This document discusses the discrete wavelet transform (DWT) and its implementation in MATLAB. It begins with an introduction to DWTs, explaining that they decompose signals into different frequency bands using low-pass and high-pass filters. It then describes how 2D DWTs are implemented by applying 1D transforms separately to rows and columns, producing subbands that emphasize different types of edges. The document provides steps for performing DWTs in MATLAB, including single-level and multi-level decomposition and reconstruction. It concludes by showing decomposed images and discussing how DWTs separate approximation and detail information.

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DISCRETE WAVELET TRANSFORM USING MATLAB
Darshana Mistry1, Asim Banerjee2
1
Asst. Professor, Computer Engineering, Indus Institute of Technology, Ahmedabad,
Gujarat, India
2
Professor, Communication Technology, DAIICT, Gandhinagar, India
ABSTRACT:
In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is
any wavelet transform for which the wavelets are discretely sampled. In this paper, there are
given fundamental of DWT and implementation in MATLAB. Image is filtered by low pass
(for smooth variation between gray level pixels) and high pass filter (for high variation
between gray level pixels). Image is decomposed into multilevel which include
approximation details (LL subband), horizontal detail (HL subband), vertical (LH subband)
and diagonal details (HH subband).
Keywords: Discrete Wavelet Transform (DWT), MATLAB, high pass filter, low pass filter.
I. INTRODUCTION
The transform of a signal is just another form of representing the signal. It does not
change the information content present in the signal.
Fourier Transmission (FT) representations do not include local information about the
original signals. Although the WFTs can provide localization information, they do not
provide flexible division of the time-frequency plane that can track slow changing
phenomena while providing more details for higher Frequencies. The wavelet representation
was introduced to correct the drawback of the former two methods using a multi-resolution
scheme.
The Wavelet Transform provides a time-frequency representation of the signal. A
wavelet series is representation of a square-integral (real or complex value) function by a
certain orthonormal (two vectors in an inner product space are orthonormal if they are
orthogonal (when two things can very independently or they are perpendicular) and all of unit
length).
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There are two classifications of wavelets [6]: (a) orthogonal (the low pass and high pass
filters have same length) and (b) biorthogonal (the low pass and high pass filters do not have
same length). Based on the application, either of them can be used.
The Wavelet transforms contribute to the desired sampling by filtering the signal with translations
and dilations of a basic function called “mother wavelet”. The mother wavelet can be used to
form orthonormal bases of wavelets, which is particularly useful for data reconstruction [2].
Daniel [1] represent the wavelet transform expands a signal, or function, into the wavelet
domain. As with any transform, like the Fourier or Gabor Transforms, the goal of expanding a
signal is to obtain information that is not apparent, or cannot be deduced, from the signal in its
original domain (usually space, time or distance).
A wavelet, in the sense of the Discrete Wavelet Transform (or DWT), is an orthogonal
function which can be applied to a finite group of data. Functionally, it is very much like the
Discrete Fourier Transform, in that the transforming function is orthogonal, a signal passed twice
through the transformation is unchanged, and the input signal is assumed to be a set of discrete-
time samples. Both transforms are convolutions.
Shripathi [6] introduce as The Discrete Wavelet Transform (DWT), which is based on sub-band
coding is found to yield a fast computation of Wavelet Transform. It is easy to implement and
reduces the computation time and resources required.
In DWT, the most prominent information in the signal appears in high amplitudes and the
less prominent information appears in very low amplitudes. Data compression can be achieved by
discarding these low amplitudes. The wavelet transforms enables high compression ratios with
good quality of reconstruction. Recently, the Wavelet Transforms have been chosen for the JPEG
2000 compression standard.
The discrete wavelet transform uses low-pass and high-pass filters, h(n) and g(n), to
expand a digital signal. They are referred to as analysis filters. The dilation performed for each
scale is now achieved by a decimator. The coefficients ܿ௞ and ݀௞ are produced by convolving the
digital signal, with each filter, and then decimating the output. The ܿ௞ coefficients are produced
by the low-pass filter, h(n), and called coarse coefficients. The ݀௞ coefficients are produced by
the high-pass filter and called detail coefficients. Coarse coefficients provide information about
low frequencies, and detail coefficients provide information about high frequencies. Coarse and
detail coefficients are produced at multiple scales by iterating the process on the coarse
coefficients of each scale. The entire process is computed using a tree-structured filter bank, as
seen in Fig. 1.
Fig. 1. Analysis filter bank. The high and low pass filters divide the signal into a series of coarse
and detail coefficients.
After analyzing, or processing, the signal in the wavelet domain it is often necessary to
return the signal back to its original domain. This is achieved using synthesis filters and
expanders. The wavelet coefficients are applied to a synthesis filter bank to restore the original
signal, as seen in Fig.2.
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Fig. 2. Synthesis Filter Bank. The high and low pass filters combine the coefficients into the
original signal.
The discrete wavelet transform has a huge number of applications in science, engineering,
and mathematics and computer science. The wavelet domain representation of an image, or any
signal, is useful for many applications, such as compression, noise reduction, image registration,
watermarking, super-resolution etc.
II. TWO DIMENSIONAL DISCRETE WAVELET TRANSFORM
The wavelet transform can be expressed by the following equation (1):
∞
Fሺa, bሻ ൌ ‫ ∞ି׬‬fሺxሻφ‫ כ‬ሺୟ,ୠሻ ሺxሻdx ……….(1)
where the * is the complex conjugate symbol and function ψ is some function.
The discrete wavelet transform (DWT) is an implementation of the wavelet transform
using a discrete set of the wavelet scales and translations obeying some defined rules.
The two dimensional discrete wavelet transform is essentially a one dimensional analysis
of a two dimensional signal. It only operates on one dimension at a time, by analyzing the rows
and columns of an image in a separable fashion. The first step applies the analysis filters to the
rows of an image. This produces two new images, where one image is set or coarse row
coefficients, and the other a set of detail row coefficients. Next analysis filters are applied to the
columns of each new image, to produce four different images called sub bands. Rows and
columns analyzed with a high pass filter are designated with an H. Likewise, rows and columns
analyzed with a low pass filter are designated with an L. For example, if a subband image was
produced using a high pass filter on the rows and a low pass filter on the columns, it is called the
HL subband. Figure 3 shows this process in its entirety.
Fig. 3. Two Dimensional Discrete Wavelet Transform. The high and low pass filters operate
separable on the rows and columns to create four different subbands. An 8x8 image is used for
example purposes only.
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Each subband provides different information about the image. The LL subband is a
coarse approximation of the image and removes all high frequency information. The LH
subband removes high frequency information along the rows and emphasizes high frequency
information along the columns. The result is an image in which vertical edges are
emphasized. The HL subband emphasizes horizontal edges, and the HH subband emphasizes
diagonal edges. To compute the DWT of the image at the next scale the process is applied
again to the LL subband (see fig. 4).
Each level of the wavelet decomposition, four new images are created from the
original N x N-pixel image. The size of these new images is reduced to ¼ of the original size,
i.e., the new size is N/2 x N/2. The new images are named according to the filter (low-pass or
high-pass) which is applied to the original image in horizontal and vertical directions. For
example, the LH image is a result of applying the low-pass filter in horizontal direction and
high-pass filter in vertical direction [2]. Thus, the four images produced from each
decomposition level are LL, LH, HL, and HH. The LL image is considered a reduced version
of the original as it retains most details. The LH image contains horizontal edge features,
while the HL contains vertical edge features. The HH contains high frequency information
only and is typically noisy and is, therefore, not useful for the registration. In wavelet
decomposition, only the LL image is used to produce the next level of decomposition (see
fig.5).
Fig.4. DWT image is based on approximate image detail (LL), horizontal details(HL),
vertical details(LH) and diagonal details(HH).
Fig. 5. Decomposed of and image level vise.
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When we apply high frequency (use high pass filter) on an image, there are
high variations in the gray level between the two adjacent pixels. So edges are
occurred in image. When we apply low frequency (use low pass filter) on an image,
there are smooth variations between the adjacent pixels. So edges are not generated or
very few edges are generated. All information of image is remaining same as real
image information (it display as approximation image).
From fig. 6 and fig. 7 represent approximate details, horizontal details, vertical details
and diagonal details of and different images. Approximate details are same as original
image details. Horizontal details construct only horizontal information (edges).
Vertical details construct only vertical information (edges). Diagonal details construct
very few information of input image. So approximation image is applied into next
level for deformation.
(a)
(b)
Fig. 6. (a) Original image, (b) DWT image based on approximate image detail (LL),
horizontal details(HL), vertical details(LH) and diagonal details(HH) in one level.
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(a)
(b)
Fig. 7. (a) original image, (b) DWT image based on approximate image detail (LL),
horizontal details(HL), vertical details(LH) and diagonal details(HH) in one level.
III. STEPS AND IMPLEMENTATION IN MATLAB
Basic steps are used to apply DWT in MATLAB (Matlab 2007b or later version).
1. Read an image.
2. Convert an input image into a gray scale image.
3. Perform a single-level wavelet decomposition(we get for information approximation,
horizontal, vertical and diagonal details of an image)
4. Construct and display approximations and details from the coefficients.
5. To display the results of the level 1 decomposition.
6. Regenerate an image by zero-level inverse Wavelet Transform.
7. Perform multilevel wavelet decomposition.
8. Extract approximation and detail coefficients. To extract the level 2 approximation
coefficients from step 5.
9. Reconstruct the Level 2 approximation and the Level 1 and 2 details.
10. Display the results of a multilevel decomposition.
11. Reconstruct the original image from the multilevel decomposition.
Fig. 8 (a) and 8(b) displayed images are decomposed into level 2 using DWT
algorithm.
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IV. CONCLUSION
Using DWT, images are decomposing into four parts: Approximate image, Horizontal
details, Vertical details and diagonal details. When we apply high frequency on an image,
there are high variations in the gray level between the two adjacent pixels. So edges are
occurred in image. When we apply low frequency on an image, there are smooth variations
between the adjacent pixels. So edges are not generated or very few edges are generated. All
information of image is remaining same as real image information (it display as
approximation image).
(a)
(b)
Fig 8.(a) and (b) different images are decomposed up to second level
ACKNOWLEDGEMENT
I am very thankful to Dr Asim Banerjee who inspired and helped me to do this work.
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