VTK：频域处理——低通滤波（理想+巴特沃斯）

1.低通滤波器

低通滤波是将频域图像中的高频部分滤除而通过低频部分。图像的边缘和噪声对应于频域图像中的高频部分，而低频滤波的作用即是减弱这部分作用，从而达到图像平滑去噪的目的。

2.理想低通滤波器

最简单的低通滤波器是理想的滤波器，基本思想是给定一个频率阈值，将高于该与值的所有部分设置为0，而低于该频率的值保持不变。
理想是指滤波器不能用电子元器件来实现，但是可以通过计算来模拟。
在VTK中定义了理想低通滤波器，
步骤：
①、首先读入一副jpg图片，通过vtkImageFFT将图像转换到频域空间。vtkImageIdealLowPass对频域图像做理想低通滤波
②、需要通过SetXCutOff（）和SetYCutOff（）。
③、执行完成后，需要通过VTKImageRFFT处理后的频域图像张翰至空域图像。
###注意1，转换后的图像的每个像素都是复数，需要vtkIamgeExtractComponents将图像的第一个分量提出出来，否在不能正常显示。
###注意2.由于傅里叶变换输入的数据类型大都是double类型，为了方便显示，还需要将其转换成Unsinged char类型，这里vtkIamgeCast负责类型转换。

#include <vtkAutoInit.h>
VTK_MODULE_INIT(vtkRenderingOpenGL2);
VTK_MODULE_INIT(vtkInteractionStyle);
#include <vtkSmartPointer.h>
#include <vtkJPEGReader.h>
#include <vtkImageFFT.h>
#include <vtkImageIdealLowPass.h>
#include <vtkImageData.h>
#include <vtkImageRFFT.h>
#include <vtkImageCast.h>
#include <vtkImageExtractComponents.h>
#include <vtkImageActor.h>
#include <vtkRenderer.h>
#include <vtkRenderWindow.h>
#include <vtkRenderWindowInteractor.h>int main()
{vtkSmartPointer<vtkJPEGReader> reader =vtkSmartPointer<vtkJPEGReader>::New();reader->SetFileName("data/Lena.jpg");reader->Update();vtkSmartPointer<vtkImageFFT> fftFilter =vtkSmartPointer<vtkImageFFT>::New();fftFilter->SetInputConnection(reader->GetOutputPort());fftFilter->Update();vtkSmartPointer<vtkImageIdealLowPass> lowPassFilter =vtkSmartPointer<vtkImageIdealLowPass>::New();lowPassFilter->SetInputConnection(fftFilter->GetOutputPort());lowPassFilter->SetXCutOff(0.05);lowPassFilter->SetYCutOff(0.05);lowPassFilter->Update();vtkSmartPointer<vtkImageRFFT> rfftFilter =vtkSmartPointer<vtkImageRFFT>::New();rfftFilter->SetInputConnection(lowPassFilter->GetOutputPort());rfftFilter->Update();vtkSmartPointer<vtkImageExtractComponents> ifftExtractReal =vtkSmartPointer<vtkImageExtractComponents>::New();ifftExtractReal->SetInputConnection(rfftFilter->GetOutputPort());ifftExtractReal->SetComponents(0);vtkSmartPointer<vtkImageCast> castFilter =vtkSmartPointer<vtkImageCast>::New();castFilter->SetInputConnection(ifftExtractReal->GetOutputPort());castFilter->SetOutputScalarTypeToUnsignedChar();castFilter->Update();vtkSmartPointer<vtkImageActor> originalActor =vtkSmartPointer<vtkImageActor>::New();originalActor->SetInputData(reader->GetOutput());vtkSmartPointer<vtkImageActor> erodedActor =vtkSmartPointer<vtkImageActor>::New();erodedActor->SetInputData(castFilter->GetOutput());double leftViewport[4] = { 0.0, 0.0, 0.5, 1.0 };double rightViewport[4] = { 0.5, 0.0, 1.0, 1.0 };vtkSmartPointer<vtkRenderer> leftRenderer =vtkSmartPointer<vtkRenderer>::New();leftRenderer->AddActor(originalActor);leftRenderer->ResetCamera();leftRenderer->SetViewport(leftViewport);leftRenderer->SetBackground(1.0, 0, 0);vtkSmartPointer<vtkRenderer> rightRenderer =vtkSmartPointer<vtkRenderer>::New();rightRenderer->AddActor(erodedActor);rightRenderer->SetViewport(rightViewport);rightRenderer->ResetCamera();rightRenderer->SetBackground(1.0, 1.0, 1.0);vtkSmartPointer<vtkRenderWindow> renderWindow =vtkSmartPointer<vtkRenderWindow>::New();renderWindow->AddRenderer(rightRenderer);renderWindow->AddRenderer(leftRenderer);renderWindow->SetSize(640, 320);renderWindow->SetWindowName("Frequency_IdealLowPassFilter");vtkSmartPointer<vtkRenderWindowInteractor> interactor =vtkSmartPointer<vtkRenderWindowInteractor>::New();interactor->SetRenderWindow(renderWindow);renderWindow->Render();interactor->Start();return 0;
}

3.巴特沃斯低通滤波器

在实际经常使用巴特沃斯滤波器。巴特沃斯滤波器对应的转移函数（可以看做是一个系数矩阵）是：
其中D(u,v)表示频域点(u,v)到频域图像原点的距离，称为截止频率，当D(u,v) = 时，H(u,v)=0.5，即对应的频域能量将为原来的一半。因为巴特沃斯低通滤波器在高低频间的过渡平滑，因此不会出现明显的振铃效应。VTK中实现巴特沃斯低通滤波器的类是vtkImageButterworthLowPass.
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代码：

#include <vtkAutoInit.h>
VTK_MODULE_INIT(vtkRenderingOpenGL2);
VTK_MODULE_INIT(vtkInteractionStyle);#include <vtkSmartPointer.h>
#include <vtkJPEGReader.h>
#include <vtkImageFFT.h>
#include <vtkImageButterworthLowPass.h>
#include <vtkImageRFFT.h>
#include <vtkImageExtractComponents.h>
#include <vtkImageCast.h>
#include <vtkImageActor.h>
#include <vtkRenderer.h>
#include <vtkRenderWindow.h>
#include <vtkRenderWindowInteractor.h>
#include <vtkInteractorStyleImage.h>int main()
{vtkSmartPointer<vtkJPEGReader> reader =vtkSmartPointer<vtkJPEGReader>::New();reader->SetFileName("data/Lena.jpg");reader->Update();vtkSmartPointer<vtkImageFFT> fftFilter =vtkSmartPointer<vtkImageFFT>::New();fftFilter->SetInputConnection(reader->GetOutputPort());fftFilter->Update();vtkSmartPointer<vtkImageButterworthLowPass> lowPassFilter =vtkSmartPointer<vtkImageButterworthLowPass>::New();lowPassFilter->SetInputConnection(fftFilter->GetOutputPort());lowPassFilter->SetXCutOff(0.05);lowPassFilter->SetYCutOff(0.05);lowPassFilter->Update();vtkSmartPointer<vtkImageRFFT> rfftFilter =vtkSmartPointer<vtkImageRFFT>::New();rfftFilter->SetInputConnection(lowPassFilter->GetOutputPort());rfftFilter->Update();vtkSmartPointer<vtkImageExtractComponents> ifftExtractReal =vtkSmartPointer<vtkImageExtractComponents>::New();ifftExtractReal->SetInputConnection(rfftFilter->GetOutputPort());ifftExtractReal->SetComponents(0);vtkSmartPointer<vtkImageCast> castFilter =vtkSmartPointer<vtkImageCast>::New();castFilter->SetInputConnection(ifftExtractReal->GetOutputPort());castFilter->SetOutputScalarTypeToUnsignedChar();castFilter->Update();vtkSmartPointer<vtkImageActor> originalActor =vtkSmartPointer<vtkImageActor>::New();originalActor->SetInputData(reader->GetOutput());vtkSmartPointer<vtkImageActor> erodedActor =vtkSmartPointer<vtkImageActor>::New();erodedActor->SetInputData(castFilter->GetOutput());///double leftViewport[4] = { 0.0, 0.0, 0.5, 1.0 };double rightViewport[4] = { 0.5, 0.0, 1.0, 1.0 };vtkSmartPointer<vtkRenderer> leftRenderer =vtkSmartPointer<vtkRenderer>::New();leftRenderer->AddActor(originalActor);leftRenderer->SetViewport(leftViewport);leftRenderer->SetBackground(1.0, 0, 0);leftRenderer->ResetCamera();vtkSmartPointer<vtkRenderer> rightRenderer =vtkSmartPointer<vtkRenderer>::New();rightRenderer->AddActor(erodedActor);rightRenderer->SetViewport(rightViewport);rightRenderer->SetBackground(1.0, 1.0, 1.0);rightRenderer->ResetCamera();//vtkSmartPointer<vtkRenderWindow> rw =vtkSmartPointer<vtkRenderWindow>::New();rw->AddRenderer(leftRenderer);rw->AddRenderer(rightRenderer);rw->SetSize(640, 320);rw->SetWindowName("ButterworthLowPassExample");vtkSmartPointer<vtkRenderWindowInteractor> rwi =vtkSmartPointer<vtkRenderWindowInteractor>::New();vtkSmartPointer<vtkInteractorStyleImage> style =vtkSmartPointer<vtkInteractorStyleImage>::New();rwi->SetInteractorStyle(style);rwi->SetRenderWindow(rw);rwi->Start();return 0;
}