電子科技大學 格拉斯哥學院 2017級 何林峰

Background

In the freshman seminar last semester, we had a preliminary understanding of some basic theories of image recognition and some discussion on its application.Inspired by this, I made some exploration on the common image sharpening problems and tried to make some preliminary solutions by using the image processing methods such as matrix mentioned in the class. Driving by inspiration, I researched some advanced graghic transformation based on least-squares method and curve fitting.

Least-Squares with respect to matrix

Intro: For a massive problem Ax=b that has no solution , when solution is demanded and none exist , the best one can do is to find an x that makes Ax as close as possible to b . The adjective “least-squares” arises from the fact that b-Ax is the square root of a sum of squares.

Solving the least-squares method with matrix

clear clc x=[2,4,5,6,6.8,7.5,9,12,13.3,15]; [~,k]=size(x); y=[-10,-6.9,-4.2,-2,0,2.1,3,5.2,6.4,4.5]; for n=1:9ANSS=polyfit(x,y,n); %用polyfit擬合曲線for i=1:n+1 %answer矩陣存儲每次求得的方程系數，按列存儲answer(i,n)=ANSS(i);endx0=0:0.01:17;y0=ANSS(1)*x0.^n ; %根據求得的系數初始化并構造多項式方程for num=2:1:n+1 y0=y0+ANSS(num)*x0.^(n+1-num);endsubplot(3,3,n)plot(x,y,'*')hold onplot(x0,y0) end suptitle('不同次數方程曲線擬合結果，從1到9階')

<經線性擬合處理后的數據曲線>

The application of least-squares method
a. Non-negative matrix factorization
The information or signal processing data collected by people at ordinary times have non-negative characteristics , such as grayscale monitoring grayscale images and CT tomography… Nonnegative matrix decomposition is a multivariate analysis method, which first decomposes the high dimensional sample set matrix and approximates the high dimensional sample by using the product of the low rank nonnegative matrix . The coefficient composition of a linear combination represents the coefficient set matrix . Under this condition , Each sample in the high dimensional sample matrix can be represented by a nonnegative linear combination of the basis vectors.


c. Face feature image extraction
Generally, the face image that people collect has a high dimension . However , in a high dimensional space , the locations of face image are divergent so that it is bad to class and increase the difficulties of calculation . Thus , we should reduce the dimension ,which is called feature extraction . Using the method of subspace analyze based on NMF is looking for a linear or nonlinear subspace to compress the original image into a lower dimensional subspace and making figures convergent in it

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