目錄

• • • 圖像內插
    • • 放大圖像

圖像內插

內插通常在圖像放大、縮小、旋轉和幾何校正等任務中使用。內插并用它來調整圖像的大小（縮小和放大），縮小和放大基本上采用圖像重取樣方法

最近鄰內插，這種方法將原圖像中最近鄰的灰度賦給了每個新位置，這種方法簡單，但會產生我們不想要的人為失真，如嚴重的直邊失真。更合適的方法是雙線性內插，它使用4個最近鄰的灰度來計算給定位置的灰度。令$(x,y)$表示待賦灰度值的位置（可將它相像為前面描述的網格點）的坐標，令$v(x,y)$表示灰度值。對于雙線性內插方法，所賦的值由如下公式得到：
$$v(x,y)=ax+by+cxy+d\text{\hspace{1em}(2.17)}$$
4 個系數可由點(x, y)的4個最近鄰點寫出的4個未知方程求出。雙線性內插的結果要比最近鄰內插的結果好得多，但計算量會隨之增大。

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 # 最近鄰插值法處理一通道的圖像 img = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH06/Fig0638(a)(lenna_RGB).tif', 0)img_up = nearest_neighbor_interpolation(img, 1000, 1000)plt.figure(figsize=(16, 8)) plt.subplot(121), plt.imshow(img, 'gray') plt.subplot(122), plt.imshow(img_up, 'gray') plt.tight_layout() plt.show()

# 最近鄰插值法處理RGB 3通過的圖像 img = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH06/Fig0638(a)(lenna_RGB).tif', -1) img = img[..., ::-1] img_up_1 = nearest_neighbor_interpolation(img[..., 0], 800, 800) img_up_2 = nearest_neighbor_interpolation(img[..., 1], 800, 800) img_up_3 = nearest_neighbor_interpolation(img[..., 2], 800, 800) img_up = np.uint8(np.dstack((img_up_1, img_up_2, img_up_3)))plt.figure(figsize=(16, 8)) plt.subplot(121), plt.imshow(img, 'gray') plt.subplot(122), plt.imshow(img_up, 'gray') plt.tight_layout() plt.show()

雙線性插值

又稱為雙線性內挺。在數學上，雙線性插值是有兩個變量的插值函數的線性插值擴展，其核心思想是在兩個方向分別進行一次線性插值。

假設我們想得到未知函數$f在點P=(x,y)$的值，假設我們已知函數$f在Q_{11}=(x_1,y_1)，Q_{12}=(x_1,y_2),Q_{21}=(x_2,y_1),Q_{22}=(x_2,y_2)$四個點的值。首先在x方向進行線性插值，得到：

$$f(R_1)\approx \frac{x_2?x}{x_2?x_1}f(Q_{11})+\frac{x?x_1}{x_2?x_1}f(Q_{21}),where{R}_1=(x,y_1)$$
$$f(R_2)\approx \frac{x_2?x}{x_2?x_1}f(Q_{12})+\frac{x?x_1}{x_2?x_1}f(Q_{22}),where{R}_2=(x,y_2)$$

然后在y方向進行線性插值，得到
$$f(P)\approx \frac{y_2?y}{y_2?y_1}f(R_1)+\frac{y?y_1}{y_2?y_1}f(R_2)$$

def bilinear_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"""src_height, src_width = img.shape[:2]if new_h == src_height and new_w == src_width:return img.copy()new_img = np.zeros([new_h, new_w])for i in range(new_h):for j in range(new_w):# 首先要找到在原圖中對應點的(x, y)x = (i+0.5) * src_height / new_h - 0.5y = (j+0.5) * src_width / new_w - 0.5# find the coordinates of the points which will be used to compute the interpolationsrc_x0 = int(np.floor(x))src_x1 = min(src_x0 + 1 , src_height - 1)src_y0 = int(np.floor(y))src_y1 = min(src_y0 + 1, src_width - 1)# calculate the interpolationtemp0 = (src_x1 - x) * img[src_x0, src_y0] + (x - src_x0) * img[src_x1, src_y0]temp1 = (src_x1 - x) * img[src_x0, src_y1] + (x - src_x0) * img[src_x1, src_y1]new_img[i, j] = int((src_y1 - y) * temp0 + (y - src_y0) * temp1)return new_img # 最近鄰插值法處理一通道的圖像 img = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH06/Fig0638(a)(lenna_RGB).tif', 0)img_up = bilinear_interpolation(img, 800, 800)plt.figure(figsize=(16, 8)) plt.subplot(121), plt.imshow(img, 'gray') plt.subplot(122), plt.imshow(img_up, 'gray') plt.tight_layout() plt.show()

雙三次內插

雙三次插值的目的就是通過找到一種關系，或者說系數，可以把這16個像素對于P處像素值的影響因子找出來，從而根據這個影響因子來獲得目標圖像對應點的像素值，達到圖像縮放的目的。

Bicubic基函數形式如下：
$$W(x)=?\begin{array}{ll}(a+2)|x{|}^3?(a+3)|x{|}^2+1,& |x|\le 1\\ a|x{|}^3?5a|x{|}^2+8a|x|?4a,& 1<|x|<2\\ 0,& \text{otherwise}\end{array}$$

$a=?1$
注 $a=?0.5$才能完美的實現內插

插值計算公式：
$$\sum _{i=0}^3\sum _{j=0}^3{a}_{ij}{x}^i{y}^i$$

def bicubic(x):"""BiCubic primary fuction"""x = abs(x)a = -0.5if x <= 1:return (a + 2) * (x**3) - (a + 3) * (x**2) + 1elif x < 2:return a * (x**3) - 5 * a * (x**2) + (8 * a * x) - (4 * a)else:return 0 def bicubic_interpolation(img, new_h, new_w):src_height, src_width = img.shape[:2]new_img = np.zeros([new_h, new_w])for h in range(new_h):for w in range(new_w):src_x = h * (src_height / new_h)src_y = w * (src_width / new_w)x = int(np.floor(src_x))y = int(np.floor(src_y))u = src_x - xv = src_y - ytemp = 0for i in range(-1, 2):for j in range(-1, 2):if x + i < 0 or y + j < 0 or x + i >= src_height or y + j >= src_width:continuetemp += img[x+i, y+j] * bicubic(i - u) * bicubic(j - v)new_img[h, w] = np.clip(temp, 0, 255)return new_img # 最近鄰插值法處理一通道的圖像 img = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH06/Fig0638(a)(lenna_RGB).tif', 0)img_up = bicubic_interpolation(img, 800, 800) # img_up = cv2.resize(img, (800, 800), cv2.INTER_CUBIC)plt.figure(figsize=(16, 8)) plt.subplot(121), plt.imshow(img, 'gray') plt.subplot(122), plt.imshow(img_up, 'gray') plt.tight_layout() plt.show()

# 先裁剪圖像，然后把裁剪的圖像縮小，再進行最近鄰、雙線內插、雙三次內插放大，對比效果 img = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH02/Fig0220(a)(chronometer 3692x2812 2pt25 inch 1250 dpi).tif', 0)img = img[1100:3500, 200:2600] img = cv2.resize(img, (200, 200), interpolation=cv2.INTER_CUBIC)new_h, new_w = 2400, 2400 img_nearest = nearest_neighbor_interpolation(img, new_h, new_w) img_bilinear = bilinear_interpolation(img, new_h, new_w) img_bicubic = bicubic_interpolation(img, new_h, new_w)plt.figure(figsize=(18, 6)) plt.subplot(131), plt.imshow(img_nearest, 'gray'), plt.title('Nearest'), #plt.xticks([]), plt.yticks([]) plt.subplot(132), plt.imshow(img_bilinear, 'gray'), plt.title('Bilinear'), #plt.xticks([]), plt.yticks([]) plt.subplot(133), plt.imshow(img_bicubic, 'gray'), plt.title('Bicubic'), #plt.xticks([]), plt.yticks([]) plt.tight_layout() plt.show()

放大圖像

def up_sample_2d(img):"""up sample 2d image, if your image is RGB, you could up sample each channel, then use np.dstack to merge a RGB imageparam: input img: it's a 2d gray imagereturn a 2x up sample image"""height, width = img.shape[:2]temp = np.zeros([height*2, width*2])temp[::2, ::2] = imgtemp[1::2, 1::2] = imgtemp[0::2, 1::2] = imgtemp[1::2, 0::2] = imgreturn temp # up sample image using Numpy img = np.random.random([12, 12]) up = up_sample_2d(img) down = np.zeros([12, 12]) down = up[::2, ::2] plt.figure(figsize=(15, 5)) plt.subplot(1,3,1), plt.imshow(img),# plt.xticks([]), plt.yticks([]) plt.subplot(1,3,2), plt.imshow(up),# plt.xticks([]), plt.yticks([]) plt.subplot(1,3,3), plt.imshow(down),# plt.xticks([]), plt.yticks([]) plt.show()

# 等比2倍放大 img_ori = cv2.imread("DIP_Figures/DIP3E_Original_Images_CH06/Fig0638(a)(lenna_RGB).tif") img_ori = img_ori[:, :, ::-1]temp = [] for i in range(img_ori.shape[-1]):temp.append(up_sample_2d(img_ori[:, :, i])) up1 = np.uint8(np.dstack(temp))temp = [] for i in range(up1.shape[-1]):temp.append(up_sample_2d(up1[:, :, i])) up2 = np.uint8(np.dstack(temp))plt.figure(figsize=(21, 7)) plt.subplot(1,3,1), plt.imshow(img_ori), plt.title("Original"),# plt.xticks([]), plt.yticks([]) plt.subplot(1,3,2), plt.imshow(up1), plt.title("2X"),# plt.xticks([]), plt.yticks([]) plt.subplot(1,3,3), plt.imshow(up2), plt.title("4X"),# plt.xticks([]), plt.yticks([]) plt.tight_layout() plt.show()

總結

以上是生活随笔為你收集整理的第2章 Python 数字图像处理(DIP) --数字图像基础3 - 图像内插 - 最近邻内插 - 双线性插值 - 双三次内插 - 图像放大的全部內容，希望文章能夠幫你解決所遇到的問題。

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