New concepts:

• Images as vectors

• Interpreting column space, null space for a collection of faces

• Face recognition by projection

facedata.zip is the MIT face database. Unzip: obtain directory 'rawdata'

These Matlab instructions display one face within the MIT face collection numbered from 1223 to 5222 (3993 faces, some indices skipped), each face is represented by a vector with ${128}^2=16384$ gray-scale values.

Faces are displayed using the function

function shwface(a)
 face=reshape(a,128,128)';
 imagesc(face); colormap(gray(256));
 axis('equal')
end

matlab>>

dir=strcat(getenv("HOME"),"/courses/MATH347");
fid=fopen(strcat(dir,"/rawdata/1223"));
face1=fread(fid);
fclose(fid);
figure(1); shwface(face1);

matlab>>

Read in another face

matlab>>

dir=strcat(getenv("HOME"),"/courses/MATH347");
fid=fopen(strcat(dir,"/rawdata/1323"));
face2=fread(fid);
fclose(fid);
figure(2); shwface(face2);

Face vectors are not orthogonal, there is overlap in the face information

matlab>>

180*acos(face1'*face2/norm(face1)/norm(face2))/pi

$\theta =0\Rightarrow$ colinear (identical face features)

$\theta ={90}^{\circ }\Rightarrow$orthogonal (completely different, not likely for members of the same species!)

The collection is gathered into a matrix $𝑭=\left[\begin{array}{lll}{𝒇}_1& \dots & {𝒇}_{3993}\end{array}\right]\in {ℝ}^{16384\times 3993}$

An “average” face vector $𝒂$ is obtained by

matlab>>

data=strcat(dir,"/faces.mat"); load(data);
figure(3); shwface(a)

Seek an economical way to reconstruct a face

• $𝒇$ reconstructed face

• $𝒂$ average face

• $𝑨\in {ℝ}^{{128}^2\times n}$ a subcollection of $n$ faces

How to choose $𝑨$?

Of all deviations from the average face choose the closest to orthogonal to $𝒂$

Choose ${𝒃}_1={𝒅}_k$ with $k$ the index for which $|{\theta }_k-\pi /2|⩽|{\theta }_j-\pi /2|$ for all $j\in \{1,2,\dots ,3993\}$

Then choose ${𝒃}_2$ closest to orthogonal to both $𝒂$ and ${𝒃}_1$

The matrix $𝑩=\left[\begin{array}{lll}{𝒃}_1& \dots & {𝒃}_n\end{array}\right]$ does not necessarily have orthonormal columns

Apply Gram-Schmidt to $𝑩$ and obtain $𝑨$ with orthonormal columns

A face can be approximately reconstructed by linear combination

How close is the reconstruction? Find the angle

matlab>>

f=A*R(:,1)+a; acos(face1'*f/norm(face1)/norm(f))*180/pi

Obtain an angle of $\theta =10.7^{\circ }$ fairly close to perfect reconstruction ($\theta =0$)

Better reconstruction accuracy would require increased $n$