MATH347

MATH347 L26: SVD applications

New concepts:
- Extracting large scale and small scale features from data
- Image processing
- Image composition
- Generating art from algorithms

Computer generated visual arts

Tate Modern Gallery in London: Electric Dreams exhibition

Art related to mathematics has a long history

Figure 1. “Op art”

M.C. Escher

Figure 2. Perception paradoxes and evolution

Analysis of paintings

A distinction is made in the visual arts between composition and technique

Matlab/Octave code

Read an image from which large-scale features will be extracted, the painting composition. Find its size, and transform from color to gray scale image and then to a matrix of real-valued components $𝑨_{1}$

>>	im1=imread("./courses/MATH347/paintings/Andy_Warhol_2.jpg");

>>	im1BW=rgb2gray(im1); [px,py,nc]=size(im1); [px py nc]

$(\begin{array}{ccc} 450 & 300 & 3 \end{array})$

>>	A1=im2double(im1BW);

>>	imwrite(im1BW,"./courses/MATH347/paintings/im1BW.jpg");

>>

Figure 3. Image from which large-scale features (composition) is extracted

Matlab/Octave code

Read a portion of another image of the same size as the first image. Small-scale features will be extracted from this second image, the painting brushwork style

>>	im2=imread("./courses/MATH347/paintings/Vincent_van_Gogh_95.jpg");

>>	im2BW=rgb2gray(im2); [qx,qy,nc]=size(im2); [qx qy nc]

$(\begin{array}{ccc} 614 & 535 & 3 \end{array})$

>>	A2=im2double(im2BW);

>>	imwrite(im2BW,"./courses/MATH347/paintings/im2BW.jpg");

>>

Figure 4. Image from which small-scale brush stroke will be extracted.

Matlab/Octave code

Compute the SVDs of the matrices whose components are the grayscale intensity values of the image
>>

[U1,S1,V1]=svd(A1); [U2,S2,V2]=svd(A2(1:px,1:py));

Form a new image from the first $j$ modes of the composition image and modes from $k$ to $l$ of the brushstyle image. Weight the contribution of the large-scale and small-scale features by $w_{1}, w_{2}$ with $w_{1} + w_{2} = 1_{}$

>>	j=25; k=50; l=75; w1=0.25; w2=0.75;

>>	composition = w1U1(:,1:j)S1(1:j,1:j)*V1(:,1:j)';

>>	style = w2U2(:,k:l)S2(k:l,k:l)*V2(:,k:l)';

>>	newimage = rescale(composition + style);

>>	imwrite(newimage,"./courses/MATH347/paintings/newimage.jpg");

>>

Linear algebra is an artist!

Figure 5. New image with same composition as first image, but brush stroke mimicking that of image 2.