十一、图像傅里叶变换

末蓝、 2023-06-15 11:03 49阅读 0赞

**目录**

一、傅里叶变换

二、Numpy实现傅里叶变换

1、实现傅里叶变换

代码

2、实现傅里叶的逆变换

代码

三、OpenCV 实现傅里叶变换

1、实现傅里叶变换

代码

2、实现傅里叶逆变换

代码

四、高通滤波和低通滤波

1、高通滤波和低通滤波概述

2、Numpy 实现高通滤波

3、OpenCV 实现低通滤波

--------------------

# 一、傅里叶变换 #

原理可以参考：[傅里叶分析之掐死教程（完整版）][Link 1]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 1][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 2][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 3][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 4][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 5][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 6][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 7][]

**任何连续周期信号，都可以用适当的一组正弦曲线组合而成**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 8][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 9][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 10][]

**相位：**

**不是同时开始的一组余弦函数，在叠加时要体现开始时间**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 11][]

# 二、Numpy实现傅里叶变换 #

## 1、实现傅里叶变换 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 12][]

**实现将低频移动到整个频谱中心**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 13][]

**重新设置傅里叶变换返回复数数组的取值范围，重新映射**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 14][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 15][]

### 代码 ###

# encoding: utf-8
    import cv2
    import matplotlib.pyplot as plt
    import numpy as np
    
    def show_img(name="test",img=None):
        plt.figure()
        channel = img.shape
        if channel == 3:
            b, g, r = cv2.split(img)
            img = cv2.merge([r, g, b])
        plt.imshow(img)
        plt.title(name)
        plt.show()
    
    
    if __name__ == '__main__':
        path = r"lena.jpg"
        img = cv2.imread(path)
        gray = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY)
        ret,binary = cv2.threshold(gray,127,255,0)
    
        # 对图像进行傅里叶变换
        f = np.fft.fft2(gray)
        # 将低频移动到中心
        fshift = np.fft.fftshift(f)
        # 重置 区间,映射到[0,255]之间，以便使用图像显示
        result = 20 * np.log(np.abs(fshift))
    
        # 原始灰度图
        plt.subplot(1,2,1 ),plt.imshow(gray,'gray'),plt.axis('off'),plt.title('original')
        # 频谱图像
        plt.subplot(1,2,2),plt.imshow(result,'gray'),plt.axis('off'),plt.title('result')
        plt.show()

## 2、实现傅里叶的逆变换 ##

使用傅里叶变换得到图像的低频和高频，

对低频（图像的细节）处理（保留、舍弃或其他），可以模糊图像等

对高频进行处理（）保留、舍弃或其他），可以保留图像边界等

![20191121235435727.png][]

将低频移动到左上角

![20191121235502106.png][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 16][]

### 代码 ###

# encoding: utf-8
    import cv2
    import matplotlib.pyplot as plt
    import numpy as np
    
    def show_img(name="test",img=None):
        plt.figure()
        channel = img.shape
        if channel == 3:
            b, g, r = cv2.split(img)
            img = cv2.merge([r, g, b])
        plt.imshow(img)
        plt.title(name)
        plt.show()
    
    
    if __name__ == '__main__':
        path = r"lena.jpg"
        img = cv2.imread(path)
        gray = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY)
        ret,binary = cv2.threshold(gray,127,255,0)
        """实现傅里叶变换"""
        # 对图像进行傅里叶变换
        f = np.fft.fft2(gray)
        # 将低频从左上角移动到中心
        fshift = np.fft.fftshift(f)
        # 重置 区间,映射到[0,255]之间，以便使用图像显示
        # result = 20 * np.log(np.abs(fshift))
    
        """实现逆傅里叶变换"""
        # 将低频从中心移动到左上角
        ishift = np.fft.ifftshift(fshift)
        # 返回一个复数数组
        iimg = np.fft.ifft2(ishift)
        # 将上述复数数组重置区间到[0,255],便于图像显示
        iimg = np.abs(iimg)
    
        # 原始灰度图
        plt.subplot(1,2,1 ),plt.imshow(gray,'gray'),plt.axis('off'),plt.title('original')
        # 频谱图像
        plt.subplot(1,2,2),plt.imshow(iimg,'gray'),plt.axis('off'),plt.title('result')
        plt.show()

# 三、OpenCV 实现傅里叶变换 #

## 1、实现傅里叶变换 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 17][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 18][]

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### 代码 ###

## 2、实现傅里叶逆变换 ##

![20191122004620398.png][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 21][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 22][]

### 代码 ###

# encoding: utf-8
    import cv2
    import matplotlib.pyplot as plt
    import numpy as np
    
    
    if __name__ == '__main__':
        path = r"lena.jpg"
        img = cv2.imread(path)
        gray = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY)
        ret,binary = cv2.threshold(gray,127,255,0)
        """实现傅里叶变换"""
        # 对图像进行傅里叶变换
        dft = cv2.dft(np.float32(gray),flags=cv2.DFT_COMPLEX_OUTPUT)
        # 将低频从左上角移动到中心
        dshift = np.fft.fftshift(dft)
        # 重置 区间,映射到[0,255]之间，以便使用图像显示
        # result = 20 * np.log(cv2.magnitude(dshift[:,:,0],dshift[:,:,1]))
    
    
        """实现逆傅里叶变换"""
        # 将低频从中心移动到左上角
        ishift = np.fft.ifftshift(dshift)
        # 返回一个复数数组
        iimg = cv2.idft(ishift)
        # 将上述复数数组重置区间到[0,255],便于图像显示
        iimg = cv2.magnitude(iimg[:,:,0],iimg[:,:,1])
    
        # 原始灰度图
        plt.subplot(1,2,1 ),plt.imshow(gray,'gray'),plt.axis('off'),plt.title('original')
        # 频谱图像
        plt.subplot(1,2,2),plt.imshow(iimg,'gray'),plt.axis('off'),plt.title('result')
        plt.show()

# 四、高通滤波和低通滤波 #

## 1、高通滤波和低通滤波概述 ##

**高通滤波器**

*  高通滤波器（HPF）是检测图像的某个区域，然后根据像素与周围像素的亮度差值来提升像素的亮度。
 *  用于：**边缘提取与增强**。
 *  注意：**通过高通滤波器进行滤波后，再和原图像叠加，可以增强图像中灰度级变化较快的部分，即锐化**。

**低通滤波器**

*  低通滤波器是像素与周围像素的亮度差值小于一个特定值时，平滑该像素的亮度。
 *  用于：**去噪和模糊化**。
 *  注意：**低通滤波器容许低频信号通过，但减弱频率高于截止频率的信号的通过。**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 23][]

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## 2、Numpy 实现高通滤波 ##

# encoding: utf-8
    import cv2
    import matplotlib.pyplot as plt
    import numpy as np
    
    if __name__ == '__main__':
        path = r"lena.jpg"
        img = cv2.imread(path)
        gray = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY)
        ret,binary = cv2.threshold(gray,127,255,0)
        """实现傅里叶变换"""
        # 对图像进行傅里叶变换
        f = np.fft.fft2(gray)
        # 将低频从左上角移动到中心
        fshift = np.fft.fftshift(f)
        # 重置 区间,映射到[0,255]之间，以便使用图像显示
        # result = 20 * np.log(np.abs(fshift))
    
        """高通滤波"""
        # 得到边缘图像
        rows,cols = gray.shape
        # 取图像中心点
        row_mid,col_mid = int(rows / 2), int(cols / 2)
        # 去掉低频区域
        fshift[row_mid-30:row_mid+30,col_mid-30:col_mid+30] = 0
    
        """实现逆傅里叶变换"""
        # 将低频从中心移动到左上角
        ishift = np.fft.ifftshift(fshift)
        # 返回一个复数数组
        iimg = np.fft.ifft2(ishift)
        # 将上述复数数组重置区间到[0,255],便于图像显示
        iimg = np.abs(iimg)
    
        # 原始灰度图
        plt.subplot(1,2,1 ),plt.imshow(gray,'gray'),plt.axis('off'),plt.title('original')
        # 频谱图像
        plt.subplot(1,2,2),plt.imshow(iimg,'gray'),plt.axis('off'),plt.title('result')
        plt.show()

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 27][]

## 3、OpenCV 实现低通滤波 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 28][]

**调节掩膜区域的大小可以得到不同模糊程度的图像，越大则越清晰，越小则越模糊。**

# encoding: utf-8
    import cv2
    import matplotlib.pyplot as plt
    import numpy as np
    
    if __name__ == '__main__':
        path = r"lena.jpg"
        img = cv2.imread(path)
        gray = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY)
        ret,binary = cv2.threshold(gray,127,255,0)
        """实现傅里叶变换"""
        # 对图像进行傅里叶变换
        dft = cv2.dft(np.float32(gray),flags=cv2.DFT_COMPLEX_OUTPUT)
        # 将低频从左上角移动到中心
        dshift = np.fft.fftshift(dft)
        # 重置 区间,映射到[0,255]之间，以便使用图像显示
        # result = 20 * np.log(cv2.magnitude(dshift[:,:,0],dshift[:,:,1]))
    
        """低通滤波"""
        rows,cols = gray.shape
        # 取图像中心点
        row_mid,col_mid = int(rows / 2), int(cols / 2)
        # 构造掩膜图像
        mask = np.zeros((rows,cols,2),np.uint8)
    
        # 去掉低频区域
        mask[row_mid-30:row_mid+30,col_mid-30:col_mid+30] = 1
        # 将频谱图像和掩膜相乘,保留低频部分，高频部分变成0
        md = dshift * mask
    
        """实现逆傅里叶变换"""
        # 将低频从中心移动到左上角
        imd = np.fft.ifftshift(md)
        # 返回一个复数数组
        iimg = cv2.idft(imd)
        # 将上述复数数组重置区间到[0,255],便于图像显示
        iimg = cv2.magnitude(iimg[:,:,0],iimg[:,:,1])
    
        # 原始灰度图
        plt.subplot(1,2,1 ),plt.imshow(gray,'gray'),plt.axis('off'),plt.title('original')
        # 频谱图像
        plt.subplot(1,2,2),plt.imshow(iimg,'gray'),plt.axis('off'),plt.title('result')
        plt.show()

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3MyOTQ4NzgzMDQ_size_16_color_FFFFFF_t_70 29][]

[Link 1]: https://zhuanlan.zhihu.com/p/19763358
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