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标题中的“dual-tree-complex-wt.rar_dual image_dual tree wt”表明这是一个关于双树复小波变换（Dual-Tree Complex Wavelet Transform, DTCWT）在图像处理，特别是图像分辨率增强方面的应用。描述中提到的“image resolution enhancement using ddddcwt”进一步确认了这个压缩包的内容是关于如何利用双树复小波来提升图像的分辨率。 **双树复小波变换（DTCWT）** 双树复小波变换是一种高级的小波分析方法，由Richard G. Baraniuk和Michael W. Berry等人在1998年提出。与传统的单树小波变换相比，DTCWT具有更高的方向选择性和更好的频域局部化特性。它通过构建两棵正交小波树来处理实数信号，这两棵树分别处理信号的实部和虚部，从而提供更精确的多分辨率分析。 **图像分辨率增强** 图像分辨率增强是指通过算法技术提高图像的细节清晰度和解析力，使得低分辨率图像能够接近或达到高分辨率图像的质量。DTCWT在这一领域有独特的优势，因为其良好的频率局部化和方向选择性使得在分解图像时能更好地保留高频信息，这对于恢复图像细节至关重要。 **DTCWT在图像处理的应用** 1. **图像去噪**：DTCWT能够有效地区分信号和噪声，通过对不同尺度和方向的系数进行阈值处理，可以有效地去除图像噪声而不影响重要信息。 2. **图像压缩**：由于DTCWT的多分辨率特性，它可以对图像进行高效编码，降低存储空间需求，同时保持图像质量。 3. **图像复原与增强**：对于失真或模糊的图像，DTCWT可以捕捉到更多的细节信息，通过重构过程可以恢复图像的原始质量，提高分辨率。 4. **边缘检测与特征提取**：DTCWT的方向敏感性有助于准确地检测图像边缘，为图像分割、目标识别等应用提供支持。 5. **医学成像**：在医疗图像分析中，DTCWT可以增强图像的对比度，改善诊断效果。压缩包内的“dual tree complex wt.pdf”可能是详细讲解DTCWT理论及其在图像分辨率增强中应用的论文或教程，包含了理论介绍、算法实现、实例分析等内容，对于理解DTCWT的工作原理和应用非常有帮助。 DTCWT是小波分析领域的一个强大工具，尤其在处理图像数据时能提供优异的表现。通过学习和掌握这一技术，我们可以更有效地进行图像处理任务，提升图像质量和应用效果。

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e-ISSN 2277-2685, p-ISSN 2320-976

IJESR/August 2014/ Vol-4/Issue-8/595-601

Aniveni Mahesh et. al./ International Journal of Engineering & Science Research

*Corresponding Author www.ijesr.org 595

DOUBLE DENSITY DUAL TREE COMPLEX WAVELET TRANSFORM BASED

SATELLITE IMAGE RESOLUTION ENHANCEMENT

Aniveni Mahesh*

, Ch. Sreedhar

, Ahmed Zeeshan

M.tech student, Global Institute of Engineering & Technology, Moinabad (A.P), India.

HOD, Project guide, Global Institute of Engineering & Technology, Moinabad (A.P), India.

Project coordinator, Assoc. Prof, Global Institute of Engineering & Technology, Moinabad (A.P), India.

ABSTRACT

Image Resolution enhancement (RE) schemes which are based on waveletshave an advantage overconventional methods

which suffer from the losing of high frequency contents which causes blurring. The discrete wavelet transform-based

(DWT) RE scheme generates artifacts due to a DWT shift-variant property. A wavelet-domain approach based on double

density dual-tree complex wavelet transform (DDDT-CWT) is proposed for RE of the satellite images and compared

with dual-tree complex wavelet transform(DT-CWT). A satellite image is decomposed by DDDT-CWT to obtain high-

frequency subbands. The high frequency subbands and the low-resolution (LR) input image are interpolated and the high

frequency subbands are passed through a low pass filter. The filtered high-frequency subbands and the LR input image

are combined using inverse DT-CWT to obtain a resolution-enhanced image. Objective and subjective analyses reveal

superiority of the proposed technique over the conventional and state-of-the-art RE techniques.

INTRODUCTION

Resolution of satellite images has always been a limiting factor for utilizing remote sensing data such as satellite imaging

etc.Spatial and spectral resolutions of satellite unprocessed images are related with each other that is a high spatial

resolution is associated with a low spectral resolution and vice versa[1]. So spatial and spectral resolution

enhancement(RE) are desirable.Another technique which is widely used for RE is interpolation [2,3].The interpolation

techniques which are commonly used are based on nearest neighbours which include nearest neighbour, bilinear, bi-

cubic, and Lanczos. These can also perform reduction of aliasing, sharpness and ringing [4].

Vector-valued image regularization based methods with partial differential equations(VVIR-PDE) [5] and inpainting and

zooming which uses sparse representations [6] are now the state of the art in the field.RE schemes which are not based on

wavelets suffer from the drawback of losing the high frequency contents which results in blurring effect.

In the research area of RE in the wavelet domain many algorithms such as discrete wavelet transform (DWT) [7],

stationary wavelet transform (SWT) [8], and dual-tree complex wavelet transform (DT-CWT) [9] have been proposed

[7–11]. Using DT-CWT an RE scheme was proposed in[9] and bicubic interpolations, and results were compared with

the conventional schemes i.e., nearest neighbour,bilinear, and bicubic interpolations and wavelet zero padding. Based on

DWT and bicubic interpolation a scheme was proposed in [7] and results were compared with conventional schemes and

the state of art schemes.Dwt is shift variant, which causes artefacts in the RE image,and has a lack of directionality,

whereas DT-CWT is almost shift and rotation invariant [13].

In this Paper a Double Density DT-CWT based lowpass filtering based RE technique is proposed using DT-CWT,

Lanczos interpolation, and lowpass filtering.Double Density DT-CWT is nearly shift invariant and directional selective,it

preserves the usual properties of perfect reconstruction with well-balanced frequency responses [13, 14]. As compared

with traditional DWT, DDDT-CWT provides promising results after the modification of the wavelet coefficients and

provides less artefacts.For spatial RE of optical images the results are compared with best performing techniques [5,7–9].

2. WAVELET TRANSFORM DOMAIN

Using Fourier Transform (FT) only the global frequency content of a signal is retrieved, the time information is lost. By

using wavelet analysis a multi-resolution analysis is possible. The frequency and time content of a signal is retrieved by

Aniveni Mahesh et. al./ International Journal of Engineering & Science Research

Wavelet Transform (WT). The types of wavelet transform are i) Continuous Wavelet Transform (CoWT) ii) Discrete

Wavelet Transform (DWT) iii) Complex Wavelet Transform (CWT). By using Fourier transform and Short Time Fourier

Transform a multi resolution analysis is not possible so there is a restriction to use these tools in image processing

systems ,particularly in image denoising applications. The multi-resolution analysis is possible by using wavelet analysis.

A Continuous Wavelet Transform (CoWT) is calculated analogous to the Fourier transform (FT), by performing

convolution between the signal and analysis function. The Discrete Wavelet Transform uses filter banks to perform the

wavelet analysis.

2.1 Complex wavelet transform

A newly introduced technique of DWT, Orthogonal wavelet decompositions, based on separable, multirate filtering

systems have been widely used in image and signal processing, largely for data compression. Dual-Tree complex wavelet

transform [12] is introduced by Kingsbury which is a very elegant computational structure which displays near shift

invariant properties.

In image processing complex wavelets have not been used due to difficulty in designing complex filters which satisfy

perfect reconstruction property.To overcome this Kingsbury proposed Dual-Tree implementation of the CWT (DT CWT)

[14], in which two trees of real filters are used to generate the real and imaginary parts of the wavelet coefficients

separately.The DWT suffers from following two problems 1] Lack of shift invariance - this results from the down

sampling operation at each level. When the input signal is shifted slightly, the amplitude of the wavelet coefficients

varies so much [2]. Lack of directional selectivity - as the DWT filters are real and separable the DWT cannot distinguish

between the opposing diagonal directions.

First problem of DWT can be avoided if the output of the filter from each level are not down sampled but this may

increase the computational cost significantly and resulting undecimated wavelet transform still cannot distinguish

between opposing diagonals since the transform is still separable. To distinguish opposing diagonals with separable

filters the filter frequency responses are required to be asymmetric for positive and negative frequencies. The way to

achieve this is to use complex wavelet filters which can be made to suppress negative frequency components.Compared

to separable DWT complex DWT has improved shift-invariance and directional selectivity[13,14].

3. BASIC CONCEPTS OF THE SYSTEM

In wavelet transform applications filter bank plays an important role.It consists of two banks namely, analysis filter bank

and synthesis filter bank. The one dimensional filter bank constructed with analysis and synthesis filter bank which is

shown in Fig. 1.

Fig 1: One dimensional Filter bank

The analysis filter bank decomposes the input signal x(n) into two sub band signals, c(n) and d(n). The signal c(n)

represents the low frequency part of x(n), while the signal d(n) represents the high frequency part of x(n). It uses filter

banks to perform the wavelet analysis. The DWT decomposes the signal into wavelet coefficients from which the

original signal can be reconstructed again. Using wavelet coefficient the signal can be represented in various frequency

bands. Using the DWT attractive properties over linear filtering the coefficients can be processed in several ways.

3.1 Basic differences between the two DWT extensions

The main differences between the dual tree Dwt and Double –Density are as follows. 1] The Dual-Tree and Double-

Density DWTs are implemented with totally different filter bank structures. 2] The Dual-Tree DWT can be interpreted as

Aniveni Mahesh et. al./ International Journal of Engineering & Science Research

a complex-valued wavelet transform which is useful for signal modelling and denoising (the Double-Density DWT

cannot be interpreted as such). 3] For the Dual-Tree DWT there are fewer degrees of freedom for design, while for the

Double-Density DWT there are more degrees of freedom for design [4]. The Dual-Tree DWT can be used to implement

two-dimensional transforms with directional wavelets, which is highly desirable for image processing [15].

4. PROPOSED SYSTEMS

Complex wavelet transforms (CWT) concept is introduced so that we can achieve Dual-Tree Complex DWT system. We

can achieve Double-Density Dual-Tree Complex DWT system by combining the Double-Density DWT and Dual-Tree

Complex DWT. Complex wavelet transforms (CWT) use complex-valued filtering (analytic filter) that decomposes the

real/complex signals into real and imaginary parts in transform domain. Amplitude and phase information are calculated

by using the real and imaginary coefficients.

4.1 Dual-Tree Complex DWT (DTCWT)

Kingsbury’s complex Dual-Tree DWT is based on (approximate) Hilbert pairs of wavelets [15]. Kingsbury found that

the Dual-Tree DWT is nearly shift-invariant when the lowpass filters of one DWT interpolate midway between the

lowpass filters of the second DWT [16]. Using two critically-sampled DWTs in parallel the Dual-Tree Complex DWT

can be implemented as shown in the Fig. 3.For N-point signal this transform gives 2N DWT coefficients.So this

transform is known as 2-times expansive.The design of the filter is done in such a way that the subband signals of the

upper DWT can be interpreted as the real part of a CWT and subbands signals of the lower DWT can be interpreted as

the imaginary part. For specially designed sets of filters, the wavelet associated with the upper DWT can be an

approximate Hilbert transform of the wavelet associated with the lower DWT. In this manner, the designed DTCWT is

nearly shift-invariant than the critically-sampled DWT [10,11].

Fig 2: Design implementation of Dual-Tree Complex DWT

Wavelets are given by DTCWT in six distinct directions.There are two wavelets in each direction. In each direction, one

of the two wavelets can be interpreted as the real part and the other wavelet can be interpreted as the imaginary part of a

complex-valued two dimensional (2D) wavelet. The DTCWT is implemented as four critically sampled separable 2D

DWTs operating in parallel. Different filter sets are used along the rows and columns [10,11]. Fig 4 shows the flowchart

of Dual-Tree Complex DWT. This gives the steps of implementation of DTCWT.

4.2 Double-Density Dual-Tree Complex DWT (DDDTCWT)

In two dimensions in terms of enhancement Double-Density DWT performs superior than the standard DWT. But here

all of the wavelets are not directional.Though more wavelets are utilized by Double-Density DWT , some lack a

dominant spatial orientation, which prevents them from being able to isolate those directions [16]. The solution for this

problem is provided by the Double-Density Dual-Tree Complex DWT, which combines the characteristics of the

Double-Density DWT and the Dual-Tree DWT. The DDDTCWT is based on two scaling functions and four distinct

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