小波构造理论及其在高光谱遥感图像去噪与压缩中的应用
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摘要
小波以其良好的时频局部化性质和多分辨特性在数学理论研究和图像处理领域得到了广泛的应用。高光谱遥感图像是最近几十年发展起来的新兴遥感技术,它能更为全面、更为详细地描述地物特征。由于高光谱图像在成像及传播中,受到很多复杂因素的影响,会引入大量的噪声,从而对图像分析带来不利影响,因此亟需研究高光谱遥感图像的去噪算法。高光谱图像较高的分辨率是以产生大量数据为代价的,它为高光谱数据的传输和储存提出了巨大挑战。因此高效压缩算法是高光谱图像处理领域研究热点之一。本文在研究小波构造理论和高维小波边界延拓算法的同时,将小波分析与高光谱遥感图像的去噪和压缩技术结合在一起展开研究。
本文研究工作和创新点主要体现在以下几个方面:
1.给定理想滤波器的形状,设计出一种逼近该理想滤波器的小波滤波器组的构造方法。首先在最小二乘意义下,对分解滤波器多项式与理想滤波器的误差最小化,求出待定参数。然后利用Bezout定理求解满足完全重构条件的综合滤波器的多项式通解。最后利用最小二乘方法求解出综合滤波器的待定参数。该方法将滤波器长度和消失矩阶数作为参数,可以根据需要自由设计。证明了用该方法构造的滤波器随着滤波器长度的增加,与理想滤波器的误差以指数速度衰减,估计出误差衰减速度的上下界。
2.基于梅花采样的不可分小波,设计出一种能够实现边界处理的完全重构的边界延拓方法。证明了这种延拓方法能实现边界的完全重构,且不需要附加信息。该延拓方法对图像压缩具有重要的意义。
3.提出一种基于软阈值函数的小波去噪方法。该方法采用迭代算法来估计软阈值的大小。证明了该算法的收敛性并估计收敛速度,分析了该算法的运算量。数值实验表明该算法估计阈值的准确性和MAD方法相当,同时计算量远远小于MAD方法。
4.针对高光谱遥感图像噪声级别低,各波段噪声的方差随各波段信号的幅值而变化的特点,提出一种导数域内空间谱间联合的高光谱图像去噪方法。首先采用光谱导数技术消除图像背景对噪声的影响,让细微的噪声更容易被提取。然后在导数域内采用基于小波变换的BayesShrink去噪方法进行空间维去噪,对光谱维采用Savitzky-Golay滤波进行平滑。最后对导数域去噪平滑处理后的图像进行光谱积分,并进行积分修正,消除光谱积分中引入的积累误差。实验结果表明该方法对高光谱图像去噪非常有效。
5.深入研究高光谱图像压缩算法,提出两种无损压缩算法及一种有损压缩算法。
(i)考虑到高光谱图像相邻的几个波段间都存在着强烈的谱间相关性,提出一种基于多波段谱间预测的无损压缩算法。通过推导多波段预测系数的求解过程,观察到求解当前波段预测系数的线性方程矩阵中会用到大量前一波段预测系数的线性方程的矩阵元素,由此设计出一种快速算法,大大减少了预测系数求解的时间复杂度。
(ii)提出一种基于递归双向预测的高光谱图像无损压缩算法。针对不同的波段,谱间相关性系数差别很大的特点,采用不同的模式进行编码。谱间相关性大的波段用递归双向预测,能够取得好的压缩效果。对谱间相关性小的波段,不再进行谱间预测,用bzip2算法直接进行编码。不同的情况分别处理,能够在节省运算时间的同时达到满意的压缩效果。
(iii)提出一种波段预测去除谱间冗余和码流预分配的高光谱图像有损压缩算法。首先用DPCM预测求出各波段的预测残差图像的标准差,然后根据标准差的大小分配对该波段进行编码所需的码流长度。最后基于均方差最小的线性预测器对图像各波段进行预测,根据事先分配的码流长度对各波段预测残差图像进行SPIHT编码。设计的分配码流长度的算法能够根据各波段信息量大小,以及相邻波段的相关性来分配码流长度,达到理想的压缩效果。
Wavelets achieve good property in both time and frequency resolution, and analyse signals in multiresolution. Therefore, they are widely applied in mathematical theory and image processing. Hyperspectral remote sensing imagery is a newly developed remote sensing technology in recent decades. It can describe ground objects more comprehensively and explicitly. In the process of imaging and transit, hyperspectral remote sensing images are interfered by many complicated factors, which will introduce a lot of noise and affect image analysis. Hence, it is emergent to research denoising algorithms for hyperspectral remote sensing images. The high resolution of hyperspectral remote sensing image achieves by the cost of massive data, which is a great challenge for the transit and storage of hyperspectral data. Hence, compression algorithm with high performance is one of the focuses in the field of hyperspectral image processing. This thesis studies wavelet construction theory and boundary extension algorithm for multi-dimesional wavelet. Wavelet analysis combined with denosing and compression technology for hyperspectral remote sensing image is studied as well.
The main work and innovation are embodied as follows.
1. Given the shape of an ideal filter, a wavelet filter construction method to approximate the ideal filter is proposed. Firstly, free parameters are determined in the meaning of least square, by minimizing the error between analysis filter and ideal filter. Then a perfect reconstructed general solution for sythesis filter is calculated by Bezout theorem. Finally, the free parameters for sythesis filter can be determined by least square design. In this method, the length of filter and the degree of vanishing moment are taken as free parameters, which can be designed at ease. We prove that the error between the constructed filter and the ideal filter is convergent at an exponential rate as the degree of filter polynomial goes large, and estimate the upper and lower boundaries of the rate.
2. Based on a family of quincunx wavelets, a boundary extension method is proposed which can achieve perfect reconstruction at boundaries. That the extension method can achieve perfect reconstruction at boundaries nonexpansionally is proved. This extension method has significant meanings for image compression.
3. A wavelet denoising method based on soft threshold function is proposed. The value of soft threshold is estimated by an iterative method. The convergence of this method is proved. The convergence rate is estimated and the computational time is analized. Numerical experiment shows that the algorithm is competitive to the MAD method, and the computational time is much less than MAD method.
4. To tackle the denosing problem that the noise level of hyperspectral remote sensing image is relatively low and the noise variance is varying with the spectral bands, a three-dimensional hybrid denoising algorithm in derivative domain is proposed. At first, hyperspectral image is transformed into spectral derivative domain where the subtle noise level is elevated, and the affect of background is eliminated effectively. And then in derivative domain, a wavelet based threshold denoising method BayesShrink algorithm is performed in the two-dimensional spacial domain, and the spectrum is smoothed by Savitzky-Golay filter. At last, the data smoothed in derivative domain is integrated along the spectral axis and corrected for the accumulated errors brought by spectral integration. Experimental results show that the proposed algorithm can reduce the noise efficiently for hyperspectral remote sensing image.
5. Hyperspectral image compression algorithms are studied. Two lossless compression algorithms and a lossy compression algorithm are proposed.
(i) Considering the significant spectral correlation in adjacent bands of hyperspectral images, a hyperspectral image lossless compression algorithm based on multi-band prediction is proposed. In the process to calculate the multi-band prediction coefficients, the matrix of linear system to solve the prediction coefficients in current band has many components which are the same as the previous band. Therefore, a fast algorithm can be designed which saves computational time for solving prediction coefficients.
(ii) A hyperspectral image lossless compression algorithm based on optimal recursive bidirection prediction is proposed. Since the spectral correlation factors in different bands are quite different, two different coding modes are chosen. For the bands whose spectral correlation is significant, using recursive bidirection prediction can achieve excellent compression performance. For the bands whose spectral correlation is not significant, bzip2 algorithm is used instead of spectral prediction. By coding with different modes, satisfactory compression performance is achieved and computational time is saved.
(iii) A hyperspectral image compression algorithm based on prediction between bands to remove spectral redundancy and rate pre-allocation is proposed. At first, the standard error of error image of each band is computed by DPCM, and then the rate for SPIHT coding of each band is allocated by the value of the standard error. At last, each band is linear predicted based on minimal square error, and coded via SPIHT algorithm according to the pre-allocated rate. The rate pre-allocation algorithm is reasonable for it makes use of both the information of each band and the correlation between neighboring bands, and achieves ideal compression efficiency.
本文研究工作和创新点主要体现在以下几个方面:
1.给定理想滤波器的形状,设计出一种逼近该理想滤波器的小波滤波器组的构造方法。首先在最小二乘意义下,对分解滤波器多项式与理想滤波器的误差最小化,求出待定参数。然后利用Bezout定理求解满足完全重构条件的综合滤波器的多项式通解。最后利用最小二乘方法求解出综合滤波器的待定参数。该方法将滤波器长度和消失矩阶数作为参数,可以根据需要自由设计。证明了用该方法构造的滤波器随着滤波器长度的增加,与理想滤波器的误差以指数速度衰减,估计出误差衰减速度的上下界。
2.基于梅花采样的不可分小波,设计出一种能够实现边界处理的完全重构的边界延拓方法。证明了这种延拓方法能实现边界的完全重构,且不需要附加信息。该延拓方法对图像压缩具有重要的意义。
3.提出一种基于软阈值函数的小波去噪方法。该方法采用迭代算法来估计软阈值的大小。证明了该算法的收敛性并估计收敛速度,分析了该算法的运算量。数值实验表明该算法估计阈值的准确性和MAD方法相当,同时计算量远远小于MAD方法。
4.针对高光谱遥感图像噪声级别低,各波段噪声的方差随各波段信号的幅值而变化的特点,提出一种导数域内空间谱间联合的高光谱图像去噪方法。首先采用光谱导数技术消除图像背景对噪声的影响,让细微的噪声更容易被提取。然后在导数域内采用基于小波变换的BayesShrink去噪方法进行空间维去噪,对光谱维采用Savitzky-Golay滤波进行平滑。最后对导数域去噪平滑处理后的图像进行光谱积分,并进行积分修正,消除光谱积分中引入的积累误差。实验结果表明该方法对高光谱图像去噪非常有效。
5.深入研究高光谱图像压缩算法,提出两种无损压缩算法及一种有损压缩算法。
(i)考虑到高光谱图像相邻的几个波段间都存在着强烈的谱间相关性,提出一种基于多波段谱间预测的无损压缩算法。通过推导多波段预测系数的求解过程,观察到求解当前波段预测系数的线性方程矩阵中会用到大量前一波段预测系数的线性方程的矩阵元素,由此设计出一种快速算法,大大减少了预测系数求解的时间复杂度。
(ii)提出一种基于递归双向预测的高光谱图像无损压缩算法。针对不同的波段,谱间相关性系数差别很大的特点,采用不同的模式进行编码。谱间相关性大的波段用递归双向预测,能够取得好的压缩效果。对谱间相关性小的波段,不再进行谱间预测,用bzip2算法直接进行编码。不同的情况分别处理,能够在节省运算时间的同时达到满意的压缩效果。
(iii)提出一种波段预测去除谱间冗余和码流预分配的高光谱图像有损压缩算法。首先用DPCM预测求出各波段的预测残差图像的标准差,然后根据标准差的大小分配对该波段进行编码所需的码流长度。最后基于均方差最小的线性预测器对图像各波段进行预测,根据事先分配的码流长度对各波段预测残差图像进行SPIHT编码。设计的分配码流长度的算法能够根据各波段信息量大小,以及相邻波段的相关性来分配码流长度,达到理想的压缩效果。
Wavelets achieve good property in both time and frequency resolution, and analyse signals in multiresolution. Therefore, they are widely applied in mathematical theory and image processing. Hyperspectral remote sensing imagery is a newly developed remote sensing technology in recent decades. It can describe ground objects more comprehensively and explicitly. In the process of imaging and transit, hyperspectral remote sensing images are interfered by many complicated factors, which will introduce a lot of noise and affect image analysis. Hence, it is emergent to research denoising algorithms for hyperspectral remote sensing images. The high resolution of hyperspectral remote sensing image achieves by the cost of massive data, which is a great challenge for the transit and storage of hyperspectral data. Hence, compression algorithm with high performance is one of the focuses in the field of hyperspectral image processing. This thesis studies wavelet construction theory and boundary extension algorithm for multi-dimesional wavelet. Wavelet analysis combined with denosing and compression technology for hyperspectral remote sensing image is studied as well.
The main work and innovation are embodied as follows.
1. Given the shape of an ideal filter, a wavelet filter construction method to approximate the ideal filter is proposed. Firstly, free parameters are determined in the meaning of least square, by minimizing the error between analysis filter and ideal filter. Then a perfect reconstructed general solution for sythesis filter is calculated by Bezout theorem. Finally, the free parameters for sythesis filter can be determined by least square design. In this method, the length of filter and the degree of vanishing moment are taken as free parameters, which can be designed at ease. We prove that the error between the constructed filter and the ideal filter is convergent at an exponential rate as the degree of filter polynomial goes large, and estimate the upper and lower boundaries of the rate.
2. Based on a family of quincunx wavelets, a boundary extension method is proposed which can achieve perfect reconstruction at boundaries. That the extension method can achieve perfect reconstruction at boundaries nonexpansionally is proved. This extension method has significant meanings for image compression.
3. A wavelet denoising method based on soft threshold function is proposed. The value of soft threshold is estimated by an iterative method. The convergence of this method is proved. The convergence rate is estimated and the computational time is analized. Numerical experiment shows that the algorithm is competitive to the MAD method, and the computational time is much less than MAD method.
4. To tackle the denosing problem that the noise level of hyperspectral remote sensing image is relatively low and the noise variance is varying with the spectral bands, a three-dimensional hybrid denoising algorithm in derivative domain is proposed. At first, hyperspectral image is transformed into spectral derivative domain where the subtle noise level is elevated, and the affect of background is eliminated effectively. And then in derivative domain, a wavelet based threshold denoising method BayesShrink algorithm is performed in the two-dimensional spacial domain, and the spectrum is smoothed by Savitzky-Golay filter. At last, the data smoothed in derivative domain is integrated along the spectral axis and corrected for the accumulated errors brought by spectral integration. Experimental results show that the proposed algorithm can reduce the noise efficiently for hyperspectral remote sensing image.
5. Hyperspectral image compression algorithms are studied. Two lossless compression algorithms and a lossy compression algorithm are proposed.
(i) Considering the significant spectral correlation in adjacent bands of hyperspectral images, a hyperspectral image lossless compression algorithm based on multi-band prediction is proposed. In the process to calculate the multi-band prediction coefficients, the matrix of linear system to solve the prediction coefficients in current band has many components which are the same as the previous band. Therefore, a fast algorithm can be designed which saves computational time for solving prediction coefficients.
(ii) A hyperspectral image lossless compression algorithm based on optimal recursive bidirection prediction is proposed. Since the spectral correlation factors in different bands are quite different, two different coding modes are chosen. For the bands whose spectral correlation is significant, using recursive bidirection prediction can achieve excellent compression performance. For the bands whose spectral correlation is not significant, bzip2 algorithm is used instead of spectral prediction. By coding with different modes, satisfactory compression performance is achieved and computational time is saved.
(iii) A hyperspectral image compression algorithm based on prediction between bands to remove spectral redundancy and rate pre-allocation is proposed. At first, the standard error of error image of each band is computed by DPCM, and then the rate for SPIHT coding of each band is allocated by the value of the standard error. At last, each band is linear predicted based on minimal square error, and coded via SPIHT algorithm according to the pre-allocated rate. The rate pre-allocation algorithm is reasonable for it makes use of both the information of each band and the correlation between neighboring bands, and achieves ideal compression efficiency.
引文
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