离散FFT和图像二维FFT变换的java实现

1.离散FFT简单介绍

    FFT是一种DFT的高效算法，称为快速傅立叶变换（fast Fourier transform）。其原理比较复杂，我们可以不关其具体

细节，值得注意的是：二维FFT可以对图像进行变换，先对每一行进行FFT变换，再对变换后的每一列进行FFT变换，

二维FFT变换的公式如下：

2.复数工具类

    FFT和二维FFT都需要复数的加减乘除，在这里给出Complex.java

ublic class Complex {    private final double re;   // the real part    private final double im;   // the imaginary part    // create a new object with the given real and imaginary parts    public Complex(double real, double imag) {        re = real;        im = imag;    }    public Complex(Complex a){        re=a.re();        im=a.im();    }    // return a string representation of the invoking Complex object    public String toString() {        if (im == 0) return re + "";        if (re == 0) return im + "i";        if (im <  0) return re + " - " + (-im) + "i";        return re + " + " + im + "i";    }    // return abs/modulus/magnitude and angle/phase/argument    public double abs()   { return Math.hypot(re, im); }  // Math.sqrt(re*re + im*im)    public double phase() { return Math.atan2(im, re); }  // between -pi and pi    // return a new Complex object whose value is (this + b)    public Complex plus(Complex b) {        Complex a = this;             // invoking object        double real = a.re + b.re;        double imag = a.im + b.im;        return new Complex(real, imag);    }    // return a new Complex object whose value is (this - b)    public Complex minus(Complex b) {        Complex a = this;        double real = a.re - b.re;        double imag = a.im - b.im;        return new Complex(real, imag);    }    // return a new Complex object whose value is (this * b)    public Complex times(Complex b) {        Complex a = this;        double real = a.re * b.re - a.im * b.im;        double imag = a.re * b.im + a.im * b.re;        return new Complex(real, imag);    }    // scalar multiplication    // return a new object whose value is (this * alpha)    public Complex times(double alpha) {        return new Complex(alpha * re, alpha * im);    }    // return a new Complex object whose value is the conjugate of this    public Complex conjugate() {  return new Complex(re, -im); }    // return a new Complex object whose value is the reciprocal of this    public Complex reciprocal() {        double scale = re*re + im*im;        return new Complex(re / scale, -im / scale);    }    // return the real or imaginary part    public double re() { return re; }    public double im() { return im; }    // return a / b    public Complex divides(Complex b) {        Complex a = this;        return a.times(b.reciprocal());    }    // return a new Complex object whose value is the complex exponential of this    public Complex exp() {        return new Complex(Math.exp(re) * Math.cos(im), Math.exp(re) * Math.sin(im));    }    // return a new Complex object whose value is the complex sine of this    public Complex sin() {        return new Complex(Math.sin(re) * Math.cosh(im), Math.cos(re) * Math.sinh(im));    }    // return a new Complex object whose value is the complex cosine of this    public Complex cos() {        return new Complex(Math.cos(re) * Math.cosh(im), -Math.sin(re) * Math.sinh(im));    }    // return a new Complex object whose value is the complex tangent of this    public Complex tan() {        return sin().divides(cos());    }    // a static version of plus    public static Complex plus(Complex a, Complex b) {        double real = a.re + b.re;        double imag = a.im + b.im;        Complex sum = new Complex(real, imag);        return sum;    }    // sample client for testing    public static void main(String[] args) {        Complex a = new Complex(5.0, 6.0);        Complex b = new Complex(-3.0, 4.0);        System.out.println("a            = " + a);        System.out.println("b            = " + b);        System.out.println("Re(a)        = " + a.re());        System.out.println("Im(a)        = " + a.im());        System.out.println("b + a        = " + b.plus(a));        System.out.println("a - b        = " + a.minus(b));        System.out.println("a * b        = " + a.times(b));        System.out.println("b * a        = " + b.times(a));        System.out.println("a / b        = " + a.divides(b));        System.out.println("(a / b) * b  = " + a.divides(b).times(b));        System.out.println("conj(a)      = " + a.conjugate());        System.out.println("|a|          = " + a.abs());        System.out.println("tan(a)       = " + a.tan());    }}

3.FFT变换——FFT.java

public class FFT {    // compute the FFT of x[], assuming its length is a power of 2    public static Complex[] fft(Complex[] x) {        int N = x.length;        // base case        if (N == 1) return new Complex[] { x[0] };        // radix 2 Cooley-Tukey FFT        if (N % 2 != 0) { throw new RuntimeException("N is not a power of 2"); }        // fft of even terms        Complex[] even = new Complex[N/2];        for (int k = 0; k < N/2; k++) {            even[k] = x[2*k];        }        Complex[] q = fft(even);        // fft of odd terms        Complex[] odd  = even;  // reuse the array        for (int k = 0; k < N/2; k++) {            odd[k] = x[2*k + 1];        }        Complex[] r = fft(odd);        // combine        Complex[] y = new Complex[N];        for (int k = 0; k < N/2; k++) {            double kth = -2 * k * Math.PI / N;            Complex wk = new Complex(Math.cos(kth), Math.sin(kth));            y[k]       = q[k].plus(wk.times(r[k]));            y[k + N/2] = q[k].minus(wk.times(r[k]));        }        return y;    }    // compute the inverse FFT of x[], assuming its length is a power of 2    public static Complex[] ifft(Complex[] x) {        int N = x.length;        Complex[] y = new Complex[N];        // take conjugate        for (int i = 0; i < N; i++) {            y[i] = x[i].conjugate();        }        // compute forward FFT        y = fft(y);        // take conjugate again        for (int i = 0; i < N; i++) {            y[i] = y[i].conjugate();        }        // divide by N        for (int i = 0; i < N; i++) {            y[i] = y[i].times(1.0 / N);        }        return y;    }    // compute the circular convolution of x and y    public static Complex[] cconvolve(Complex[] x, Complex[] y) {        // should probably pad x and y with 0s so that they have same length        // and are powers of 2        if (x.length != y.length) { throw new RuntimeException("Dimensions don't agree"); }        int N = x.length;        // compute FFT of each sequence        Complex[] a = fft(x);        Complex[] b = fft(y);        // point-wise multiply        Complex[] c = new Complex[N];        for (int i = 0; i < N; i++) {            c[i] = a[i].times(b[i]);        }        // compute inverse FFT        return ifft(c);    }    // compute the linear convolution of x and y    public static Complex[] convolve(Complex[] x, Complex[] y) {        Complex ZERO = new Complex(0, 0);        Complex[] a = new Complex[2*x.length];        for (int i = 0;        i <   x.length; i++) a[i] = x[i];        for (int i = x.length; i < 2*x.length; i++) a[i] = ZERO;        Complex[] b = new Complex[2*y.length];        for (int i = 0;        i <   y.length; i++) b[i] = y[i];        for (int i = y.length; i < 2*y.length; i++) b[i] = ZERO;        return cconvolve(a, b);    }    // display an array of Complex numbers to standard output    public static void show(Complex[] x, String title) {        System.out.println(title);        System.out.println("-------------------");        for (int i = 0; i < x.length; i++) {            System.out.println(x[i]);        }        System.out.println();    }    /*********************************************************************     *  Test client and sample execution     *     *  % java FFT 4     *  x     *  -------------------     *  -0.03480425839330703     *  0.07910192950176387     *  0.7233322451735928     *  0.1659819820667019     *     *  y = fft(x)     *  -------------------     *  0.9336118983487516     *  -0.7581365035668999 + 0.08688005256493803i     *  0.44344407521182005     *  -0.7581365035668999 - 0.08688005256493803i     *     *  z = ifft(y)     *  -------------------     *  -0.03480425839330703     *  0.07910192950176387 + 2.6599344570851287E-18i     *  0.7233322451735928     *  0.1659819820667019 - 2.6599344570851287E-18i     *     *  c = cconvolve(x, x)     *  -------------------     *  0.5506798633981853     *  0.23461407150576394 - 4.033186818023279E-18i     *  -0.016542951108772352     *  0.10288019294318276 + 4.033186818023279E-18i     *     *  d = convolve(x, x)     *  -------------------     *  0.001211336402308083 - 3.122502256758253E-17i     *  -0.005506167987577068 - 5.058885073636224E-17i     *  -0.044092969479563274 + 2.1934338938072244E-18i     *  0.10288019294318276 - 3.6147323062478115E-17i     *  0.5494685269958772 + 3.122502256758253E-17i     *  0.240120239493341 + 4.655566391833896E-17i     *  0.02755001837079092 - 2.1934338938072244E-18i     *  4.01805098805014E-17i     *     *********************************************************************/    public static void main(String[] args) {//        int N = Integer.parseInt(args[0]);//        Complex[] x = new Complex[N];////        // original data//        for (int i = 0; i < N; i++) {//            x[i] = new Complex(i, 0);//            x[i] = new Complex(-2*Math.random() + 1, 0);//        }//        show(x, "x");////        long starTime=System.currentTimeMillis();//        long Time=0;//        // FFT of original data//        Complex[] y = fft(x);//        show(y, "y = fft(x)");////        long endTime=System.currentTimeMillis();//        Time=endTime-starTime;//        System.out.println(Time);////        // take inverse FFT//        Complex[] z = ifft(y);//        show(z, "z = ifft(y)");//        endTime=System.currentTimeMillis();//        Time=endTime-starTime;//        System.out.println(Time);////        // circular convolution of x with itself//        Complex[] c = cconvolve(x, x);//        show(c, "c = cconvolve(x, x)");//        endTime=System.currentTimeMillis();//        Time=endTime-starTime;//        System.out.println(Time);////        // linear convolution of x with itself//        Complex[] d = convolve(x, x);//        show(d, "d = convolve(x, x)");//        endTime=System.currentTimeMillis();//        Time=endTime-starTime;//        System.out.println(Time);        Complex x[]={new Complex(224.0,-224.0),new Complex(-32.0,32),new Complex(0,32.0),new Complex(32,31.999999999999996)};        Complex[] z=fft(x);        show(z,"test:");    }}

4.二维FFT变换——FFT2D.java

public class FFT2D {    public static Complex[][] fft2d(Complex[][] x) {        int N =x.length;        for (int i = 0; i < N; i++) {            Complex[] temp = new Complex[N];            for (int j = 0; j < N; j++)                temp[j] = x[i][j];            Complex[] reslutTemp=FFT.fft(temp);            for(int j=0;j<N;j++)                x[i][j]=reslutTemp[j];        }        for (int i = 0; i < N; i++) {            Complex[] temp = new Complex[N];            for (int j = 0; j < N; j++)                temp[j] = x[j][i];            Complex[] resultTemp=FFT.fft(temp);            for(int j=0;j<N;j++)                x[j][i]=resultTemp[j];        }        Complex[][] y = new Complex[N][N];        for(int i=0;i<N;i++)            for(int j=0;j<N;j++)                y[i][j]=x[i][j];        return y;    }    public static Complex[][] ifft2d(Complex[][] x) {        int N=x.length;        for (int i = 0; i < N; i++) {            Complex[] temp = new Complex[N];            for (int j = 0; j < N; j++)                temp[j] = x[i][j].conjugate();            Complex[] resultTemp=FFT.fft(temp);            for(int j=0;j<N;j++)                x[i][j]=resultTemp[j];        }        for (int i = 0; i < N; i++) {            Complex[] temp = new Complex[N];            for (int j = 0; j < N; j++)                temp[j] = x[j][i];            Complex[] resultTemp=FFT.fft(temp);            for(int j=0;j<N;j++)                x[j][i]=resultTemp[j].conjugate();        }        Complex[][] y = new Complex[N][N];       for(int i=0;i<N;i++)           for(int j=0;j<N;j++)               y[i][j]=x[i][j].times(1.0/(N*N));        return y;    }    public static void show(Complex[][] x, String title) {        int N = x.length;        System.out.println(title);        System.out.println("-------------------");        for (int i = 0; i < N; i++) {            for(int j=0;j<N;j++)            System.out.print(x[i][j]+"  ");            System.out.println();        }        System.out.println();    }    public static int compare(Complex[][] x,Complex[][] y){        int N=x.length;        for (int i = 0; i < N; i++) {            for(int j=0;j<N;j++){                if(x[i][j].re()==y[i][j].re()&&x[i][j].im()==y[i][j].im())                  continue;                else{                    System.out.println("两数组不相等 "+i+" "+j);                    return 0;                }            }        }        System.out.println("两数组相等");        return 1;    }    public static void main(String[] args) {        int N = Integer.parseInt(args[0]);        Complex[][] x = new Complex[N][N];        // original data        for (int i = 0; i < N; i++)            for(int j=0;j<N;j++)            x[i][j] = new Complex(i, j);        show(x, "x");        long starTime=System.currentTimeMillis();        long Time=0;        // FFT of original data        Complex[][] y = fft2d(x);      show(y, "y = fft(x)");        long endTime=System.currentTimeMillis();        Time=endTime-starTime;        System.out.println(Time);        // take inverse FFT        Complex[][] z = ifft2d(y);        show(z, "z = ifft(y)");        endTime=System.currentTimeMillis();        Time=endTime-starTime;        System.out.println(Time);    }}