Introduction to linear interpolation methods and image scaling

1 Linear interpolation

Let the function $y = f (x)$ at two points $x_{0}$ 、 $x_{1}$ The values on are respectively $y_{0}$ 、 $y_{1}$ . Find the polynomial $y=p_{1}(x)=a_{0}+a_{1}x$ ，使满足 $p_{1}(x_{0})=y_{0},p_{1}(x_{1})=y_{1}$ . Its geometric meaning is found through $A(x_{0},y_{0})$ ， $B(x_{1},y_{1})$ is a straight line. From analytical geometry we know: $y=p_{1}(x)=y_{0}+\frac{ y_{1}-y_{0}}{x_{1}-x_{0}}(x-x_{0})$ Rearrange the above formula to get: $p_{1}(x)=\frac{x -x_{1}}{x_{0}-x_{1}}y_{0}+\frac{x-x_{0}}{x_{1}-x_{0}}y_{1}$

2 Bilinear interpolation

2.1 Calculation formula

Bilinear interpolation is a linear interpolation extension of the interpolation function of two variables. Its core idea is to perform linear interpolation in two directions. Here is the following figure as an example:
Suppose now you want to know the unknown function $f$ is at point $P (x, y)$ value, currently known function $f$ 在 $Q_{11}=(x_{1},y_{1})$ 、 $Q_{12}=(x_{1},y_{2})$ 、 $Q_{21}=(x_{2},y_{1})$ 、 $Q_{22}=(x_{2},y_{2})$ value.
Insert image description here
The calculation steps of bilinear interpolation are as follows:

First in Linear interpolation in the $x$ $f$ point $R_{1}$ and $R_{2}$ value on. The calculation formula is as follows: $f(R_{1})=\frac{ x_{2}-x}{x_{2}-x_{1}}f(Q_{11})+\frac{x-x_{1}}{x_{2}-x_{1}}f(Q_ {twenty one})$ $f(R_{2})=\frac{x_{2}-x}{x_{2}-x_{1}}f(Q_{12})+\frac{x-x_{1}}{x_{2}-x_{1}}f(Q_{22})$
Secondly, use $R_{1}$ and $R_{2}$ Calculate $P$ 。其计算公式如下： $f(P)=\frac{y_{2}-y}{y_{2}-y_{1}}f(R_{1})+\frac{y-y_{1}}{y_{2}-y_{1}}f(R_{2})$

2.2 Use cases

The following is a method of modifying image pixels using bilinear interpolation.

from PIL import Image
import numpy as np
import math

def get_pixel(path):
    img=Image.open(path)
    pixel=img.load()
    data=[]
    for i in range(img.size[0]):
        for j in range(img.size[1]):
            data.extend(list(pixel[i,j]))
    data=np.reshape(np.array(data),(img.size[0],img.size[1],3))
    return img,data
    
def img_resize(height,width,img,data):
    #生成在图片上取样的x坐标、y坐标
    lin_height=np.linspace(0,img.size[0]-1,height)
    lin_width=np.linspace(0,img.size[1]-1,width)
              
    new_img=Image.new('RGB',(height,width),(0,0,0))
    
    x1=np.array(list(map(lambda x:x if x<(img.size[0]-1) else x-1,[math.floor(i) for i in lin_height])))
    x2=np.array(list(map(lambda x:x-1 if x==img.size[0] else x,[math.floor(i+1) for i in lin_height])))
    
    y1=np.array(list(map(lambda x:x if x<(img.size[1]-1) else x-1,[math.floor(i) for i in lin_width])))
    y2=np.array(list(map(lambda x:x-1 if x==img.size[1] else x,[math.floor(i+1) for i in lin_width])))
    pixel_list=[]
    for i in range(height):    
        Q_r1=data[x1[i],y1,:]*(x2[i]-lin_height[i])+data[x2[i],y1,:]*(lin_height[i]-x1[i])
        Q_r2=data[x1[i],y2,:]*(x2[i]-lin_height[i])+data[x2[i],y2,:]*(lin_height[i]-x1[i])
        Q=Q_r1*np.reshape(y2-lin_width,(-1,1))+\
            Q_r2*np.reshape(lin_width-y1,(-1,1))
        Q=Q.astype(int)
        for j in range(Q.shape[0]):
            new_img.putpixel((i,j),tuple(Q[j]))
    return new_img
if __name__=="__main__":
    path=r'/Users/sherry/Downloads/杀生丸.jpeg'
    img,data=get_pixel(path)
    new_img=img_resize(2000,2000,img,data)