Applications of the SVD - Semantic Scholar
E3101 2002
SVD Fun 1
Applications of the SVD Marc Spiegelman Detail from Durer’s Melancolia, dated 1514., 359x371 image
E3101 2002
SVD Fun 2
Image Compression Given an original image (here 359 × 371 pixels) Detail from Durer’s Melancolia, dated 1514., 359x371 image
We can write it as a 359 × 371 matrix A which can then be decomposed via the singular value decomposition as
A = U ΣV T where U is 359 × 359, Σ is 359 × 371 and V is 371 × 371.
E3101 2002
SVD Fun 3
The matrix A however can also be written as a sum of rank 1 matrices
A = σ1 u1 vT1 + σ2 u2 vT2 + . . . + σn un vTn where each rank 1 matrix u i vT i is the size of the original matrix. Each one of these matrices is a mode. Because the singular values σ i are ordered σ1
≥ σ2 ≥ . . . ≥ σn ≥ 0,
however, significant compression of the image is possible if the spectrum of singular values has only a few very strong entries.
E3101 2002
SVD Fun 4
Spectrum of Singular values for A Singular Values
5
10
4
10
3
10
2
10
1
10
0
10
0
50
100
150
200
250
300
350
400
Here the spectrum is contained principally in the first 100–200 modes (max).
E3101 2002
SVD Fun 5
We can therefore reconstruct the image from just a subset of modes. For example in matlabese we can write just the first mode as
[U,S,V]=svd(A) B=U(:,1)*S(1,1)*V(:,1)’ Detail from Durer’s Melancolia, dated 1514., 359x371 image
EOF reconstruction with 1 modes
E3101 2002
SVD Fun 6
Or as a sum of the first 10 modes as
B=U(:,1:10)*S(1:10,1:10)*V(:,1:10)’ Detail from Durer’s Melancolia, dated 1514., 359x371 image
EOF reconstruction with 10 modes
which only uses 5% of the storage (10 × 359 + 10 × 371 + 10 vs 359 × 371
= 133189 pixels.
= 7310 pixels
E3101 2002
SVD Fun 7
Adding modes, just adds resolution
B=U(:,1:20)*S(1:20,1:20)*V(:,1:20)’ Detail from Durer’s Melancolia, dated 1514., 359x371 image
EOF reconstruction with 20 modes
E3101 2002
SVD Fun 8
Adding modes, just adds resolution
B=U(:,1:30)*S(1:30,1:30)*V(:,1:30)’ Detail from Durer’s Melancolia, dated 1514., 359x371 image
EOF reconstruction with 30 modes
E3101 2002
SVD Fun 9
Adding modes, just adds resolution
B=U(:,1:40)*S(1:40,1:40)*V(:,1:40)’ Detail from Durer’s Melancolia, dated 1514., 359x371 image
EOF reconstruction with 40 modes
E3101 2002
SVD Fun 10
Adding modes, just adds resolution
B=U(:,1:50)*S(1:50,1:50)*V(:,1:50)’ Detail from Durer’s Melancolia, dated 1514., 359x371 image
EOF reconstruction with 50 modes
E3101 2002
SVD Fun 11
Adding modes, just adds resolution
B=U(:,1:100)*S(1:100,1:100)*V(:,1:100)’ Detail from Durer’s Melancolia, dated 1514., 359x371 image
EOF reconstruction with 100 modes
E3101 2002
SVD Fun 12
Adding modes, just adds resolution
B=U(:,1:200)*S(1:200,1:200)*V(:,1:200)’ Detail from Durer’s Melancolia, dated 1514., 359x371 image
EOF reconstruction with 200 modes
E3101 2002
SVD Fun 13
Application 2: EOF analysis Pattern extraction—Mid-ocean ridge topography Here we consider a real research use of the SVD by Chris Small (LDEO) A Global Analysis Of Midocean Ridge Axial Topography GEOPHYSICAL JOURNAL INTERNATIONAL 116 (1): 64-84 JAN 1994
E3101 2002
SVD Fun 14
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
80˚
80˚
60˚
60˚
40˚
40˚
20˚
20˚
0˚
0˚
-20˚
-20˚
-40˚
-40˚
-60˚
-60˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
E3101 2002
SVD Fun 15
The data: cross axis topography profiles from different spreading rates Cross axis topography of mid−ocean ridges 1500 slow medium fast
relative Elevation (m)
1000
500
0
−500
−1000
−1500 0
10
20
30 40 50 Km across axis
60
70
80
E3101 2002
SVD Fun 16
Form a matrix A (179 × 80) of elevation vs. distance across the ridge Cross axis topography of mid−ocean ridges 2000
20 1500
40 spreading rate
1000
60 500
80 0
100
−500
120
−1000
140
20 40 60 Km across axis
= U ΣV T . Here U is the same size as A and Σ and V are both square 80 × 80 matrices. and again take the SVD A
E3101 2002
SVD Fun 17
Now the rows of V T form an orthonormal basis for the row space of A, i.e. each profile (row of A) can be written as a linear combination of the rows of V T or
A = CV T which by inspection of the SVD shows that C
= U Σ. Here, the rows of V T are
known as Empirical Orthogonal Functions or EOFs.
E3101 2002
SVD Fun 18
Again, if the spectrum of Singular values contains a few large values and a long tail of very small values, it may be possible to reconstruct the rows of A with only a small number of EOFs. The spectrum for this data looks like Singular Values
5
10
4
10
3
10
2
10
1
10
0
10
20
30
40
50
60
70
80
which suggests that you only need about 4 EOF’s to explain most of the data.
E3101 2002
SVD Fun 19
The first 4 EOFs First 4 EOFs 0.4 EOF1 EOF2 EOF3 EOF4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3 0
10
20
30 40 50 Km across axis
60
70
80
E3101 2002
SVD Fun 20
And we can reconstruct individual profiles as combinations of the first 4 EOF’s. For example here is one for a slow spreading rate EOF reconstruction, sample 10=(−2994.2,−3475.3,−2403.3,−148.1) 1500
relative Elevation (m)
1000
500
0
−500
−1000
−1500 0
10
20
30 40 50 Km across axis
60
70
80
E3101 2002
SVD Fun 21
Intermediate spreading rate EOF reconstruction, sample 60=(−557.5,699.8,−299.1,676.9) 700 600
relative Elevation (m)
500 400 300 200 100 0 −100 −200 −300 0
10
20
30 40 50 Km across axis
60
70
80
E3101 2002
SVD Fun 22
Fast spreading rate EOF reconstruction, sample 100=(−617.3,1316.3,−557.2,135.8) 500
relative Elevation (m)
400
300
200
100
0
−100 0
10
20
30 40 50 Km across axis
60
70
80
SVD Fun 1
Applications of the SVD Marc Spiegelman Detail from Durer’s Melancolia, dated 1514., 359x371 image
E3101 2002
SVD Fun 2
Image Compression Given an original image (here 359 × 371 pixels) Detail from Durer’s Melancolia, dated 1514., 359x371 image
We can write it as a 359 × 371 matrix A which can then be decomposed via the singular value decomposition as
A = U ΣV T where U is 359 × 359, Σ is 359 × 371 and V is 371 × 371.
E3101 2002
SVD Fun 3
The matrix A however can also be written as a sum of rank 1 matrices
A = σ1 u1 vT1 + σ2 u2 vT2 + . . . + σn un vTn where each rank 1 matrix u i vT i is the size of the original matrix. Each one of these matrices is a mode. Because the singular values σ i are ordered σ1
≥ σ2 ≥ . . . ≥ σn ≥ 0,
however, significant compression of the image is possible if the spectrum of singular values has only a few very strong entries.
E3101 2002
SVD Fun 4
Spectrum of Singular values for A Singular Values
5
10
4
10
3
10
2
10
1
10
0
10
0
50
100
150
200
250
300
350
400
Here the spectrum is contained principally in the first 100–200 modes (max).
E3101 2002
SVD Fun 5
We can therefore reconstruct the image from just a subset of modes. For example in matlabese we can write just the first mode as
[U,S,V]=svd(A) B=U(:,1)*S(1,1)*V(:,1)’ Detail from Durer’s Melancolia, dated 1514., 359x371 image
EOF reconstruction with 1 modes
E3101 2002
SVD Fun 6
Or as a sum of the first 10 modes as
B=U(:,1:10)*S(1:10,1:10)*V(:,1:10)’ Detail from Durer’s Melancolia, dated 1514., 359x371 image
EOF reconstruction with 10 modes
which only uses 5% of the storage (10 × 359 + 10 × 371 + 10 vs 359 × 371
= 133189 pixels.
= 7310 pixels
E3101 2002
SVD Fun 7
Adding modes, just adds resolution
B=U(:,1:20)*S(1:20,1:20)*V(:,1:20)’ Detail from Durer’s Melancolia, dated 1514., 359x371 image
EOF reconstruction with 20 modes
E3101 2002
SVD Fun 8
Adding modes, just adds resolution
B=U(:,1:30)*S(1:30,1:30)*V(:,1:30)’ Detail from Durer’s Melancolia, dated 1514., 359x371 image
EOF reconstruction with 30 modes
E3101 2002
SVD Fun 9
Adding modes, just adds resolution
B=U(:,1:40)*S(1:40,1:40)*V(:,1:40)’ Detail from Durer’s Melancolia, dated 1514., 359x371 image
EOF reconstruction with 40 modes
E3101 2002
SVD Fun 10
Adding modes, just adds resolution
B=U(:,1:50)*S(1:50,1:50)*V(:,1:50)’ Detail from Durer’s Melancolia, dated 1514., 359x371 image
EOF reconstruction with 50 modes
E3101 2002
SVD Fun 11
Adding modes, just adds resolution
B=U(:,1:100)*S(1:100,1:100)*V(:,1:100)’ Detail from Durer’s Melancolia, dated 1514., 359x371 image
EOF reconstruction with 100 modes
E3101 2002
SVD Fun 12
Adding modes, just adds resolution
B=U(:,1:200)*S(1:200,1:200)*V(:,1:200)’ Detail from Durer’s Melancolia, dated 1514., 359x371 image
EOF reconstruction with 200 modes
E3101 2002
SVD Fun 13
Application 2: EOF analysis Pattern extraction—Mid-ocean ridge topography Here we consider a real research use of the SVD by Chris Small (LDEO) A Global Analysis Of Midocean Ridge Axial Topography GEOPHYSICAL JOURNAL INTERNATIONAL 116 (1): 64-84 JAN 1994
E3101 2002
SVD Fun 14
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
80˚
80˚
60˚
60˚
40˚
40˚
20˚
20˚
0˚
0˚
-20˚
-20˚
-40˚
-40˚
-60˚
-60˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
E3101 2002
SVD Fun 15
The data: cross axis topography profiles from different spreading rates Cross axis topography of mid−ocean ridges 1500 slow medium fast
relative Elevation (m)
1000
500
0
−500
−1000
−1500 0
10
20
30 40 50 Km across axis
60
70
80
E3101 2002
SVD Fun 16
Form a matrix A (179 × 80) of elevation vs. distance across the ridge Cross axis topography of mid−ocean ridges 2000
20 1500
40 spreading rate
1000
60 500
80 0
100
−500
120
−1000
140
20 40 60 Km across axis
= U ΣV T . Here U is the same size as A and Σ and V are both square 80 × 80 matrices. and again take the SVD A
E3101 2002
SVD Fun 17
Now the rows of V T form an orthonormal basis for the row space of A, i.e. each profile (row of A) can be written as a linear combination of the rows of V T or
A = CV T which by inspection of the SVD shows that C
= U Σ. Here, the rows of V T are
known as Empirical Orthogonal Functions or EOFs.
E3101 2002
SVD Fun 18
Again, if the spectrum of Singular values contains a few large values and a long tail of very small values, it may be possible to reconstruct the rows of A with only a small number of EOFs. The spectrum for this data looks like Singular Values
5
10
4
10
3
10
2
10
1
10
0
10
20
30
40
50
60
70
80
which suggests that you only need about 4 EOF’s to explain most of the data.
E3101 2002
SVD Fun 19
The first 4 EOFs First 4 EOFs 0.4 EOF1 EOF2 EOF3 EOF4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3 0
10
20
30 40 50 Km across axis
60
70
80
E3101 2002
SVD Fun 20
And we can reconstruct individual profiles as combinations of the first 4 EOF’s. For example here is one for a slow spreading rate EOF reconstruction, sample 10=(−2994.2,−3475.3,−2403.3,−148.1) 1500
relative Elevation (m)
1000
500
0
−500
−1000
−1500 0
10
20
30 40 50 Km across axis
60
70
80
E3101 2002
SVD Fun 21
Intermediate spreading rate EOF reconstruction, sample 60=(−557.5,699.8,−299.1,676.9) 700 600
relative Elevation (m)
500 400 300 200 100 0 −100 −200 −300 0
10
20
30 40 50 Km across axis
60
70
80
E3101 2002
SVD Fun 22
Fast spreading rate EOF reconstruction, sample 100=(−617.3,1316.3,−557.2,135.8) 500
relative Elevation (m)
400
300
200
100
0
−100 0
10
20
30 40 50 Km across axis
60
70
80
