How to implement low-pass filter blur image function in Python

Use low-pass filter to blur the image

0. Introduction

Low Pass Filter(Low Pass Filter,LPF) filters out the high-frequency parts of the image and allows only the low-frequency parts to pass. Therefore, applyingLPFon an image removes details/edges and noise/outliers in the image. This process is also called image blurring (or smoothing). Image smoothing can be used as a precursor to complex image processing tasks. processing part.

1. Different types of kernels and convolutions in the frequency domain

1.1 Image blur classification

Image blur usually includes the following types:

• Edge Blur (Edge) This type of blur is usually applied explicitly to the image through convolution, such as a linear filter kernel or a Gaussian kernel, etc., using these filter kernels it can be smoothed/removed Unnecessary detail/noise in the image.

• Motion blur (Motion) is usually caused by camera shake when capturing images, that is, the camera or the subject being photographed is moving. We can use the point spread function to simulate this blur.

• Defocus blur (de-focus) This type of blur occurs when the object captured by the camera is out of focus; we can use blur (blur) kernel to simulate this blur.

Next, we create the above three different types of kernels and apply them to the image to observe the results of different types of kernels processing the image.

1.2 Perform image blur using different kernels

(1)We first define the functionget_gaussian_edge_blur_kernel()to return2DGaussian blur kernel is used for edge blur. This function accepts the standard deviation of the Gaussian ( σ σ σ) and the size of the created2Dkernel (for example,sz = 15will create a kernel) as a parameter of the function. As shown below, a1DGaussian kernel is first created, and then the outer product of two1DGaussian kernels is calculated to return the2Dkernel:

import numpy as np import numpy.fft as fp from skimage.io import imread from skimage.color import rgb2gray import matplotlib.pyplot as plt import cv2 def get_gaussian_edge_blur_kernel(sigma, sz=15): # First create a 1-D Gaussian kernel x = np.linspace(-10, 10, sz) kernel_1d = np.exp(-x**2/sigma**2) kernel_1d /= np.trapz(kernel_1d) # normalize the sum to 1.0 # create a 2-D Gaussian kernel from the 1-D kernel kernel_2d = kernel_1d[:, np.newaxis] * kernel_1d[np.newaxis, :] return kernel_2d

(2)

Next, define the functionget_motion_blur_kernel()to generate a motion blur kernel, and obtain a line of a given length and a specific direction (angle) as a convolution kernel to simulate the input Motion blur effect of the image:

def get_motion_blur_kernel(ln, angle, sz=15): kern = np.ones((1, ln), np.float32) angle = -np.pi*angle/180 c, s = np.cos(angle), np.sin(angle) A = np.float32([[c, -s, 0], [s, c, 0]]) sz2 = sz // 2 A[:,2] = (sz2, sz2) - np.dot(A[:,:2], ((ln-1)*0.5, 0)) kern = cv2.warpAffine(kern, A, (sz, sz), flags=cv2.INTER_CUBIC) return kern

Function

get_motion_blur_kernel()

takes the blur length and angle and the size of the blur kernel as parameters, and the function usesOpenCVThe warpaffine()function returns the kernel matrix (taking the center of the matrix as the midpoint, using the given length and the given angle to obtain the kernel).(3)

Finally, define the function

get_out_of_focus_kernel()to generate an out-of-focus kernel (simulating image out-of-focus blur), which creates a circle based on a given radius as Convolution kernel, this function accepts the radiusR(Deocus Radius) and the size of the kernel to be generated as input parameters:

def get_out_of_focus_kernel(r, sz=15): kern = np.zeros((sz, sz), np.uint8) cv2.circle(kern, (sz, sz), r, 255, -1, cv2.LINE_AA, shift=1) kern = np.float32(kern) / 255 return kern

(4)

Next, implement the function

dft_convolve(), which performs frequency domain convolution using element-wise multiplication of the image and a convolution kernel in the frequency domain (based on the convolution theorem). This function also plots the input image, the kernel, and the output image resulting from the convolution calculation:

def dft_convolve(im, kernel): F_im = fp.fft2(im) #F_kernel = fp.fft2(kernel, s=im.shape) F_kernel = fp.fft2(fp.ifftshift(kernel), s=im.shape) F_filtered = F_im * F_kernel im_filtered = fp.ifft2(F_filtered) cmap = 'RdBu' plt.figure(figsize=(20,10)) plt.gray() plt.subplot(131), plt.imshow(im), plt.axis('off'), plt.title('input image', size=10) plt.subplot(132), plt.imshow(kernel, cmap=cmap), plt.title('kernel', size=10) plt.subplot(133), plt.imshow(im_filtered.real), plt.axis('off'), plt.title('output image', size=10) plt.tight_layout() plt.show()

(5)

Apply the

get_gaussian_edge_blur_kernel()kernel function to the image, and plot the input, kernel and output blur images:

im = rgb2gray(imread('3.jpg')) kernel = get_gaussian_edge_blur_kernel(25, 25) dft_convolve(im, kernel)

(6)

Next, apply the

get_motion_blur_kernel()function to the image and plot the input, kernel and Output blurred image:

kernel = get_motion_blur_kernel(30, 60, 25) dft_convolve(im, kernel)

(7)

Finally, apply the

get_out_of_focus_kernel()function to the image and draw the input, kernel and output blurred image:

kernel = get_out_of_focus_kernel(15, 20) dft_convolve(im, kernel)

2. Use the scipy.ndimage filter to blur the image

scipy.ndimage

The module provides a series of functions that can apply low-pass filters to images in the frequency domain. In this section, we learn how to use some of these filters through several examples.

2.1 Using the fourier_gaussian() function

Use the

fourier_gaussian()

function in the

scipy.ndimagelibrary to perform the volume using a Gaussian kernel in the frequency domain accumulation operation.(1)

First, read the input image, convert it to a grayscale image, and obtain its frequency domain representation by using

FFT:

import numpy as np import numpy.fft as fp from skimage.io import imread import matplotlib.pyplot as plt from scipy import ndimage im = imread('1.png', as_gray=True) freq = fp.fft2(im)

(2)

Next, use the

fourier_gaussian()function to perform a blur operation on the image, using two Gaussian kernels with different standard deviations, plot the input, output images and Power spectrum:

fig, axes = plt.subplots(2, 3, figsize=(20,15)) plt.subplots_adjust(0,0,1,0.95,0.05,0.05) plt.gray() # show the filtered result in grayscale axes[0, 0].imshow(im), axes[0, 0].set_title('Original Image', size=10) axes[1, 0].imshow((20*np.log10( 0.1 + fp.fftshift(freq))).real.astype(int)), axes[1, 0].set_title('Original Image Spectrum', size=10) i = 1 for sigma in [3,5]: convolved_freq = ndimage.fourier_gaussian(freq, sigma=sigma) convolved = fp.ifft2(convolved_freq).real # the imaginary part is an artifact axes[0, i].imshow(convolved) axes[0, i].set_title(r'Output with FFT Gaussian Blur, $\sigma$={}'.format(sigma), size=10) axes[1, i].imshow((20*np.log10( 0.1 + fp.fftshift(convolved_freq))).real.astype(int)) axes[1, i].set_title(r'Spectrum with FFT Gaussian Blur, $\sigma$={}'.format(sigma), size=10) i += 1 for a in axes.ravel(): a.axis('off') plt.show()

2.2 Use the fourier_uniform() function

scipy.ndimage

The function of the module

fourier_uniform()implements the multi-dimensional uniform Fourier filter . The frequency array is multiplied by the Fourier transform of a square kernel of given size. Next, we learn how to blur an input grayscale image usingLPF(Mean Filter).(1)

First, read the input image and use

DFTto obtain its frequency domain representation:

im = imread('1.png', as_gray=True) freq = fp.fft2(im)

(2)然后，使用函数fourier_uniform()应用10x10方形核(由功率谱上的参数指定)，以获取平滑输出：

freq_uniform = ndimage.fourier_uniform(freq, size=10)

(3)绘制原始输入图像和模糊后的图像：

fig, (axes1, axes2) = plt.subplots(1, 2, figsize=(20,10)) plt.gray() # show the result in grayscale im1 = fp.ifft2(freq_uniform) axes1.imshow(im), axes1.axis('off') axes1.set_title('Original Image', size=10) axes2.imshow(im1.real) # the imaginary part is an artifact axes2.axis('off') axes2.set_title('Blurred Image with Fourier Uniform', size=10) plt.tight_layout() plt.show()

(4)最后，绘制显示方形核的功率谱：

plt.figure(figsize=(10,10)) plt.imshow( (20*np.log10( 0.1 + fp.fftshift(freq_uniform))).real.astype(int)) plt.title('Frequency Spectrum with fourier uniform', size=10) plt.show()

2.3 使用 fourier_ellipsoid() 函数

与上一小节类似，通过将方形核修改为椭圆形核，我们可以使用椭圆形核生成模糊的输出图像。

(1)类似的，我们首先在图像的功率谱上应用函数fourier_ellipsoid()，并使用IDFT在空间域中获得模糊后的输出图像：

freq_ellipsoid = ndimage.fourier_ellipsoid(freq, size=10) im1 = fp.ifft2(freq_ellipsoid)

(2)接下来，绘制原始输入图像和模糊后的图像：

fig, (axes1, axes2) = plt.subplots(1, 2, figsize=(20,10)) axes1.imshow(im), axes1.axis('off') axes1.set_title('Original Image', size=10) axes2.imshow(im1.real) # the imaginary part is an artifact axes2.axis('off') axes2.set_title('Blurred Image with Fourier Ellipsoid', size=10) plt.tight_layout() plt.show()

(3)最后，显示应用椭圆形核后图像的频谱：

plt.figure(figsize=(10,10)) plt.imshow( (20*np.log10( 0.1 + fp.fftshift(freq_ellipsoid))).real.astype(int)) plt.title('Frequency Spectrum with Fourier ellipsoid', size=10) plt.show()

3. 使用 scipy.fftpack 实现高斯模糊

我们已经学习了如何在实际应用中使用numpy.fft模块的2D-FFT。在本节中，我们将介绍scipy.fftpack模块的fft2()函数用于实现高斯模糊。

(1)使用灰度图像作为输入，并使用FFT从图像中创建2D频率响应数组：

import numpy as np import numpy.fft as fp from skimage.color import rgb2gray from skimage.io import imread import matplotlib.pyplot as plt from scipy import signal from matplotlib.ticker import LinearLocator, FormatStrFormatter im = rgb2gray(imread('1.png')) freq = fp.fft2(im)

(2)通过计算两个1D高斯核的外积，在空间域中创建高斯2D核用作LPF：

kernel = np.outer(signal.gaussian(im.shape[0], 1), signal.gaussian(im.shape[1], 1))

(3)使用DFT获得高斯核的频率响应：

freq_kernel = fp.fft2(fp.ifftshift(kernel))

(4)使用卷积定理通过逐元素乘法在频域中将LPF与输入图像卷积：

convolved = freq*freq_kernel # by the Convolution theorem

(5)使用IFFT获得输出图像，需要注意的是，要正确显示输出图像，需要缩放输出图像：

im_blur = fp.ifft2(convolved).real im_blur = 255 * im_blur / np.max(im_blur)

(6)绘制图像、高斯核和在频域中卷积后获得图像的功率谱，可以使用matplotlib.colormap绘制色，以了解不同坐标下的频率响应值：

plt.figure(figsize=(20,20)) plt.subplot(221), plt.imshow(kernel, cmap='coolwarm'), plt.colorbar() plt.title('Gaussian Blur Kernel', size=10) plt.subplot(222) plt.imshow( (20*np.log10( 0.01 + fp.fftshift(freq_kernel))).real.astype(int), cmap='inferno') plt.colorbar() plt.title('Gaussian Blur Kernel (Freq. Spec.)', size=10) plt.subplot(223), plt.imshow(im, cmap='gray'), plt.axis('off'), plt.title('Input Image', size=10) plt.subplot(224), plt.imshow(im_blur, cmap='gray'), plt.axis('off'), plt.title('Output Blurred Image', size=10) plt.tight_layout() plt.show()

(7)要绘制输入/输出图像和3D核的功率谱，我们定义函数plot_3d()，使用mpl_toolkits.mplot3d模块的plot_surface()函数获取3D功率谱图，给定相应的Y和Z值作为2D阵列传递：

def plot_3d(X, Y, Z, cmap=plt.cm.seismic): fig = plt.figure(figsize=(20,20)) ax = fig.gca(projection='3d') # Plot the surface. surf = ax.plot_surface(X, Y, Z, cmap=cmap, linewidth=5, antialiased=False) #ax.plot_wireframe(X, Y, Z, rstride=10, cstride=10) #ax.set_zscale("log", nonposx='clip') #ax.zaxis.set_scale('log') ax.zaxis.set_major_locator(LinearLocator(10)) ax.zaxis.set_major_formatter(FormatStrFormatter('%.02f')) ax.set_xlabel('F1', size=15) ax.set_ylabel('F2', size=15) ax.set_zlabel('Freq Response', size=15) #ax.set_zlim((-40,10)) # Add a color bar which maps values to colors. fig.colorbar(surf) #, shrink=0.15, aspect=10) #plt.title('Frequency Response of the Gaussian Kernel') plt.show()

(8)在3D空间中绘制高斯核的频率响应，并使用plot_3d()函数：

Y = np.arange(freq.shape[0]) #-freq.shape[0]//2,freq.shape[0]-freq.shape[0]//2) X = np.arange(freq.shape[1]) #-freq.shape[1]//2,freq.shape[1]-freq.shape[1]//2) X, Y = np.meshgrid(X, Y) Z = (20*np.log10( 0.01 + fp.fftshift(freq_kernel))).real plot_3d(X,Y,Z)

下图显示了3D空间中高斯LPF核的功率谱：

(9)绘制3D空间中输入图像的功率谱：

Z = (20*np.log10( 0.01 + fp.fftshift(freq))).real plot_3d(X,Y,Z)

(10)最后，绘制输出图像的功率谱(通过将高斯核与输入图像卷积获得)：

Z = (20*np.log10( 0.01 + fp.fftshift(convolved))).real plot_3d(X,Y,Z)

从输出图像的频率响应中可以看出，高频组件被衰减，从而导致细节的平滑/丢失，并导致输出图像模糊。

4. 彩色图像频域卷积

在本节中，我们将学习使用scipy.signal模块的fftconvolve()函数，用于与RGB彩色输入图像进行频域卷积，从而生成RGB彩色模糊输出图像：

scipy.signal.fftconvolve(in1, in2, mode='full', axes=None)

函数使用FFT卷积两个n维数组in1和in2，并由mode参数确定输出大小。卷积模式mode具有以下类型：

• 输出是输入的完全离散线性卷积，默认情况下使用此种卷积模式

• 输出仅由那些不依赖零填充的元素组成，in1或in2的尺寸必须相同

• 输出的大小与in1相同，并以输出为中心

4.1 基于 scipy.signal 模块的彩色图像频域卷积

接下来，我们实现高斯低通滤波器并使用Laplacian高通滤波器执行相应操作。

(1)首先，导入所需的包，并读取输入RGB图像：

from skimage import img_as_float from scipy import signal import numpy as np import matplotlib.pyplot as plt im = img_as_float(plt.imread('1.png'))

(2)实现函数get_gaussian_edge_kernel()，并根据此函数创建一个尺寸为15x15的高斯核：

def get_gaussian_edge_blur_kernel(sigma, sz=15): # First create a 1-D Gaussian kernel x = np.linspace(-10, 10, sz) kernel_1d = np.exp(-x**2/sigma**2) kernel_1d /= np.trapz(kernel_1d) # normalize the sum to 1.0 # create a 2-D Gaussian kernel from the 1-D kernel kernel_2d = kernel_1d[:, np.newaxis] * kernel_1d[np.newaxis, :] return kernel_2d kernel = get_gaussian_edge_blur_kernel(sigma=10, sz=15)

(3)然后，使用np.newaxis将核尺寸重塑为15x15x1，并使用same模式调用函数signal.fftconvolve()：

im1 = signal.fftconvolve(im, kernel[:, :, np.newaxis], mode='same') im1 = im1 / np.max(im1)

在以上代码中使用的mode='same'，可以强制输出形状与输入阵列形状相同，以避免边框效应。

(4)接下来，使用laplacian HPF内核，并使用相同函数执行频域卷积。需要注意的是，我们可能需要缩放/裁剪输出图像以使输出值保持像素的浮点值范围[0,1]内：

kernel = np.array([[0,-1,0],[-1,4,-1],[0,-1,0]]) im2 = signal.fftconvolve(im, kernel[:, :, np.newaxis], mode='same') im2 = im2 / np.max(im2) im2 = np.clip(im2, 0, 1)

(5)最后，绘制输入图像和使用卷积创建的输出图像。

plt.figure(figsize=(20,10)) plt.subplot(131), plt.imshow(im), plt.axis('off'), plt.title('original image', size=10) plt.subplot(132), plt.imshow(im1), plt.axis('off'), plt.title('output with Gaussian LPF', size=10) plt.subplot(133), plt.imshow(im2), plt.axis('off'), plt.title('output with Laplacian HPF', size=10) plt.tight_layout() plt.show()

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