alireza tavakoli targhi [email protected]
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Singular Value Decompositions with applications to 1. Texture differentiation 2. Detection of an extraneous object in a texture environment 3. Segmentation of images 4. Locating eyes in facial images. Alireza Tavakoli Targhi [email protected] - PowerPoint PPT PresentationTRANSCRIPT
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
2International Workshop on Computer Vision April 26-30, 2004 Tehran,Iran
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.
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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
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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
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Classification Classification
Diagonal entries reflect image Diagonal entries reflect image characteristics .characteristics .
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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
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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
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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.
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SVD-Distance Classifier
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Detection of Extraneous Object:Detection of Extraneous Object:
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Even Small objects Even Small objects
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Interest Point Detector
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The technique identifies The technique identifies the bug even the location the bug even the location
of its legsof its legs
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Added two coinsAdded two coins
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No DifferentiationNo Differentiation
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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.
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SVD TransformSVD Transform
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Original Image Original Image SVD SVD TransformTransform
SVD surface
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Segmentation via SVD TransformSegmentation via SVD Transform
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Segmentation via SVD TransformSegmentation via SVD Transform
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Segmentation via SVD TransformSegmentation via SVD Transform
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Segmentation via SVD TransformSegmentation via SVD Transform
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Segmentation via SVD Segmentation via SVD TransformTransform
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Segmentation via SVD TransformSegmentation via SVD Transform
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Application of Feature Vector IIApplication of Feature Vector IISVD Transform, Segmentation SVD Transform, Segmentation
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Segmentation via SVD TransformSegmentation via SVD Transform
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Sensitivity to TextureSensitivity to Texture
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Sensitivity to TextureSensitivity to Texture
The Berkeley Segmentation Data Base. Computer Vision Group.
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Segmentation via SVD TransformSegmentation via SVD Transform
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Segmentation via SVD Segmentation via SVD transformtransform
The Berkeley Segmentation Data Base. Computer Vision Group.
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The Berkeley Segmentation Data Base. Computer Vision Group.
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Effect of change of a Effect of change of a parameterparameter
The Berkeley Segmentation Data Base. Computer Vision Group.
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Effect of change of a Effect of change of a parameterparameter
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Segmentation (cont.)Segmentation (cont.)
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Segmentation (cont.)Segmentation (cont.)
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Segmentation (cont.)Segmentation (cont.)
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Segmentation (cont.)Segmentation (cont.)
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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
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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
Understanding Understanding SVDSVD
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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 an decomposition of an image into that of another image into that of another image.image.
ws=32ws=32
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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
Understanding Understanding SVDSVD
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Invariance relative to Invariance relative to inversioninversion
These images are negatives of each other.
Their SVD surfaces are identical.
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An SVD SurfaceAn SVD Surface
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Detecting cracks and Detecting cracks and defectsdefects
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Detecting cracks and Detecting cracks and defectsdefects
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Detecting cracks and Detecting cracks and defectsdefects
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Detecting cracks and defectsDetecting cracks and defects
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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.
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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.