java使用傅里葉變換，得到變換之後的傅里葉頻譜圖像。

使用傅里葉變換來處理圖像，可以濾波，去噪。 但是網絡上java實現很少，可以運行的就更少，我總結了一些代碼，並加以調試，

保證速度的情況下得以運行成功。 這些代碼僅對灰度圖驗證通過。學習之用。

調用結果可以用http://cns-alumni.bu.edu/~slehar/fourier/fourier.html#filtering 網站的圖片驗證：

     

輸入圖像：ff8.bmp                  輸出圖像fft_result.bmp

     

輸入圖像：ff8.bmp                  輸出圖像fft_result.bmp

所有FFT變換都已經驗證通過。  但是傅里葉逆變換沒有成功。  有高手可以指點一下。 如何使逆變換的圖片顯示出來。

FFT 類負責一維變換：

package app.fourier;

public class FFT {

//	double r_data[] = null;
//	double i_data[] = null;
	
	// compute the FFT of x[], assuming its length is a power of 2
	public static void fft(Complex[] src, int row, int width, Complex[] dest) {
		
		Complex[] temp = new Complex[width];
		for (int k = 0; k < width; k++) {
			temp[k] = src[row * width + k];
		}
		temp = fft(temp);
		//set output
		for (int k = 0; k < width; k++) {
			dest[row * width + k] = temp[k];
		}
	}
	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;
	}
}

Complex類是複數類：

public class Complex {
    public  double re;   // the real part
    public  double im;   // the imaginary part

    public Complex() {
        re = 0;
        im = 0;
    }
    
    // create a new object with the given real and imaginary parts
    public Complex(double real, double imag) {
        re = real;
        im = imag;
    }

    // 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;
    }
}

FourierTransformer 爲主調類：

import java.awt.Graphics;
import java.awt.Image;
import java.awt.image.BufferedImage;
import java.awt.image.ColorModel;
import java.awt.image.PixelGrabber;
import java.awt.image.WritableRaster;
import java.io.File;
import java.io.IOException;

import javax.imageio.ImageIO;
import javax.swing.JFrame;

@SuppressWarnings("serial")
public class FourierTransformer  extends JFrame{
	Image im;
	BufferedImage imageAuth = null;
	int iw;
	int ih;
	int[] pixels;
	int[] newPixels;

	public FourierTransformer() {
		
		try {
			this.im = ImageIO.read(getClass().getResource("fft8.bmp"));
		} catch (IOException e1) {
			e1.printStackTrace();
		}
		
		this.iw = im.getWidth(null);
		this.ih = im.getHeight(null);
		pixels = new int[iw * ih];
		try {
			PixelGrabber pg = new PixelGrabber(im, 0, 0, iw, ih, pixels, 0, iw);
			pg.grabPixels();
		} catch (InterruptedException e3) {
			e3.printStackTrace();
		}
		
	}

	public void paint(Graphics g) {
		super.paint(g);
				
		g.drawImage(this.im, 0, 100, this.iw, this.ih, this);
		if(imageAuth != null)
			g.drawImage(imageAuth, 250, 100, imageAuth.getWidth(), imageAuth.getHeight(), this);
		
	}
	
	public static void main(String[] args){
		FourierTransformer frame = new FourierTransformer();
		
		frame.setSize(600, 500);
		frame.setTitle("ImageMenu");
		frame.setName("hello my dongjing");
		
		frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);		
		
		
		
		frame.setVisible(true);
		frame.convert(frame.getGraphics());
	}
	public Image convert(Graphics g) {
		// 賦初值
		int w = 1;
		int h = 1;
		// 計算進行付立葉變換的寬度和高度（2的整數次方）
		while (w * 2 <= iw) {
			w *= 2;
		}
		while (h * 2 <= ih) {
			h *= 2;
		}
		// 分配內存
		Complex[] src = new Complex[h * w];
		Complex[] dest = new Complex[h * w];
		newPixels = new int[h * w];
		// 初始化newPixels
		for (int i = 0; i < h; i++) {
			for (int j = 0; j < w; j++) {
				newPixels[i * w + j] = pixels[i * iw + j] & 0xff;
			}
		}
		// 初始化src,dest
		for (int i = 0; i < h; i++) {
			for (int j = 0; j < w; j++) {
				dest[i * w + j] = new Complex();
				src[i * w + j] = new Complex(newPixels[i * w + j], 0);
			}
		}
		// 在y方向上進行快速傅立葉變換
		for (int i = 0; i < h; i++) {
			FFT.fft(src, i, w, dest);
		}
		/**
		 * 以下一定要進行轉換，高手指點一下原因 (^ - ^)
		 */
		for (int i = 0; i < h; i++) {
			for (int j = 0; j < w; j++) {
				src[j * h + i] = dest[i * w + j];
//				System.out.println("dest " + j*h+i + ",  src " + i*w+j);
			}
		}
		// 對x方向進行傅立葉變換
		for (int i = 0; i < w; i++) {
			FFT.fft(src, i, h, dest);
		}
		/**
		 * 將圖像看做二維函數，圖像灰度值爲函數在相應XY處的函數值，對其進行二維快速傅里葉變換，
		 * 得到一個複數矩陣，將此矩陣水平循環移動半寬，垂直循環移動半高。
		 */
		for (int i = 0; i < h; i++) {
			for (int j = 0; j < w; j++) {
				double re = dest[j * h + i].re;
				double im = dest[j * h + i].im;
				int ii = 0, jj = 0;
				int temp = (int) (Math.sqrt(re * re + im * im) / 100);
				if (temp > 255) {
					temp = 255;
				}
				if (i < h / 2) {
					ii = i + h / 2;
				} else {
					ii = i - h / 2;
				}
				if (j < w / 2) {
					jj = j + w / 2;
				} else {
					jj = j - w / 2;
				}
				newPixels[ii * w + jj] = temp;
			}
		}

		imageAuth = new BufferedImage(w, h, BufferedImage.TYPE_BYTE_GRAY);
		ColorModel colorModel = imageAuth.getColorModel();
		WritableRaster raster = colorModel.createCompatibleWritableRaster(w, h);	
		raster.setPixels(0, 0, w, h, newPixels);
		imageAuth.setData(raster);
		
		try {
			ImageIO.write(imageAuth, "bmp", new File("fft_result.bmp"));
		} catch (IOException e) {
			e.printStackTrace();
		}
		this.update(g);
		
		
		return imageAuth;
	}
}