hungpham2002 / svd-and-its-application

Image compression & Denoise Image with SVD

SVD-and-Its-Application

Image compression & Denoise Image with SVD

Outline

• Explain Vector
• Explain Eigenvector
• Explain Eigenvalue
• Combination of Eigenvector & Eigenvalue
• Explain SVD

References: AI Viet Nam

Vector:

• Vector is an object has both a magnitude and a direction.
• Vector is a matrix with single column or single row.
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Eigenvector

• A eigenvector v, is a non-zero vector that sastifies the following equation:

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• Vector v is called eigenvector of matrix A if we multiply matrix A by vector v, the new vector (lamda)v does not change direction after the transformation
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Eigenvalue

• Eigenvalue tell us how much the eigenvector changes in size when mutiplied with the matrix
• New column vector (lamda)v has same direction eigenvector.
• New column vector (lamda)v maybe either longer or shorter than eigenvector because of eigenvalue Lamda.
• Eigenvalue lamda might be negative. The direction of new vector is reversed but still on the same line
• We might find many eigenvectors of matrix A
• All vector with the same direction are actually eigenvector of matrix A
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Combination of Eigenventor & Eigenvalue

Singular Value Decomposition

Since mathematics is just the art of assigning different names to the same concept, SVD is nothing more than decomposing vectors onto orthogonal axes.

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• Support u and v are vectors
  • Vector u decomposed into orthogonal components w1 and w2
  • Want to decompose u as: u = w1 + w2
• w1 is parallel to vector v and w1 is perpendicular/orthogonal to w2
• The vector component w1 is also called the projection of vector u onto vector v:
  • w1 = proj(v)(u) (projection of vector u onto vector v).
  • w2 = u - w1

Decomposing a Vector into two Orthogonal Vectors

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Vector Projection

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Extend this conclusion for handling a matrix (a bunch of vectors)

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Recall Vector Projection

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• Extend to more than 1 vectors.
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• Generalize to any number of points and dimensions
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• Any set of vectors (A) can be expressed in terms of their lengths of projections (S) on some set of orthogonal axes (V).
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We are here!

Convention SVD

Keep going on decomposing S

• Normalize these comlumn vectors to make them of unit length
• Dividing each column vector by its magnitude, but in matrix form
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S - Explanation

Sigma - Explanation

• Sigma determines most points (decomposed vector) are closer to v1 or v2.

Another Approaching SVD (mapping to Eigenvector / Eigenvalue)

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• For Example:
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This is the end of Convention SVD

Thank you!