五、c++實現離散傅里葉變換


C++離散傅里葉變換

一、序言:

該教程基於之前的圖像處理類MYCV,是對其的補充。

 

二、設計目標

對圖像進行簡單的離散傅里葉變換,並輸出生成的頻譜圖。

 

三、需要提前掌握的知識

二維傅里葉變換公式:

 

 

四、詳細步驟

1.首先定義一個方法,該方法對輸入的圖像進行傅里葉變換

輸入:MyImage 源圖像

輸出:ComplexNu 進行離散傅里葉變換后的復數數組

定義:

static ComplexNumber* Dft2(MyImage const &Scr);

 

實現:

ComplexNumber* MyCV::Dft2(MyImage const &Scr)

{

    int width = Scr.m_width;

    int height = Scr.m_height;

 

    // 將 scr_data 轉化為灰度

    MyImage *grayimage = Gray(Scr);

    unsigned char* gray_data = grayimage->m_data;

    int gray_bytesPerLine = grayimage->m_bytesPerLine;

 

    // 將 gray_data 轉化為 double 型,並去掉用於填充的多余空間

    double *double_data = new double[width*height];

 

    for(int i=0;i<height;i++)

        for(int j=0;j<width;j++)

        {

            double_data[i*width+j]=(double)gray_data[i*gray_bytesPerLine+j];

        }

 

    // 對 double_data 進行傅里葉變換

    ComplexNumber *dft2_data = new ComplexNumber[width*height];

    double fixed_factor_for_axisX = (-2 * PI) / height;

    // evaluate -i2π/N of -i2πux/N, and store the value for computing efficiency

    double fixed_factor_for_axisY = (-2 * PI) / width;

    // evaluate -i2π/N of -i2πux/N, and store the value for computing efficiency

 

    for (int u = 0; u<height; u++) {

            for (int v = 0; v<width; v++) {

                for (int x = 0; x<height; x++) {

                    for (int y = 0; y<width; y++) {

                        double powerX = u * x * fixed_factor_for_axisX; // evaluate -i2πux/N

                        double powerY = v * y * fixed_factor_for_axisY; // evaluate -i2πux/N

                        ComplexNumber cplTemp;

                        cplTemp.m_rl = double_data[y + x*width] * cos(powerX + powerY);

                        // evaluate f(x) * e^(-i2πux/N), which is equal to f(x) * (cos(-i2πux/N)+sin(-i2πux/N)i) according to Euler's formula

                        cplTemp.m_im = double_data[y + x*width] * sin(powerX + powerY);

                        dft2_data[v + u*width] = dft2_data[v + u*width] + cplTemp;

                    }

                }

            }

        }

 

    // 返回傅里葉數組

    return dft2_data;

}

 

2.為了讓傅里葉變換可視化,旭陽對其進行標准化和中性化

輸入:ComplexNumber 離散傅里葉變換生成的復數數組

輸出:MyImage 可視化后的圖像

定義:

static MyImage* Dft22MyImage(ComplexNumber *Scr,int width,int height);

 

實現:

MyImage* MyCV::Dft22MyImage(ComplexNumber *Scr, int const width, int const height)

{

    // 將傅里葉數組歸一化

    // 取模

    double mold[width*height];

    for(int i = 0 ;i<width*height;i++)

    {

        mold[i] = Scr[i].get_mold();

    }

 

    // 獲取最小值

    double min = mold[0];

    for(int i = 0;i<width*height;i++)

    {

        if(mold[i]<min)

            min = mold[i];

    }

 

    // 獲取去掉前幾大值的最大值

    double maxqueue[20] = {0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},max;

    for(int i = 0;i<width*height;i++){

        if(mold[i]>maxqueue[0])

            maxqueue[0] = mold[i];

    }

    for(int j =1;j<20;j++){

        for(int i = 0;i<width*height;i++){

            if(mold[i]>maxqueue[j]&&mold[i]<maxqueue[j-1])

                maxqueue[j] = mold[i];

        }

    }

    max = maxqueue[19];

 

    unsigned char *normalized_data = new unsigned char[width*height];

 

    for(int i=0;i<height;i++)

        for(int j=0;j<width;j++)

        {

            unsigned char t = (unsigned char)((mold[i*width+j]-min)/(max-min)*255);

            if(t>255)

                t = 255;

            normalized_data[i*width+j]=t;

        }

 

 

 

    // 將圖像中心化

    unsigned char* center_data = new unsigned char[width*height];

 

    for (int u = 0; u<height; u++)

        {

        for (int v = 0; v<width; v++) {

            if ((u<(height / 2)) && (v<(width / 2))) {

                center_data[v + u*width] =

                    normalized_data[width / 2 + v + (height / 2 + u)*width];

            }

            else if ((u<(height / 2)) && (v >= (width / 2))) {

                center_data[v + u*width] =

                    normalized_data[(v - width / 2) + (height / 2 + u)*width];

            }

            else if ((u >= (height / 2)) && (v<(width / 2))) {

                center_data[v + u*width] =

                    normalized_data[(width / 2 + v) + (u - height / 2)*width];

            }

            else if ((u >= (height / 2)) && (v >= (width / 2))) {

                center_data[v + u*width] =

                    normalized_data[(v - width / 2) + (u - height / 2)*width];

            }

        }

    }

 

    // 向中心化的數組填充空間

    int bytesPerLine = (width*8+31)/32*4;

    unsigned char *dst_data = new unsigned char[bytesPerLine*height];

 

    for(int i=0;i<height;i++)

        for(int j=0;j<width;j++)

        {

            dst_data[i*bytesPerLine+j] = center_data[i*width+j];

        }

 

    return new MyImage(dst_data,width,height,MyImage::format::GRAY8);

}

 

至此,離散傅里葉變換的方法實現完成,效果圖如下:

 

 

如果上述教程或代碼中有任何錯誤,歡迎批評和指證。

 


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