1 International Workshop on Computer Vision April 26-30, 2004 Tehran,Iran Singular Value Decompositions with applications to Singular Value Decompositions

1International Workshop on Computer Vision April 26-30, 2004 Tehran,Iran

Singular Value Decompositions with Singular Value Decompositions with applications toapplications to

1. Texture differentiation2. Detection of an extraneous object in a texture

environment3. Segmentation of images

4. Locating eyes in facial images

Alireza Tavakoli TarghiAlireza Tavakoli [email protected]

Institute for Studies in Theoretical Physics and Mathematics (IPM), Iran

and

Royal Institute of Technology (KTH), Sweden


IntroductionIntroductionWe propose new measures for texture classification based on a local version of Singular Value Decomposition (SVD) .

The proposed measures classify textures by their roughness and structure.

Experimental results show that these measures are suitable for texture clustering and image segmentation and they are robust to changes in local lighting, orientation etc.


Singular Value DecompositionSingular Value Decomposition

A=UA=U11*D*U*D*U22

UUi i Orthogonal Matrix , D Diagonal Matrix with Orthogonal Matrix , D Diagonal Matrix with Diagonal Entries in descending order: Diagonal Entries in descending order:

dd1 1 >d>d2 2 > …>=0> …>=0

Overview of SVDOverview of SVD


To find the SVD curve of an arbitrary row of To find the SVD curve of an arbitrary row of the texture:the texture:

1.1. Scan a row with overlapping w*w Scan a row with overlapping w*w windows Wwindows Wa a , a=1,2,3,….. ; w ≈ 32 , a=1,2,3,….. ; w ≈ 32

2.2. Calculate the SVD Decomposition Calculate the SVD Decomposition

3.3. WWaa =U =U1,a1,a*D*Daa *U *U2,a2,a

4.4. As windows scan the image we As windows scan the image we obtain w curves (obtain w curves (ii) ) corresponding to diagonal entries corresponding to diagonal entries dd1,a 1,a >d>d2,a2,a>….>d>….>dw,aw,a

SVD CurvesSVD Curves


Classification Classification

Diagonal entries reflect image Diagonal entries reflect image characteristics .characteristics .


The sizes of the first few The sizes of the first few coefficients are considerably coefficients are considerably

larger than the remaininglarger than the remaining


SVD Curve Classifiers

We introduce two measures, obtained from SVD curves, which capture some of the perceptual and conceptual features in an image.

SVD Curve mean classifier SVD Curve distance classifier


SVD Curve Mean Classifier Our experiments show that the smaller coefficients d a,j ,

i.e., da,j with 23<j<32, are more representative of the structure of the texture and less dependent on inessential features.

In practice, we set l = 22 and k = 10.



SVD-Distance Classifier


Detection of Extraneous Object:Detection of Extraneous Object:


Even Small objects Even Small objects


Interest Point Detector


The technique identifies The technique identifies the bug even the location the bug even the location

of its legsof its legs


Added two coinsAdded two coins


No DifferentiationNo Differentiation


SVD TransformSVD Transform

We slide a w*w window across the We slide a w*w window across the image. image.

We identify each window by its upper We identify each window by its upper left corner coordinates (x,y).left corner coordinates (x,y).

Let F be a function of w variables. Let F be a function of w variables. The SVD surface is the graph of the The SVD surface is the graph of the

function (x,y) → the value of F on Dfunction (x,y) → the value of F on D(x,y )(x,y ).. The SVD transform is the representation The SVD transform is the representation

of the of the SVD surface as a 2D image. of the of the SVD surface as a 2D image.


SVD TransformSVD Transform


Original Image Original Image SVD SVD TransformTransform

SVD surface


Segmentation via SVD TransformSegmentation via SVD Transform








Segmentation via SVD Segmentation via SVD TransformTransform




Application of Feature Vector IIApplication of Feature Vector IISVD Transform, Segmentation SVD Transform, Segmentation




Sensitivity to TextureSensitivity to Texture


Sensitivity to TextureSensitivity to Texture

The Berkeley Segmentation Data Base. Computer Vision Group.




Segmentation via SVD Segmentation via SVD transformtransform





Effect of change of a Effect of change of a parameterparameter



Effect of change of a Effect of change of a parameterparameter


Segmentation (cont.)Segmentation (cont.)








Images show the effect of Images show the effect of substituting the diagonal substituting the diagonal part or the orthogonal part or the orthogonal parts from the SVD parts from the SVD decomposition of an decomposition of an image into that of another image into that of another image.image.

ws=5ws=5

Understanding Understanding SVDSVD


Images show the effect of Images show the effect of substituting the diagonal substituting the diagonal part or the orthogonal part or the orthogonal parts from the SVD parts from the SVD decomposition of Lena decomposition of Lena into that of a randomly into that of a randomly generated image.generated image.

ws=5ws=5




Images show the effect of Images show the effect of substituting the diagonal substituting the diagonal part or the orthogonal part or the orthogonal parts from the SVD parts from the SVD decomposition of an decomposition of an image into that of another image into that of another image.image.

ws=32ws=32


Images show the effect of Images show the effect of substituting the diagonal substituting the diagonal part or the orthogonal part or the orthogonal parts from the SVD parts from the SVD decomposition of Lena decomposition of Lena into that of a randomly into that of a randomly generated image.generated image.

ws=32ws=32



Invariance relative to Invariance relative to inversioninversion

These images are negatives of each other.

Their SVD surfaces are identical.


An SVD SurfaceAn SVD Surface


Detecting cracks and Detecting cracks and defectsdefects






Detecting cracks and defectsDetecting cracks and defects


Theoretical FrameworkTheoretical Framework We do not have a definitive answer why SVD works to the extent We do not have a definitive answer why SVD works to the extent

that it does. However, on the basis of our experimentations we that it does. However, on the basis of our experimentations we can assert the following:can assert the following:

1. The diagonal entries of SVD capture some features of an image 1. The diagonal entries of SVD capture some features of an image which are not encoded by the correlations of nearby pixels in an which are not encoded by the correlations of nearby pixels in an image.image.

2. This may explain why images constructed on the basis local 2. This may explain why images constructed on the basis local correlations virtually never exhibit features similar to ones in real correlations virtually never exhibit features similar to ones in real images. Images constructed on the basis of Markov random images. Images constructed on the basis of Markov random fields or similar procedures generally look very random.fields or similar procedures generally look very random.

3. As the size of the sliding window increases the importance of 3. As the size of the sliding window increases the importance of the orthogonal parts of SVD decomposition relative to the the orthogonal parts of SVD decomposition relative to the diagonal part increases.diagonal part increases.


Work in ProgressWork in Progress We are in the process of applying SVD for We are in the process of applying SVD for

material/surface classification.material/surface classification. The SVD transform can be applied to the test The SVD transform can be applied to the test

case of differentiating between a dog and a cat. case of differentiating between a dog and a cat. The results are preliminary and require further The results are preliminary and require further tests. tests.

SVD transforms are also being tested on movie SVD transforms are also being tested on movie images.images.

Our methods are being tested on medical Our methods are being tested on medical images by IRMA (Image Retrieval in Medical images by IRMA (Image Retrieval in Medical Applications) of Institut fuer Medizinische Applications) of Institut fuer Medizinische Informatik Universitaetklinikum der RWTH in Informatik Universitaetklinikum der RWTH in Aachen, Germany.Aachen, Germany.

Documents

1 International Workshop on Computer Vision April 26-30, 2004 Tehran,Iran Singular Value Decompositions with applications to Singular Value Decompositions