eee508_Transforms

eee508_Transforms - Transforms • Basic tools in image...

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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 ¾ In general, forward and inverse transforms are not necessarily similar ¡ is not necessarily ¾ 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 ¾ In general, forward and inverse transforms are not necessarily similar ¡ is not necessarily ¾ 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 ¾ Unitary transforms are desirable since they can result in simpler implementations and less complexity since forward and inverse transforms are the same. ¾ Unitary transforms have also nice properties. EEE 508 1 Transforms • Matrix representation of images ¾ 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 2-1 columns n 2 x ( n 1 , n 2 ) Row Col index index N 2 column #1 N 1 N 1-1 n EEE 508 1 1 Transforms ¾ Rearrange image in matrix form ( 90 o rotation of x ( n 1, n 2) ) ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ » » º « « ª £ , 1 1 , 1 , , 2 x N x x x ¡ ¢ £ X ¤ Input image ¡ ¢ ¡ ¢ » » ¼ « « ¬ £ £ £ 1 , 1 , 1 2 1 1 N N x N x £ ¡ X ¾ Let = marix representation of X ( K 1 , K 2 ) = Transformed image ¡ ¢ ¡ ¢ ¡ ¢ » º « ª £ 1 1 , 1 , , 2 X N X X X £ X ¡ ¡ ¢ ¡ ¢ ¡ ¢ » » » ¼ « « « ¬ £ £ £ 1 , 1 , 1 , 1 ¡ 2 1 1 N...
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eee508_Transforms - Transforms • Basic tools in image...

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