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提示：内容整理自：https://github.com/gzr2017/ImageProcessing100Wen


CV小白从0开始学数字图像处理

27 双三次插值（ Bicubic Interpolation ）

使用双三次插值将图像放大1.5倍

双三次插值是双线性插值的扩展，使用邻域16像素进行插值。

I(x-1,y-1)  I(x,y-1)  I(x+1,y-1)  I(x+2,y-1)
I(x-1,y)    I(x,y)    I(x+1,y)    I(x+2,y)
I(x-1,y+1)  I(x,y+1)  I(x+1,y+1)  I(x+2,y+1)
I(x-1,y+2)  I(x,y+2)  I(x+1,y+2)  I(x+2,y+2)

各自像素间的距离由下式决定：

dx1 = x'/a - (x-1) , dx2 = x'/a - x , dx3 = (x+1) - x'/a , dx4 = (x+2) - x'/a
dy1 = y'/a - (y-1) , dy2 = y'/a - y , dy3 = (y+1) - y'/a , dy4 = (y+2) - y'/a

基于距离的权重函数由以下函数取得，a在大部分时候取-1：

h(t) = { (a+2)|t|^3 - (a+3)|t|^2 + 1    (when |t|<=1)
         a|t|^3 - 5a|t|^2 + 8a|t| - 4a  (when 1<|t|<=2)
         0                              (when 2<|t|) 

利用上面得到的权重，通过下面的式子扩大图像。将每个像素与权重的乘积之和除以权重的和。

I'(x', y') = (Sum{i=-1:2}{j=-1:2} I(x+i,y+j) * wxi * wyj) / Sum{i=-1:2}{j=-1:2} wxi * wyj

代码如下：

1.引入库

CV2计算机视觉库

import cv2
import numpy as np
import matplotlib.pyplot as plt

2.读入数据

img = cv2.imread("imori.jpg").astype(np.float32)
H, W, C = img.shape

3.双三次插值

a = 1.5
aH = int(a * H)
aW = int(a * W)

y = np.arange(aH).repeat(aW).reshape(aW, -1)
x = np.tile(np.arange(aW), (aH, 1))
y = (y / a)
x = (x / a)

ix = np.floor(x).astype(np.int)
iy = np.floor(y).astype(np.int)

ix = np.minimum(ix, W-1)
iy = np.minimum(iy, H-1)

dx2 = x - ix
dy2 = y - iy
dx1 = dx2 + 1
dy1 = dy2 + 1
dx3 = 1 - dx2
dy3 = 1 - dy2
dx4 = 1 + dx3
dy4 = 1 + dy3

dxs = [dx1, dx2, dx3, dx4]
dys = [dy1, dy2, dy3, dy4]

def weight(t):
    a = -1.
    at = np.abs(t)
    w = np.zeros_like(t)
    ind = np.where(at <= 1)
    w[ind] = ((a+2) * np.power(at, 3) - (a+3) * np.power(at, 2) + 1)[ind]
    ind = np.where((at > 1) & (at <= 2))
    w[ind] = (a*np.power(at, 3) - 5*a*np.power(at, 2) + 8*a*at - 4*a)[ind]
    return w

w_sum = np.zeros((aH, aW, C), dtype=np.float32)
out = np.zeros((aH, aW, C), dtype=np.float32)

for j in range(-1, 3):
    for i in range(-1, 3):
        ind_x = np.minimum(np.maximum(ix + i, 0), W-1)
        ind_y = np.minimum(np.maximum(iy + j, 0), H-1)

        wx = weight(dxs[i+1])
        wy = weight(dys[j+1])
        wx = np.repeat(np.expand_dims(wx, axis=-1), 3, axis=-1)
        wy = np.repeat(np.expand_dims(wy, axis=-1), 3, axis=-1)

        w_sum += wx * wy
        out += wx * wy * img[ind_y, ind_x]

out /= w_sum
out[out>255] = 255
out = out.astype(np.uint8)

4.保存结果

cv2.imshow("result", out)
cv2.waitKey(0)
cv2.imwrite("out.jpg", out)

5.结果