基于IDCT/DCT域的全相位数字滤波和图像内插
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摘要
本文将IDCT/DCT和全相位思想结合起来研究数字滤波和图像内插问题。
本文把基于DFT、WHT、IDCT和DCT域的全相位滤波器(APDF)设计公式纳入到一个统一的框架中，在原有基于正交变换域APDF的设计公式的基础上得出了逆过渡矩阵G~(-1)的表达式。基于正交域的APDF设计方法可以统一起来，加窗方法和不加窗方法也能实现统一。这是通过过渡矩阵G和逆过渡矩阵G~(-1)实现的。在全相位滤波器设计理论的基础上，本文证明了基于IDCT/DCT域的APDF有零相位特性和子带互补特性。
为了使基于IDCT/DCT域的APDF应用于实际，本文提出了三种快速实现结构，它们都能实时地调节滤波器频率响应，可用于列率域自适应滤波等需要实时改变滤波器响应的场合。在列率域直接实现结构中充分利用了全相位数据空间的时移特点和IDCT/DCT的递推性质，推导了由输入数据计算IDCT/DCT的递推算法，适用于连续数据流的场合。在时域等价结构和列率域等价结构中利用了过渡矩阵变换的表达式，列率域等价结构尤其适用于窄带滤波的场合。在一般情况下，时域等价结构的计算复杂度是最低的。
为了改善直接基于IDCT/DCT域设计的APDF幅频特性，本文推导了IDCT/DCT域加窗APDF的设计公式。实验表明，无论在IDCT还是在DCT域，加凯塞窗都能使滤波器达到相对而言最好的幅频特性。在相同的频率采样点数的情况下，IDCT/DCT域加凯赛窗(β=3.2)APDF的幅频特性较频率采样法设计的FIR低通滤波器其通带、阻带更平坦，过渡带更窄。
本文分析了基于IDCT域设计的菱形半带滤波器在应用于图像采样栅格转换时的内插特性。将一维傅立叶特性分析方法扩展到二维可以得出：在有预滤波的情况下，阶数较大的内插滤波器有较好的内插结果，滤波器很好地保持了边缘，但对图像的纹理部分内插效果稍差一些。
作为DCT和全相位思想的结合，本文构造了基于DCT域的可分离内插函数，这是一族由余弦函数组成的内插核函数。为了实时应用，本文提出了查找表实现方法。实验表明，采用不同阶数的全相位DCT域可分离内插函数，都能够达到优于6×6的立方内插的图像质量。
作为对内插器性能评价准则的研究，本文定义了连续消失矩曲线，它比传统的基于核函数的傅立叶特性评价方法能更好地解释实验数据。
IDCT/DCT and All Phase philosophy are integrated in research on digital filtering and image interpolation.
Firstly, the design formula of all phase digital filter (APDF) based on DFT, WHT, IDCT and DCT are unified in one frame, and based on the existed theory of All Phase digital filtering, the formula of inverse transition matrix G~(-1) is derived. Then through the transition matrix G and the inverse transition matrix G~(-1), the design method of APDF based on DFT, WHT, IDCT and DCT is unified, also the design method of APDF and windowed APDF is unified.
Secondly, to find the suited application, the characteristics of APDF based on IDCT/DCT are discussed. Zero-phase and sub-band filtering with complementary are two characters of APDF.
Thirdly, to put the APDF based on IDCT/DCT into practice, three fast realizations are put forward. The amplitude frequency response characteristics of the filter all can be tuned in real time, and are especially suited in adaptive filtering where the response of filter is changing in real time. The recursive formula of the DCT/IDCT of all phase input data space is given in direct sequency field realization, and transition matrix is also given in time and sequency field equivalence realization. Direct sequency field recursive realization is especially suited in continuous data processing, and sequency field equivalence realization is especially suited in narrow band filtering. In most condition, the calculation complication of time field equivalence realization is the least.
Fourthly, to improve the amplitude frequency response characteristics of APDF based on IDCT/DCT, the design formula of windowed APDF based on IDCT/DCT is derived. Experiments show that when using the window of Kaiser better amplitude frequency response characteristics of APDF are achieved. On the condition of using the same number of frequency samples, the amplitude frequency response characteristics of Kaiser windowed APDF based on IDCT/DCT is better than of FIR digital filter designed by frequency sampling method.
Fifthly, the interpolation characters of diamond half band filter which designed based on IDCT applied in sampling structure conversion are discussed. Fourier analysis method is extended to 2-D. It shows that higher order can achieve better interpolation results, and the interpolator keeps the edges well but keeps the texture worse.
本文把基于DFT、WHT、IDCT和DCT域的全相位滤波器(APDF)设计公式纳入到一个统一的框架中，在原有基于正交变换域APDF的设计公式的基础上得出了逆过渡矩阵G~(-1)的表达式。基于正交域的APDF设计方法可以统一起来，加窗方法和不加窗方法也能实现统一。这是通过过渡矩阵G和逆过渡矩阵G~(-1)实现的。在全相位滤波器设计理论的基础上，本文证明了基于IDCT/DCT域的APDF有零相位特性和子带互补特性。
为了使基于IDCT/DCT域的APDF应用于实际，本文提出了三种快速实现结构，它们都能实时地调节滤波器频率响应，可用于列率域自适应滤波等需要实时改变滤波器响应的场合。在列率域直接实现结构中充分利用了全相位数据空间的时移特点和IDCT/DCT的递推性质，推导了由输入数据计算IDCT/DCT的递推算法，适用于连续数据流的场合。在时域等价结构和列率域等价结构中利用了过渡矩阵变换的表达式，列率域等价结构尤其适用于窄带滤波的场合。在一般情况下，时域等价结构的计算复杂度是最低的。
为了改善直接基于IDCT/DCT域设计的APDF幅频特性，本文推导了IDCT/DCT域加窗APDF的设计公式。实验表明，无论在IDCT还是在DCT域，加凯塞窗都能使滤波器达到相对而言最好的幅频特性。在相同的频率采样点数的情况下，IDCT/DCT域加凯赛窗(β=3.2)APDF的幅频特性较频率采样法设计的FIR低通滤波器其通带、阻带更平坦，过渡带更窄。
本文分析了基于IDCT域设计的菱形半带滤波器在应用于图像采样栅格转换时的内插特性。将一维傅立叶特性分析方法扩展到二维可以得出：在有预滤波的情况下，阶数较大的内插滤波器有较好的内插结果，滤波器很好地保持了边缘，但对图像的纹理部分内插效果稍差一些。
作为DCT和全相位思想的结合，本文构造了基于DCT域的可分离内插函数，这是一族由余弦函数组成的内插核函数。为了实时应用，本文提出了查找表实现方法。实验表明，采用不同阶数的全相位DCT域可分离内插函数，都能够达到优于6×6的立方内插的图像质量。
作为对内插器性能评价准则的研究，本文定义了连续消失矩曲线，它比传统的基于核函数的傅立叶特性评价方法能更好地解释实验数据。
IDCT/DCT and All Phase philosophy are integrated in research on digital filtering and image interpolation.
Firstly, the design formula of all phase digital filter (APDF) based on DFT, WHT, IDCT and DCT are unified in one frame, and based on the existed theory of All Phase digital filtering, the formula of inverse transition matrix G~(-1) is derived. Then through the transition matrix G and the inverse transition matrix G~(-1), the design method of APDF based on DFT, WHT, IDCT and DCT is unified, also the design method of APDF and windowed APDF is unified.
Secondly, to find the suited application, the characteristics of APDF based on IDCT/DCT are discussed. Zero-phase and sub-band filtering with complementary are two characters of APDF.
Thirdly, to put the APDF based on IDCT/DCT into practice, three fast realizations are put forward. The amplitude frequency response characteristics of the filter all can be tuned in real time, and are especially suited in adaptive filtering where the response of filter is changing in real time. The recursive formula of the DCT/IDCT of all phase input data space is given in direct sequency field realization, and transition matrix is also given in time and sequency field equivalence realization. Direct sequency field recursive realization is especially suited in continuous data processing, and sequency field equivalence realization is especially suited in narrow band filtering. In most condition, the calculation complication of time field equivalence realization is the least.
Fourthly, to improve the amplitude frequency response characteristics of APDF based on IDCT/DCT, the design formula of windowed APDF based on IDCT/DCT is derived. Experiments show that when using the window of Kaiser better amplitude frequency response characteristics of APDF are achieved. On the condition of using the same number of frequency samples, the amplitude frequency response characteristics of Kaiser windowed APDF based on IDCT/DCT is better than of FIR digital filter designed by frequency sampling method.
Fifthly, the interpolation characters of diamond half band filter which designed based on IDCT applied in sampling structure conversion are discussed. Fourier analysis method is extended to 2-D. It shows that higher order can achieve better interpolation results, and the interpolator keeps the edges well but keeps the texture worse.
引文
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