深圳集团网站开发网站开发公司,防止网站扫描,wordpress 精简,开发游戏需要多少钱目录二变量函数的傅里叶变换二维冲激及其取样性质二维连续傅里叶变换对二维取样和二维取样定理图像中的混叠二维离散傅里叶变换及其反变换二变量函数的傅里叶变换
二维冲激及其取样性质
两个连续变量的冲激函数定义为#xff1a; δ(t,z){1,tz00,others(4.52)\delta(t, z) …
目录二变量函数的傅里叶变换二维冲激及其取样性质二维连续傅里叶变换对二维取样和二维取样定理图像中的混叠二维离散傅里叶变换及其反变换二变量函数的傅里叶变换
二维冲激及其取样性质
两个连续变量的冲激函数定义为 δ(t,z){1,tz00,others(4.52)\delta(t, z) \begin{cases} 1, tz0 \\ 0, \text{others} \end{cases} \tag{4.52}δ(t,z){1,0,tz0others(4.52) ∫−∞∞∫−∞∞δ(t,z)dtdz(4.53)\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \delta(t, z) \text{d}t \text{d}z\tag{4.53}∫−∞∞∫−∞∞δ(t,z)dtdz(4.53)
二维冲激在积分下展现了取样性质 ∫−∞∞∫−∞∞f(t,z)δ(t,z)dtdzf(0,0)(4.54)\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(t,z) \delta(t, z) \text{d}t \text{d}z\ f(0, 0) \tag{4.54}∫−∞∞∫−∞∞f(t,z)δ(t,z)dtdz f(0,0)(4.54) 一般的情况 ∫−∞∞∫−∞∞f(t,z)δ(t−t0,z−z0)dtdzf(t0,z0)(4.55)\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(t,z) \delta(t - t_0, z - z_0) \text{d}t \text{d}z\ f(t_0, z_0) \tag{4.55}∫−∞∞∫−∞∞f(t,z)δ(t−t0,z−z0)dtdz f(t0,z0)(4.55)
二维离散单位冲激定义为 δ(x,y){1,xy00,others(4.56)\delta(x, y) \begin{cases} 1, xy0 \\ 0, \text{others} \end{cases} \tag{4.56}δ(x,y){1,0,xy0others(4.56) 取样性质为 ∑−∞∞∑−∞∞f(x,y)δ(x−x0,y−y0)dxdyf(x0,y0)(4.58)\sum_{-\infty}^{\infty} \sum_{-\infty}^{\infty} f(x,y) \delta(x - x_0, y - y_0) \text{d}x \text{d}y\ f(x_0, y_0) \tag{4.58}−∞∑∞−∞∑∞f(x,y)δ(x−x0,y−y0)dxdy f(x0,y0)(4.58)
二维连续傅里叶变换对
F(μ,v)∫−∞∞∫−∞∞f(t,z)e−j2π(μtvz)dtdz(4.59)F(\mu, v) \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(t, z) e^{-j2\pi(\mu t vz)} \text{d}t \text{d}z\tag{4.59}F(μ,v)∫−∞∞∫−∞∞f(t,z)e−j2π(μtvz)dtdz(4.59)
f(t,z)∫−∞∞∫−∞∞F(μ,v)ej2π(μtvz)dμdv(4.60)f(t, z) \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} F(\mu, v) e^{j2\pi(\mu t vz)} \text{d}\mu \text{d}v\tag{4.60}f(t,z)∫−∞∞∫−∞∞F(μ,v)ej2π(μtvz)dμdv(4.60)
μ\muμ和vvv是频率变量涉及图像时ttt和zzz解释为连续空间变量。变量μ\muμ和vvv的域定义了连续频率域
# 二维盒式函数的傅里叶变换
height, width 128, 128
m int((height - 1) / 2)
n int((width - 1) / 2)
f np.zeros([height, width])
# T 控制方格的大小
T 5
f[m-T:mT, n-T:nT] 1fft np.fft.fft2(f)
shift_fft np.fft.fftshift(fft)
amp np.log(1 np.abs(shift_fft))plt.figure(figsize(16, 16))
plt.subplot(2, 2, 1), plt.imshow(f, gray), plt.title(Box filter), plt.xticks([]), plt.yticks([])
plt.subplot(2, 2, 2), plt.imshow(amp, gray), plt.title(FFT Spectrum), plt.xticks([]), plt.yticks([])# 不同的盒式函数对应的傅里叶变换
height, width 128, 128
m int((height - 1) / 2)
n int((width - 1) / 2)
f np.zeros([height, width])
T 20
f[m-T:mT, n-T:nT] 1fft np.fft.fft2(f)
shift_fft np.fft.fftshift(fft)
amp np.abs(shift_fft)
plt.subplot(2, 2, 3), plt.imshow(f, gray), plt.title(Box filter), plt.xticks([]), plt.yticks([])
plt.subplot(2, 2, 4), plt.imshow(amp, gray), plt.title(FFT Spectrum), plt.xticks([]), plt.yticks([])
plt.tight_layout()
plt.show()二维取样和二维取样定理
sΔTΔZ(t,z)∑m−∞∞∑n−∞∞δ(t−mΔT,z−nΔZ)(4.61)s_{\Delta T \Delta Z}(t, z) \sum_{m-\infty}^{\infty} \sum_{n-\infty}^{\infty} \delta(t - m \Delta T, z - n \Delta Z) \tag{4.61}sΔTΔZ(t,z)m−∞∑∞n−∞∑∞δ(t−mΔT,z−nΔZ)(4.61)
在区间[−μmax,μmax][-\mu_{max}, \mu_{max}][−μmax,μmax]和[−vmax,vmax][-v_{max}, v_{max}][−vmax,vmax]建立的频率域矩形之外函数f(t,z)f(t,z)f(t,z)的傅里叶变换为零即时 F(μ,v)0,∣μ∣≥μmax且∣v∣≥vmax(4.62)F(\mu, v) 0, \quad |\mu| \ge \mu_{max} 且|v| \ge v_{max} \tag{4.62}F(μ,v)0,∣μ∣≥μmax且∣v∣≥vmax(4.62) 称该函数为带限函数。
二维取样定理 ΔT12μmax(4.63)\Delta T \frac{1}{2\mu_{max}} \tag{4.63}ΔT2μmax1(4.63) 和 ΔZ12vmax(4.64)\Delta Z \frac{1}{2 v_{max}} \tag{4.64}ΔZ2vmax1(4.64)
或者是 1ΔT2μmax(4.65)\frac{1} {\Delta T} {2\mu_{max}} \tag{4.65}ΔT12μmax(4.65) 和 1ΔZ2vmax(4.66)\frac{1} {\Delta Z} {2 v_{max}} \tag{4.66}ΔZ12vmax(4.66)
则连续带限函数f(t,z)f(t,z)f(t,z)可由基一组样本无误地复原。
图像中的混叠
def get_check(height, width, check_size(5, 5), lower130, upper255):create check pattern imageheight: input, height of the image you wantwidth: input, width of the image you wantcheck_size: the check size you want, default is 5x5lower: dark color of the check, default is 130, which is dark gray, 0 is black, 255 is whiteupper: light color of the check, default is 255, which is white, 0 is blackreturn uint8[0, 255] grayscale check pattern imagem, n check_sizeblack np.zeros((m, n), np.uint8)white np.zeros((m, n), np.uint8)black[:] lower # darkwhite[:] upper # whiteblack_white np.concatenate([black, white], axis1)white_black np.concatenate([white, black], axis1)black_white_black_white np.vstack((black_white, white_black))tile_times_h int(np.ceil(height / m / 2))tile_times_w int(np.ceil(width / n / 2))img_temp np.tile(black_white_black_white, (tile_times_h, tile_times_w))img_dst np.zeros([height, width])img_dst img_temp[:height, :width]return img_dst# 混叠
img_16 get_check(512, 800, check_size(16, 16), lower10, upper255)
img_16_show img_16[:48, :96]
img_6 get_check(512, 800, check_size(6, 6), lower10, upper255)
img_6_show img_6[:48, :96]# 16 * 0.95 15.2
img_095 img_16[::15, ::15]
img_095 np.concatenate((img_095, img_095[:, 3:]), axis1)
img_095 np.concatenate((img_095, img_095[:, 3:]), axis1)
img_095 np.concatenate((img_095, img_095[:, 3:]), axis1)
img_095 np.concatenate((img_095, img_095[3:, :]), axis0)
img_095 np.concatenate((img_095, img_095[3:, :]), axis0)
img_095 img_095[2:50, 2:98]# 16 * 0.48 7.68
img_05 img_16[::2, ::2] # 为了显示这里的步长选了2
img_05 img_05[:48, :96]fig plt.figure(figsize(15, 8))
plt.subplot(2, 2, 1), plt.imshow(img_16_show, gray, vmin0, vmax255), plt.xticks([]), plt.yticks([])
plt.subplot(2, 2, 2), plt.imshow(img_6_show, gray, vmin0, vmax255), plt.xticks([]), plt.yticks([])
plt.subplot(2, 2, 3), plt.imshow(img_095, gray, vmin0, vmax255), plt.xticks([]), plt.yticks([])
plt.subplot(2, 2, 4), plt.imshow(img_05, gray, vmin0, vmax255), plt.xticks([]), plt.yticks([])
plt.tight_layout()
plt.show()图像重取样和刚插
在图像被取样之前必须在前端进行搞混叠滤波但对于违反取样定理导致的混叠效应事实上不存在事后能降低它的抗混叠滤波的软件。多数抗混叠的功能主要是模糊数字图像进行降低由重取样导致的其他混叠伪影而不能降低原取样图像中的混叠。
对数字图像降低混叠在重取样之前 需要使用低通滤波器来平滑以衰减数字图像的高频分量。但实际上只是平滑图像而减少了那些令人讨厌的混叠现象。
def nearest_neighbor_interpolation(img, new_h, new_w):get nearest_neighbor_interpolation for image, can up or down scale image into any ratioparam: img: input image, grady image, 1 channel, shape like [512, 512]param: new_h: new image height param: new_w: new image widthreturn a nearest_neighbor_interpolation up or down scale imagenew_img np.zeros([new_h, new_w])src_height, src_width img.shape[:2]r new_h / src_heightl new_w / src_widthfor i in range(new_h):for j in range(new_w):x0 int(i / r)y0 int(j / l)new_img[i, j] img[x0, y0]return new_imgdef box_filter(image, kernel)::param image: input image:param kernel: input kernel:return: image after convolutionimg_h image.shape[0]img_w image.shape[1]m kernel.shape[0]n kernel.shape[1]# paddingpadding_h int((m -1)/2)padding_w int((n -1)/2)image_pad np.zeros((image.shape[0]padding_h*2, image.shape[1]padding_w*2), np.uint8)image_pad[padding_h:padding_himg_h, padding_w:padding_wimg_w] imageimage_convol image.copy()for i in range(padding_h, img_h padding_h):for j in range(padding_w, img_w padding_w):temp np.sum(image_pad[i-padding_h:ipadding_h1, j-padding_w:jpadding_w1] * kernel)image_convol[i - padding_h][j - padding_w] temp # 1/(m * n) * tempreturn image_convol# 抗混叠
img_ori cv2.imread(DIP_Figures/DIP3E_Original_Images_CH04/Fig0417(a)(barbara).tif, 0)# 先缩小图像再放大图像
img_down nearest_neighbor_interpolation(img_ori, int(img_ori.shape[0]*0.33), int(img_ori.shape[1]*0.33))
img_up nearest_neighbor_interpolation(img_down, img_ori.shape[0], img_ori.shape[1])# 先对原图像进行5x5的平均滤波再缩小图像再放大图像
kernel np.ones([5, 5])
kernel kernel / kernel.size
img_box_filter box_filter(img_ori, kernelkernel)
img_down_1 nearest_neighbor_interpolation(img_box_filter, int(img_ori.shape[0]*0.33), int(img_ori.shape[1]*0.33))
img_up_1 nearest_neighbor_interpolation(img_down_1, img_ori.shape[0], img_ori.shape[1])fig plt.figure(figsize(15, 10))
plt.subplot(1, 3, 1), plt.imshow(img_ori, gray), plt.xticks([]), plt.yticks([])
plt.subplot(1, 3, 2), plt.imshow(img_up, gray), plt.xticks([]), plt.yticks([])
plt.subplot(1, 3, 3), plt.imshow(img_up_1, gray), plt.xticks([]), plt.yticks([])plt.tight_layout()
plt.show()混叠和莫尔模式
莫尔模式是由近似等间隔的两个光栅叠加所产生的一种视觉现象。
def rotate_image(img, angle45):height, width img.shape[:2]if img.ndim 3:channel 3else:channel Noneif int(angle / 90) % 2 0:reshape_angle angle % 90else:reshape_angle 90 - (angle % 90)reshape_radian np.radians(reshape_angle) # 角度转弧度# 三角函数计算出来的结果会有小数所以做了向上取整的操作。new_height int(np.ceil(height * np.cos(reshape_radian) width * np.sin(reshape_radian)))new_width int(np.ceil(width * np.cos(reshape_radian) height * np.sin(reshape_radian)))if channel:new_img np.zeros((new_height, new_width, channel), dtypenp.uint8)else:new_img np.zeros((new_height, new_width), dtypenp.uint8)radian np.radians(angle)cos_radian np.cos(radian)sin_radian np.sin(radian)# dx 0.5 * new_width 0.5 * height * sin_radian - 0.5 * width * cos_radian# dy 0.5 * new_height - 0.5 * width * sin_radian - 0.5 * height * cos_radian# ---------------前向映射--------------------# for y0 in range(height):# for x0 in range(width):# x x0 * cos_radian - y0 * sin_radian dx# y x0 * sin_radian y0 * cos_radian dy# new_img[int(y) - 1, int(x) - 1] img[int(y0), int(x0)] # 因为整体映射的结果会比偏移一个单位所以这里x,y做减一操作。# ---------------后向映射--------------------dx_back 0.5 * width - 0.5 * new_width * cos_radian - 0.5 * new_height * sin_radiandy_back 0.5 * height 0.5 * new_width * sin_radian - 0.5 * new_height * cos_radianfor y in range(new_height):for x in range(new_width):x0 x * cos_radian y * sin_radian dx_backy0 y * cos_radian - x * sin_radian dy_backif 0 int(x0) width and 0 int(y0) height: # 计算结果是这一范围内的x0y0才是原始图像的坐标。new_img[int(y), int(x)] img[int(y0) - 1, int(x0) - 1] # 因为计算的结果会有偏移所以这里做减一操作。# # ---------------双线性插值--------------------
# if channel:
# fill_height np.zeros((height, 2, channel), dtypenp.uint8)
# fill_width np.zeros((2, width 2, channel), dtypenp.uint8)
# else:
# fill_height np.zeros((height, 2), dtypenp.uint8)
# fill_width np.zeros((2, width 2), dtypenp.uint8)
# img_copy img.copy()
# # 因为双线性插值需要得到x1y1位置的像素映射的结果如果在最边缘的话会发生溢出所以给图像的右边和下面再填充像素。
# img_copy np.concatenate((img_copy, fill_height), axis1)
# img_copy np.concatenate((img_copy, fill_width), axis0)
# for y in range(new_height):
# for x in range(new_width):
# x0 x * cos_radian y * sin_radian dx_back
# y0 y * cos_radian - x * sin_radian dy_back
# x_low, y_low int(x0), int(y0)
# x_up, y_up x_low 1, y_low 1
# u, v np.modf(x0)[0], np.modf(y0)[0] # 求x0和y0的小数部分
# x1, y1 x_low, y_low
# x2, y2 x_up, y_low
# x3, y3 x_low, y_up
# x4, y4 x_up, y_up
# if 0 int(x0) width and 0 int(y0) height:
# pixel (1 - u) * (1 - v) * img_copy[y1, x1] (1 - u) * v * img_copy[y2, x2] u * (1 - v) * img_copy[y3, x3] u * v * img_copy[y4, x4] # 双线性插值法求像素值。
# new_img[int(y), int(x)] pixelreturn new_img# 混叠和莫尔模式
# 竖线原图
img_lines np.ones([129, 129]) * 255
img_lines[:, ::3] 0# 旋转时使用刚内插产生了混叠
rotate_matrix cv2.getRotationMatrix2D((int(img_lines.shape[0]*0.5), int(img_lines.shape[1]*0.5)), -5, 1)
img_lines_r cv2.warpAffine(img_lines, rotate_matrix, dsize(img_lines.shape[0], img_lines.shape[1]), flagscv2.INTER_CUBIC, borderValue255)# 相加后产生的混叠更明显
img_add img_lines img_lines_r
img_add np.uint8(normalize(img_add) * 255)fig plt.figure(figsize(15, 10))
plt.subplot(2, 3, 1), plt.imshow(img_lines, gray, vmin0, vmax255), plt.xticks([]), plt.yticks([])
plt.subplot(2, 3, 2), plt.imshow(img_lines_r, gray, vmin0, vmax255), plt.xticks([]), plt.yticks([])
plt.subplot(2, 3, 3), plt.imshow(img_add, gray, vmin0, vmax255), plt.xticks([]), plt.yticks([])# 格子原图
img_check np.ones([129, 129]) * 255
img_check[::2, ::2] 0# 旋转时使用刚内插产生了混叠
rotate_matrix cv2.getRotationMatrix2D((int(img_check.shape[0]*0.5), int(img_check.shape[1]*0.5)), -5, 1)
img_check_r cv2.warpAffine(img_check, rotate_matrix, dsize(img_check.shape[0], img_check.shape[1]), flagscv2.INTER_CUBIC, borderValue255)# 相加后产生的混叠更明显
img_add img_check img_check_r
img_add np.uint8(normalize(img_add) * 255)plt.subplot(2, 3, 4), plt.imshow(img_check, gray, vmin0, vmax255), plt.xticks([]), plt.yticks([])
plt.subplot(2, 3, 5), plt.imshow(img_check_r, gray, vmin0, vmax255), plt.xticks([]), plt.yticks([])
plt.subplot(2, 3, 6), plt.imshow(img_add, gray, vmin0, vmax255), plt.xticks([]), plt.yticks([])plt.tight_layout()
plt.show()# 印刷采用欠取样时产生莫尔模式效应
img_ori cv2.imread(DIP_Figures/DIP3E_Original_Images_CH04/Fig0421(car_newsprint_sampled_at_75DPI).tif, 0)fig plt.figure(figsize(15, 10))
plt.subplot(1, 3, 1), plt.imshow(img_ori, gray), plt.xticks([]), plt.yticks([])plt.tight_layout()
plt.show()二维离散傅里叶变换及其反变换
二维DFT Fu,v∑x0M−1∑y0N−1f(x,y)e−j2π(ux/Mvy/N)(4.67)F_{u, v} \sum_{x 0}^{M - 1} \sum_{y 0}^{N - 1} f(x, y) e^{-j2\pi(u x/M v y /N)} \tag{4.67}Fu,vx0∑M−1y0∑N−1f(x,y)e−j2π(ux/Mvy/N)(4.67)
二维IDFT f(x,y)1MN∑u0M−1∑v0N−1F(u,v)ej2π(ux/Mvy/N)(4.68)f(x, y) \frac{1}{MN}\sum_{u 0}^{M - 1} \sum_{v 0}^{N - 1} F(u, v) e^{j2\pi(u x /M vy /N)} \tag{4.68}f(x,y)MN1u0∑M−1v0∑N−1F(u,v)ej2π(ux/Mvy/N)(4.68)
上式u0,1,2,⋯,M−1u 0, 1, 2, \cdots, M-1u0,1,2,⋯,M−1和v0,1,2,⋯,N−1v 0, 1, 2, \cdots, N-1v0,1,2,⋯,N−1x0,1,2,⋯,M−1x 0, 1, 2, \cdots, M-1x0,1,2,⋯,M−1和y0,1,2,⋯,N−1y 0, 1, 2, \cdots, N-1y0,1,2,⋯,N−1
