

combination of different scalings and translations of some defined wavelet function

known as the mother wavelet. This wavelet function must have periodicity and its value

must decrease as some function of the distance from its centre. Note that the transform

does not specify exactly what the mother wavelet function should be. In practice, the

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function is customized to suit the particular application. The following equation in Figure

9 states the one-dimensional wavelet transform. In two-dimensional functions, such as

images, “the two-dimensional version of a wavelet transform can be expressed in terms of

a number of one-dimensional transforms.” (Parker, 1997, p. 269). Wavelet transforms are

often used for applications such as image filtering and compression.

Figure 9: 2-D Wavelet Transform

γ f (t ) ( s,„ ) = « f (t )ψ s ,„ (t )dt

where the mother wavelet is defined as ψ(t) and the series of wavelet is defined as ψ(t)s,„

is defined by the equation noted in Figure 10.

Figure 10: Mother Wavelet Equation

« t ’„

1

ψ ψ¬

(t ) = ·

s

s ,„

s

Hough's transform represents each pixel in the standard spatial representation as a line in

what is known as the Hough space. The equation for a line in the standard domain is y =

mx + b where m and b are constants. In the Hough space, this equation is rearranged to

be b = -xm + y where x and y are constant terms. Lines that intersect in the Hough space

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represent pixels in the standard domain that reside on the same line. This provides a

novel way of line detection in images.

This section provided the reader with some background knowledge about image

processing. This information will be useful when reading rest of this paper.

OCR

Computer applications can write human readable information to paper using a printing

device such as a plotter or printer. When a computer is instructed to print a character to

an output device, it simply looks up the pre-defined bitmap set used to generate the

desired character. A defined set of bitmaps (or an equivalent representation) is stored

within the computer system for each character in a language™s alphabet. The character

lookup is done using the machine™s representation of the character. The actual bitmap

retrieved within the selected set is based on the visual attributes desired. Once the actual

bitmap is found, appropriate instructions are sent to a printing device to render the map on

paper. Assuming that the output is printed in a colour with sufficient contrast to the colour

of the paper, it can be easily interpreted by humans. Translation in the opposite direction

has proven to be a much more difficult problem to solve. Collecting character

information from paper is done using a scanning device. This device produces a digital

snapshot of the sheet of paper. In order for the computer to extract the required character

information, it must simulate the process of a human reading and interpreting the

snapshot. The field of optical character recognition (OCR) involves the study of

techniques used by a computer to extract character-based information from such

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snapshots. This discipline draws from the fields of artificial intelligence, image

processing and other areas of computer science in order to define algorithms for

accomplishing its goals. The rest of this section will present a general process for