ECE468_22 - ECE 468 Digital Image Processing Lecture 21...

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ECE 468: Digital Image Processing Lecture 21 Prof. Sinisa Todorovic [email protected] 1
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Outline Wavelet functions (Textbook 7.2.3) Wavelet transform in 1D (Textbook 7.3) 2
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Wavelet Function Spaces 3
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Scaling Functions ϕ j,k ( x ) = 2 j/ 2 ϕ (2 j x - k ) Given: ϕ ( x ) 4
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Wavelet Functions ψ j,k ( x ) = 2 j 2 ψ (2 j x - k ) V j +1 = V j W J ψ j,k ( x ) , ϕ j,l ( x ) = 0 5
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Any Function If f ( x ) V j 6
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Any Function If f ( x ) V j f ( x ) = k α k ϕ j - 1 ,k ( x ) V j - 1 + k β k ψ j - 1 ,k ( x ) V j 6
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Any Function If f ( x ) V j f ( x ) = k α k ϕ j - 2 ,k ( x ) V j - 2 + k β k ψ j - 2 ,k ( x ) V j - 1 + k β k ψ j - 1 ,k ( x ) V j f ( x ) = k α k ϕ j - 1 ,k ( x ) V j - 1 + k β k ψ j - 1 ,k ( x ) V j 6
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Any Function If f ( x ) V j ... f ( x ) = k α k ϕ j 0 ,k ( x ) + j - 1 l = j 0 k β k ψ l,k ( x ) f ( x ) = k α k ϕ j - 2 ,k ( x ) V j - 2 + k β k ψ j - 2 ,k ( x ) V j - 1 + k β k ψ j - 1 ,k ( x ) V j f ( x ) = k α k ϕ j - 1 ,k ( x ) V j - 1 + k β k ψ j - 1 ,k ( x ) V j 6
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Haar Wavelet Functions 7
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Haar Wavelet Functions 7
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Haar Wavelet Functions Function in V 1 7
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Haar Wavelet Functions Function in V 1 f ( x ) = f a ( x ) + f d ( x ) 7
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Haar Wavelet Functions Function in V 1 Expressed via V 0 f ( x ) = f a ( x ) + f d ( x ) 7
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