图像内插

内插通常在图像放大、缩小、旋转和几何校正等任务中使用。内插并用它来调整图像的大小（缩小和放大），缩小和放大基本上采用图像重取样方法

最近邻内插，这种方法将原图像中最近邻的灰度赋给了每个新位置，这种方法简单，但会产生我们不想要的人为失真，如严重的直边失真。更合适的方法是双线性内插，它使用4个最近邻的灰度来计算给定位置的灰度。令$(x,y)$表示待赋灰度值的位置（可将它相像为前面描述的网格点）的坐标，令$v(x,y)$表示灰度值。对于双线性内插方法，所赋的值由如下公式得到：
$$v(x,y)=ax+by+cxy+d\text{\hspace{1em}(2.17)}$$
4 个系数可由点(x, y)的4个最近邻点写出的4个未知方程求出。双线性内插的结果要比最近邻内插的结果好得多，但计算量会随之增大。

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

# 最近邻插值法处理一通道的图像
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()

# 最近邻插值法处理RGB 3通过的图像
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()

双线性插值

又称为双线性内挺。在数学上，双线性插值是有两个变量的插值函数的线性插值扩展，其核心思想是在两个方向分别进行一次线性插值。

假设我们想得到未知函数$f在点P=(x,y)$的值，假设我们已知函数$f在Q_{11}=(x_1,y_1)，Q_{12}=(x_1,y_2),Q_{21}=(x_2,y_1),Q_{22}=(x_2,y_2)$四个点的值。首先在x方向进行线性插值，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方向进行线性插值，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):# 首先要找到在原图中对应点的(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

# 最近邻插值法处理一通道的图像
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()

双三次内插

双三次插值的目的就是通过找到一种关系，或者说系数，可以把这16个像素对于P处像素值的影响因子找出来，从而根据这个影响因子来获得目标图像对应点的像素值，达到图像缩放的目的。

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

$a=-1$
注 $a=-0.5$才能完美的实现内插

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

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

# 最近邻插值法处理一通道的图像
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()

# 先裁剪图像，然后把裁剪的图像缩小，再进行最近邻、双线内插、双三次内插放大，对比效果
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()

放大图像

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()