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Unformatted text preview: Transforms Basic tools in image processing Represent images by a series of coefficients which can be used for processing and analysis General form for N 1 xN 2 image 1 ; , , , 1 1 1 1 2 1 2 1 2 1 1 2 2 1 d d d d ) N K N K K K n n x K K X N N n n 1 2 2 Transform) (Image Kernel Transform 1 2 2 1 d d N K n n 1 1 1 1 1 1 2 d d < N n K K K K X n n N N 1 ; , , , 2 2 Basis Image 2 1 2 1 2 1 1 2 2 1 d d < N n K K K K X n n x K K n n EEE 508 1 Transforms In general, forward and inverse transforms are not necessarily similar is not necessarily In image processing, transforms are typically orthogonal and form 2 1 , 2 1 K K n n < 2 1 * , 2 1 K K n n ) a set of complete orthonormal discrete basis functions Orthonormality 2 2 1 1 2 1 2 1 , , , 1 2 2 1 2 1 L K L K L L K K n n n n n n < ) G Completeness 1 2 2 2 1 1 2 1 2 1 , , , 1 2 2 1 2 1 m n m n K K K K K K m m n n < ) G Orthonormality and completeness ensure invertibility. EEE 508 Transforms In general, forward and inverse transforms are not necessarily similar is not necessarily In image processing, transforms are typically orthogonal and form 2 1 , 2 1 K K n n < 2 1 * , 2 1 K K n n ) a set of complete orthonormal discrete basis functions Orthonormality 2 2 1 1 2 1 2 1 , , , 1 2 2 1 2 1 L K L K L L K K n n n n n n < ) G Completeness 1 2 2 2 1 1 2 1 2 1 , , , 1 2 2 1 2 1 m n m n K K K K K K m m n n < ) G Orthonormality and completeness ensure invertibility. EEE 508 1 Transforms Most often, we are interested in transforms where Unitary transforms 2 1 * 2 1 , , 2 1 2 1 K K K K n n n n ) < Unitary transforms Unitary transforms are desirable since they can result in simpler implementations and less complexity since forward and inverse transforms are the same. Unitary transforms have also nice properties. EEE 508 1 Transforms Matrix representation of images It is convenient to view signals and transforms as matrices and matrix operations represent these in matrix form. Example.: Image as a matrix n n N 21 columns n 2 x ( n 1 , n 2 ) Row Col index index N 2 column #1 N 1 N 11 n EEE 508 1 1 Transforms Rearrange image in matrix form ( 90 o rotation of x ( n 1, n 2) ) , 1 1 , 1 , , 2 x N x x x X Input image 1 , 1 , 1 2 1 1 N N x N x X Let = marix representation of X ( K 1 , K 2 ) = Transformed image 1 1 , 1 , , 2 X N X X X X 1 , 1 , 1 , 1 2 1 1 N...
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 Fall '09

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