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
;
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
;

// 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
;
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
;
//
//        // 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
;
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
;
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

;
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
;
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
;
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

;
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

;

// 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);
}
}