Image Compression Using SVD

Compress a matrix or image using singular value decomposition (SVD)

Define a

33

matrix:

In[1]:=

(mat={{0.1,0.2,0.3},{0.4,0.5,0.6},{0.7,0.8,0.9}})//MatrixForm

Out[1]//MatrixForm=

0.1	0.2	0.3
0.4	0.5	0.6
0.7	0.8	0.9

Use Image constructor to create a black and white image from this matrix. The entries of the matrix indicate shades in the image:

represents black,

represents white and numbers in between represent different shades of gray:

In[2]:=

Image[mat,ImageSize100]

Out[2]=

Every image corresponds to either a matrix or a set of matrices and possibly some vectors; and matrix operations become tools to handle them.

Consider the following (carefully chosen)

1010

matrix. Going down a column increases each entry by

0.1

, and going right on a row increases each entry by

0.01

In[3]:=

(mat=Table[0.1i+0.01j,{i,0,9},{j,0,9}])//MatrixForm

Out[3]//MatrixForm=

0.	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
0.1	0.11	0.12	0.13	0.14	0.15	0.16	0.17	0.18	0.19
0.2	0.21	0.22	0.23	0.24	0.25	0.26	0.27	0.28	0.29
0.3	0.31	0.32	0.33	0.34	0.35	0.36	0.37	0.38	0.39
0.4	0.41	0.42	0.43	0.44	0.45	0.46	0.47	0.48	0.49
0.5	0.51	0.52	0.53	0.54	0.55	0.56	0.57	0.58	0.59
0.6	0.61	0.62	0.63	0.64	0.65	0.66	0.67	0.68	0.69
0.7	0.71	0.72	0.73	0.74	0.75	0.76	0.77	0.78	0.79
0.8	0.81	0.82	0.83	0.84	0.85	0.86	0.87	0.88	0.89
0.9	0.91	0.92	0.93	0.94	0.95	0.96	0.97	0.98	0.99

Find the SVD for this matrix:

In[4]:=

{u,Σ,v}=SingularValueDecomposition[mat];

Matrix Σ, which includes the largest singular values along its diagonal:

In[5]:=

Σ//MatrixForm

Out[5]//MatrixForm=

5.72837	0.	0.	0.	0.	0.	0.	0.	0.	0.
0.	0.14402	0.	0.	0.	0.	0.	0.	0.	0.
0.	0.	0.	0.	0.	0.	0.	0.	0.	0.
0.	0.	0.	0.	0.	0.	0.	0.	0.	0.
0.	0.	0.	0.	0.	0.	0.	0.	0.	0.
0.	0.	0.	0.	0.	0.	0.	0.	0.	0.
0.	0.	0.	0.	0.	0.	0.	0.	0.	0.
0.	0.	0.	0.	0.	0.	0.	0.	0.	0.
0.	0.	0.	0.	0.	0.	0.	0.	0.	0.
0.	0.	0.	0.	0.	0.	0.	0.	0.	0.

The following function uses the first

singular values and corresponding vectors to approximate

mat

In[6]:=

compressed[k_]:=Sum[Σ〚i,i〛{u〚All,i〛}.{v〚All,i〛},{i,k}]

Approximate the matrix using only the first singular value and corresponding vectors:

In[7]:=

Round[compressed[1],.001]//MatrixForm

Out[7]//MatrixForm=

0.043	0.044	0.044	0.045	0.046	0.047	0.047	0.048	0.049	0.049
0.136	0.138	0.14	0.143	0.145	0.147	0.149	0.151	0.154	0.156
0.229	0.233	0.236	0.24	0.244	0.248	0.251	0.255	0.259	0.263
0.322	0.327	0.333	0.338	0.343	0.348	0.353	0.359	0.364	0.369
0.415	0.422	0.429	0.435	0.442	0.449	0.456	0.462	0.469	0.476
0.508	0.516	0.525	0.533	0.541	0.549	0.558	0.566	0.574	0.582
0.601	0.611	0.621	0.63	0.64	0.65	0.66	0.669	0.679	0.689
0.694	0.705	0.717	0.728	0.739	0.75	0.762	0.773	0.784	0.796
0.787	0.8	0.813	0.825	0.838	0.851	0.864	0.877	0.889	0.902
0.88	0.894	0.909	0.923	0.937	0.952	0.966	0.98	0.994	1.009

This approximation only stores

10+10+1=21

numbers (1 singular value which is a scalar and two vectors with 10 entries each). Now, approximate

mat

using two singular values:

In[8]:=

{Σ〚1,1〛,Σ〚2,2〛}

Out[8]=

{5.72837,0.14402}

In[9]:=

compressed[2]//Chop//MatrixForm

Out[9]//MatrixForm=

0	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
0.1	0.11	0.12	0.13	0.14	0.15	0.16	0.17	0.18	0.19
0.2	0.21	0.22	0.23	0.24	0.25	0.26	0.27	0.28	0.29
0.3	0.31	0.32	0.33	0.34	0.35	0.36	0.37	0.38	0.39
0.4	0.41	0.42	0.43	0.44	0.45	0.46	0.47	0.48	0.49
0.5	0.51	0.52	0.53	0.54	0.55	0.56	0.57	0.58	0.59
0.6	0.61	0.62	0.63	0.64	0.65	0.66	0.67	0.68	0.69
0.7	0.71	0.72	0.73	0.74	0.75	0.76	0.77	0.78	0.79
0.8	0.81	0.82	0.83	0.84	0.85	0.86	0.87	0.88	0.89
0.9	0.91	0.92	0.93	0.94	0.95	0.96	0.97	0.98	0.99

This is the original matrix but only

20+20+2=42

numbers are needed to store it as opposed to the original one which required

1010=100

numbers.

Consider now a greyscale image:

In[10]:=

mat=ImageData[ColorConvert[ResourceData["Sample Image: White Dog on a Beach"],"Grayscale"]];

In[11]:=

Image[mat]

Out[11]=

Find the SVD for the matrix representation of this image:

In[12]:=

{u,Σ,v}=SingularValueDecomposition[mat];

Compress the image using different numbers of singular values to get multiple resultant images with different compression rates:

In[13]:=

GraphicsRow[Table[Graphics[Image[compressed[i]],PlotLabel"Compression rate:"<>ToString[(106800.*i)/273280]<>"%"],{i,20,100,20}],10,FrameAll,ImageSizeFull]

Out[13]=

External Links

Wolfram U Introduction to Linear Algebra

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Contributed by: Juan Ortiz

Wolfram Language Example Repository

Image Compression Using SVD

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Image Compression Using SVD

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