目錄

• • 二變量函數的傅里葉變換
  • • 二維沖激及其取樣性質
    • 二維連續傅里葉變換對
    • 二維取樣和二維取樣定理
    • 圖像中的混疊
    • 二維離散傅里葉變換及其反變換

二變量函數的傅里葉變換

二維沖激及其取樣性質

兩個連續變量的沖激函數定義為：
$$\delta (t,z)=\{\begin{array}{ll}1,& t=z=0\\ 0,& \text{others}\end{array}\text{\hspace{1em}(4.52)}$$
$${\int }_{?\infty }^{\infty }{\int }_{?\infty }^{\infty }\delta (t,z)\text{d}t\text{d}z\text{\hspace{1em}(4.53)}$$

二維沖激在積分下展現了取樣性質
$${\int }_{?\infty }^{\infty }{\int }_{?\infty }^{\infty }f(t,z)\delta (t,z)\text{d}t\text{d}z=f(0,0)\text{\hspace{1em}(4.54)}$$
一般的情況
$${\int }_{?\infty }^{\infty }{\int }_{?\infty }^{\infty }f(t,z)\delta (t?t_0,z?z_0)\text{d}t\text{d}z=f(t_0,z_0)\text{\hspace{1em}(4.55)}$$

二維離散單位沖激定義為
$$\delta (x,y)=\{\begin{array}{ll}1,& x=y=0\\ 0,& \text{others}\end{array}\text{\hspace{1em}(4.56)}$$
取樣性質為
$$\sum _{?\infty }^{\infty }\sum _{?\infty }^{\infty }f(x,y)\delta (x?x_0,y?y_0)\text{d}x\text{d}y=f(x_0,y_0)\text{\hspace{1em}(4.58)}$$

二維連續傅里葉變換對

$$F(\mu ,v)={\int }_{?\infty }^{\infty }{\int }_{?\infty }^{\infty }f(t,z)e^{?j2\pi (\mu t+vz)}\text{d}t\text{d}z\text{\hspace{1em}(4.59)}$$

$$f(t,z)={\int }_{?\infty }^{\infty }{\int }_{?\infty }^{\infty }F(\mu ,v)e^{j2\pi (\mu t+vz)}\text{d}\mu \text{d}v\text{\hspace{1em}(4.60)}$$

$\mu$和$v$是頻率變量，涉及圖像時，$t$和$z$解釋為連續空間變量。變量$\mu$和$v$的域定義了連續頻率域

# 二維盒式函數的傅里葉變換 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:m+T, n-T:n+T] = 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([])# 不同的盒式函數對應的傅里葉變換 height, width = 128, 128 m = int((height - 1) / 2) n = int((width - 1) / 2) f = np.zeros([height, width]) T = 20 f[m-T:m+T, n-T:n+T] = 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()

二維取樣和二維取樣定理

$$s_{\Delta T\Delta Z}(t,z)=\sum _{m=?\infty }^{\infty }\sum _{n=?\infty }^{\infty }\delta (t?m\Delta T,z?n\Delta Z)\text{\hspace{1em}(4.61)}$$

在區間$[?{\mu }_{max},{\mu }_{max}]$和$[?v_{max},v_{max}]$建立的頻率域矩形之外，函數$f(t,z)$的傅里葉變換為零，即時，
$$F(\mu ,v)=0,|\mu |\ge {\mu }_{max}且|v|\ge v_{max}\text{\hspace{1em}(4.62)}$$
稱該函數為帶限函數。

二維取樣定理：
$$\Delta T<\frac{1}{2{\mu }_{max}}\text{\hspace{1em}(4.63)}$$
和
$$\Delta Z<\frac{1}{2v_{max}}\text{\hspace{1em}(4.64)}$$

或者是：
$$\frac{1}{\Delta T}>2{\mu }_{max}\text{\hspace{1em}(4.65)}$$
和
$$\frac{1}{\Delta Z}>2v_{max}\text{\hspace{1em}(4.66)}$$

則連續帶限函數$f(t,z)$可由基一組樣本無誤地復原。

圖像中的混疊

def get_check(height, width, check_size=(5, 5), lower=130, upper=255):"""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 image"""m, 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], axis=1)white_black = np.concatenate([white, black], axis=1)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 # 混疊 img_16 = get_check(512, 800, check_size=(16, 16), lower=10, upper=255) img_16_show = img_16[:48, :96] img_6 = get_check(512, 800, check_size=(6, 6), lower=10, upper=255) 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:]), axis=1) img_095 = np.concatenate((img_095, img_095[:, 3:]), axis=1) img_095 = np.concatenate((img_095, img_095[:, 3:]), axis=1) img_095 = np.concatenate((img_095, img_095[3:, :]), axis=0) img_095 = np.concatenate((img_095, img_095[3:, :]), axis=0) img_095 = img_095[2:50, 2:98]# 16 * 0.48 = 7.68 img_05 = img_16[::2, ::2] # 為了顯示這里的步長選了2 img_05 = img_05[:48, :96]fig = plt.figure(figsize=(15, 8)) plt.subplot(2, 2, 1), plt.imshow(img_16_show, 'gray', vmin=0, vmax=255), plt.xticks([]), plt.yticks([]) plt.subplot(2, 2, 2), plt.imshow(img_6_show, 'gray', vmin=0, vmax=255), plt.xticks([]), plt.yticks([]) plt.subplot(2, 2, 3), plt.imshow(img_095, 'gray', vmin=0, vmax=255), plt.xticks([]), plt.yticks([]) plt.subplot(2, 2, 4), plt.imshow(img_05, 'gray', vmin=0, vmax=255), plt.xticks([]), plt.yticks([]) plt.tight_layout() plt.show()

圖像重取樣和剛插

在圖像被取樣之前，必須在前端進行搞混疊濾波，但對于違反取樣定理導致的混疊效應，事實上不存在事后能降低它的抗混疊濾波的軟件。多數抗混疊的功能主要是模糊數字圖像，進行降低由重取樣導致的其他混疊偽影，而不能降低原取樣圖像中的混疊。

對數字圖像降低混疊，在重取樣之前 需要使用低通濾波器來平滑，以衰減數字圖像的高頻分量。但實際上只是平滑圖像，而減少了那些令人討厭的混疊現象。

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 image"""new_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_img def box_filter(image, kernel):""":param image: input image:param kernel: input kernel:return: image after convolution"""img_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_h+img_h, padding_w:padding_w+img_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:i+padding_h+1, j-padding_w:j+padding_w+1] * kernel)image_convol[i - padding_h][j - padding_w] = temp # 1/(m * n) * tempreturn image_convol # 抗混疊 img_ori = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH04/Fig0417(a)(barbara).tif', 0)# 先縮小圖像，再放大圖像 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])# 先對原圖像進行5x5的平均濾波，再縮小圖像，再放大圖像 kernel = np.ones([5, 5]) kernel = kernel / kernel.size img_box_filter = box_filter(img_ori, kernel=kernel) 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()

混疊和莫爾模式

莫爾模式是由近似等間隔的兩個光柵疊加所產生的一種視覺現象。

def rotate_image(img, angle=45):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) # 角度轉弧度# 三角函數計算出來的結果會有小數，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), dtype=np.uint8)else:new_img = np.zeros((new_height, new_width), dtype=np.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)] # 因為整體映射的結果會比偏移一個單位，所以這里x,y做減一操作。# ---------------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: # 計算結果是這一范圍內的x0，y0才是原始圖像的坐標。new_img[int(y), int(x)] = img[int(y0) - 1, int(x0) - 1] # 因為計算的結果會有偏移，所以這里做減一操作。# # ---------------雙線性插值-------------------- # if channel: # fill_height = np.zeros((height, 2, channel), dtype=np.uint8) # fill_width = np.zeros((2, width + 2, channel), dtype=np.uint8) # else: # fill_height = np.zeros((height, 2), dtype=np.uint8) # fill_width = np.zeros((2, width + 2), dtype=np.uint8) # img_copy = img.copy() # # 因為雙線性插值需要得到x+1，y+1位置的像素，映射的結果如果在最邊緣的話會發生溢出，所以給圖像的右邊和下面再填充像素。 # img_copy = np.concatenate((img_copy, fill_height), axis=1) # img_copy = np.concatenate((img_copy, fill_width), axis=0) # 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的小數部分 # 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] # 雙線性插值法，求像素值。 # new_img[int(y), int(x)] = pixelreturn new_img # 混疊和莫爾模式 # 豎線原圖 img_lines = np.ones([129, 129]) * 255 img_lines[:, ::3] = 0# 旋轉時使用剛內插，產生了混疊 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]), flags=cv2.INTER_CUBIC, borderValue=255)# 相加后，產生的混疊更明顯 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', vmin=0, vmax=255), plt.xticks([]), plt.yticks([]) plt.subplot(2, 3, 2), plt.imshow(img_lines_r, 'gray', vmin=0, vmax=255), plt.xticks([]), plt.yticks([]) plt.subplot(2, 3, 3), plt.imshow(img_add, 'gray', vmin=0, vmax=255), plt.xticks([]), plt.yticks([])# 格子原圖 img_check = np.ones([129, 129]) * 255 img_check[::2, ::2] = 0# 旋轉時使用剛內插，產生了混疊 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]), flags=cv2.INTER_CUBIC, borderValue=255)# 相加后，產生的混疊更明顯 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', vmin=0, vmax=255), plt.xticks([]), plt.yticks([]) plt.subplot(2, 3, 5), plt.imshow(img_check_r, 'gray', vmin=0, vmax=255), plt.xticks([]), plt.yticks([]) plt.subplot(2, 3, 6), plt.imshow(img_add, 'gray', vmin=0, vmax=255), plt.xticks([]), plt.yticks([])plt.tight_layout() plt.show()

# 印刷采用欠取樣時產生莫爾模式效應 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()

二維離散傅里葉變換及其反變換

二維DFT
$$F_{u,v}=\sum _{x=0}^{M?1}\sum _{y=0}^{N?1}f(x,y)e^{?j2\pi (ux/M+vy/N)}\text{\hspace{1em}(4.67)}$$

二維IDFT
$$f(x,y)=\frac{1}{MN}\sum _{u=0}^{M?1}\sum _{v=0}^{N?1}F(u,v)e^{j2\pi (ux/M+vy/N)}\text{\hspace{1em}(4.68)}$$

上式，$u=0,1,2,?,M?1$和$v=0,1,2,?,N?1$，$x=0,1,2,?,M?1$和$y=0,1,2,?,N?1$

總結

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