Continuous Wavelet Transform: ECG Recognition Based on Phase and - PowerPoint PPT Presentation

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE E XAMPLES C ONCLUSION R EFERENCES Continuous Wavelet Transform: ECG Recognition Based on Phase and Modulus Representations E DGAR G ONZALEZ Based on the paper by: Lofti Senhadjii, Laurent Thoroval, and Guy Carrault [2] May 12, 2009 G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE E XAMPLES C ONCLUSION R EFERENCES Outline Introduction The Square Modulus and Phase Square Modulus Phase Behavior Examples Symmetrical Properties Without Symmetrical Properties ECG Signal Conclusion References G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE E XAMPLES C ONCLUSION R EFERENCES Introduction Biomedical signals: ◮ Fundamental to Analyzing Diseases ◮ Generally Time-Varying ◮ Non-stationary ◮ Usually Noisy G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE E XAMPLES C ONCLUSION R EFERENCES The Analyzing Tools: ◮ Fourier Transform ◮ Continuous Wavelet Transform (CWT) G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE E XAMPLES C ONCLUSION R EFERENCES The Analyzing Tools: ◮ Fourier Transform ◮ Continuous Wavelet Transform (CWT) G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE E XAMPLES C ONCLUSION R EFERENCES Electrocardiography (ECG) ECG is the "recording of the electrical activity of the heart over time via skin electrodes." [1] Fig. 1: Electrocardiogram and leads [1] G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE E XAMPLES C ONCLUSION R EFERENCES Facts of ECG: ◮ Voltage measured between pairs of electrodes ◮ Usually 12-Leads ◮ Diagnose a wide range of heart conditions ◮ and much more... G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE E XAMPLES C ONCLUSION R EFERENCES Facts of ECG: ◮ Voltage measured between pairs of electrodes ◮ Usually 12-Leads ◮ Diagnose a wide range of heart conditions ◮ and much more... G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE E XAMPLES C ONCLUSION R EFERENCES Facts of ECG: ◮ Voltage measured between pairs of electrodes ◮ Usually 12-Leads ◮ Diagnose a wide range of heart conditions ◮ and much more... G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE E XAMPLES C ONCLUSION R EFERENCES Facts of ECG: ◮ Voltage measured between pairs of electrodes ◮ Usually 12-Leads ◮ Diagnose a wide range of heart conditions ◮ and much more... G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE E XAMPLES C ONCLUSION R EFERENCES Electrocardiograph For a normal heart beat, the ECG usually looks like below: Fig. 2: Normal ECG [1] G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE E XAMPLES C ONCLUSION R EFERENCES Wavelets are used on ECG to: ◮ Enhance late potentials ◮ Reduce noise ◮ QRS detection ◮ Normal & abnormal beat recognition G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE E XAMPLES C ONCLUSION R EFERENCES Wavelets are used on ECG to: ◮ Enhance late potentials ◮ Reduce noise ◮ QRS detection ◮ Normal & abnormal beat recognition G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE E XAMPLES C ONCLUSION R EFERENCES Wavelets are used on ECG to: ◮ Enhance late potentials ◮ Reduce noise ◮ QRS detection ◮ Normal & abnormal beat recognition G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE E XAMPLES C ONCLUSION R EFERENCES Wavelets are used on ECG to: ◮ Enhance late potentials ◮ Reduce noise ◮ QRS detection ◮ Normal & abnormal beat recognition G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE E XAMPLES C ONCLUSION R EFERENCES Layout of Presentation 1. Theoretical ◮ CWT with complex analysis function ◮ CWT square modulus ( scalogram ) ◮ Local Symmetric Properties 2. See some examples G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE E XAMPLES C ONCLUSION R EFERENCES Layout of Presentation 1. Theoretical ◮ CWT with complex analysis function ◮ CWT square modulus ( scalogram ) ◮ Local Symmetric Properties 2. See some examples G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE S QUARE M ODULUS E XAMPLES P HASE B EHAVIOR C ONCLUSION R EFERENCES The Continuous Wavelet Transform � ∞ � t − b � 1 ( W Ψ f )( a , b ) = √ a f ( t ) · Ψ dt a −∞ ◮ Ψ is complex, compactly supported, and hermitian ( Ψ( t ) = Ψ( − t ) ) ◮ Ψ and f are at least twice continuous differentiable G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE S QUARE M ODULUS E XAMPLES P HASE B EHAVIOR C ONCLUSION R EFERENCES The Square Modulus We attempt to find an approximation to the square modulus. The square modulus of the CWT is defined as: | ( W Ψ f )( a , b ) | 2 = ( W Ψ f )( a , b )( W Ψ f )( a , b ) and ∂ | ( W Ψ f )( a , b ) | 2 = ∂ ( W Ψ f )( a , b ) ( W Ψ f )( a , b ) ∂ b ∂ b +( W Ψ f )( a , b ) ∂ ( W Ψ f )( a , b ) ∂ b G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE S QUARE M ODULUS E XAMPLES P HASE B EHAVIOR C ONCLUSION R EFERENCES A complex valued function Ψ( x ) can written as: Ψ( x ) = a ( x ) + ib ( x ) and d = d � � Ψ( x )Ψ( x ) dx [( a ( x ) + ib ( x )) · ( a ( x ) − ib ( x ))] dx = d � � a 2 ( x ) + b 2 ( x ) dx = 2 ( a ( x ) a ′ ( x ) + b ( x ) b ′ ( x )) = 2 · Re (Ψ ′ ( x ) · Ψ( x )) G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE S QUARE M ODULUS E XAMPLES P HASE B EHAVIOR C ONCLUSION R EFERENCES A complex valued function Ψ( x ) can written as: Ψ( x ) = a ( x ) + ib ( x ) and d = d � � Ψ( x )Ψ( x ) dx [( a ( x ) + ib ( x )) · ( a ( x ) − ib ( x ))] dx = d � � a 2 ( x ) + b 2 ( x ) dx = 2 ( a ( x ) a ′ ( x ) + b ( x ) b ′ ( x )) = 2 · Re (Ψ ′ ( x ) · Ψ( x )) G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE S QUARE M ODULUS E XAMPLES P HASE B EHAVIOR C ONCLUSION R EFERENCES The derivative of ( W Ψ f ) with respect to b is: � t − b � ∞ � f ( t ) · ∂ Ψ ∂ ( W Ψ f )( a , b ) 1 a = √ a dt ∂ b ∂ b −∞ � ∞ � t − b � 1 f ( t ) · Ψ ′ = √ dt a a 3 −∞ Using partial integration, � ∞ ∂ ( W Ψ f )( a , b ) � t − b � 1 f ′ ( t ) · Ψ √ a = dt ∂ b a −∞ G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE S QUARE M ODULUS E XAMPLES P HASE B EHAVIOR C ONCLUSION R EFERENCES The derivative of ( W Ψ f ) with respect to b is: � t − b � ∞ � f ( t ) · ∂ Ψ ∂ ( W Ψ f )( a , b ) 1 a = √ a dt ∂ b ∂ b −∞ � ∞ � t − b � 1 f ( t ) · Ψ ′ = √ dt a a 3 −∞ Using partial integration, � ∞ ∂ ( W Ψ f )( a , b ) � t − b � 1 f ′ ( t ) · Ψ √ a = dt ∂ b a −∞ G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE S QUARE M ODULUS E XAMPLES P HASE B EHAVIOR C ONCLUSION R EFERENCES Working out ∂ | ( W Ψ f )( a , b ) | 2 , we get ∂ b ∂ | ( W Ψ f )( a , b ) | 2 � ∂ ( W Ψ f )( a , b ) � = 2Re ( W Ψ f )( a , b ) ∂ b ∂ b and using ∂ ( W Ψ f )( a , b ) above, ∂ b � ∞ ∂ | ( W Ψ f )( a , b ) | 2 � t − b � = 2 f ′ ( t ) · Ψ a Re dt ∂ b a −∞ � ∞ � t − b � f ( t ) · Ψ dt a −∞ G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE S QUARE M ODULUS E XAMPLES P HASE B EHAVIOR C ONCLUSION R EFERENCES Working out ∂ | ( W Ψ f )( a , b ) | 2 , we get ∂ b ∂ | ( W Ψ f )( a , b ) | 2 � ∂ ( W Ψ f )( a , b ) � = 2Re ( W Ψ f )( a , b ) ∂ b ∂ b and using ∂ ( W Ψ f )( a , b ) above, ∂ b � ∞ ∂ | ( W Ψ f )( a , b ) | 2 � t − b � = 2 f ′ ( t ) · Ψ a Re dt ∂ b a −∞ � ∞ � t − b � f ( t ) · Ψ dt a −∞ G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE S QUARE M ODULUS E XAMPLES P HASE B EHAVIOR C ONCLUSION R EFERENCES For sufficiently smooth function U ( t ) and small a (fine-scale), � ∞ � ∞ � t − b � U ( t ) · Ψ dt = a U ( ax + b ) · Ψ( x ) dx a −∞ −∞ � ∞ ≈ a 2 U ′ ( b ) x · Ψ( x ) dx −∞ G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG

I NTRODUCTION T HE S QUARE M ODULUS AND P HASE S QUARE M ODULUS E XAMPLES P HASE B EHAVIOR C ONCLUSION R EFERENCES Applying the above approximation to ∂ | ( W Ψ f )( a , b ) | 2 , ∂ b ∂ | ( W Ψ f )( a , b ) | 2 ≈ 2 a 3 f ′ ( b ) · f ′′ ( b ) · | m | 2 ∂ b where � ∞ m = x · Ψ( x ) dx −∞ G ONZALEZ C ONTINUOUS W AVELET T RANSFORM IN ECG