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Base de la Ondoleta de Haar

Module by: Roy Ha, Justin Romberg. E-mail the authorsTranslated By: Fara Meza, Erika Jackson

Based on: Haar Wavelet Basis by Roy Ha, Justin Romberg

Summary: Este módulo nos da una descripción de las ondoletas y su utilidad como base en el procesamiento de imagenes. En particular veremos las propiedades de la base de la ondoleta de Haar.

Introducción

Las series de Fourier es una útil representación ortonormal en L 2 0 T L 2 0 T especialmente para entradas en sistemas LTI. Sin embargo es útil para algunas aplicaciones, es decir, procesamiento de imagenes (recordando el fenomeno de Gibb).

Las ondoletas, descubiertas en los pasados 15 años, son otro tipos de base para L 2 0 T L 2 0 T y tiene varias propiedades.

Comparación de Base

Las series de Fourier - c n c n dan información frecuente. Las funciones de la base duran todo el intervalo entero.

Figura 1: Funciones de la base de Fourier
Figura 1 (fig1.png)

Ondoletas - las funciones de la base con frecuencia nos dan información pero es local en el tiempo.

Figura 2: Funciones de la Base de la Ondoleta
Figura 2 (fig2.png)

En la base de Fourier, las funciones de la base son armónicas multiples de ej ω 0 t ω 0 t

Figura 3: base=1Tej ω 0 nt base 1 T ω 0 n t
Figura 3 (fig3.png)

En la base de la ondoleta de Haar , las funciones de la base son escaladas y trasladadas de la version de la "ondoleta madre" ψt ψ t .

Figura 4
Figura 4 (fig4s.png)

Funciones base ψ j , k t ψ j , k t se les pone un índice por un escalar j y un desplazamiento k.

Sea φt=1  ,   0t<T    0 t T φ t 1 Entonces φt 2j2ψ2jtk φt2j2ψ2jtk j and (k= 0 , 1 , 2 , , 2 j - 1 ) φ t 2 j 2 ψ 2 j t k j k 0 , 1 , 2 , , 2 j - 1 φ t 2 j 2 ψ 2 j t k

Figura 5
Figura 5 (fig5a.png)
ψt={1  if  0t<T2-1  if  0T2<T ψ t 1 0 t T 2 -1 0 T 2 T
(1)
Figura 6
Figura 6 (fig6.png)

Sea ψ j , k t=2j2ψ2jtk ψ j , k t 2 j 2 ψ 2 j t k

Figura 7
Figura 7 (fig7a.png)

Más grande jj → "delgado" la función de la base , j=012 j 0 1 2 , 2j 2 j cambia a cada escala: k= 0 , 1 , , 2 j - 1 k 0 , 1 , , 2 j - 1

Checar: cada ψ j , k t ψ j , k t tiene energia unitaria

Figura 8
Figura 8 (fig8s.png)
( ψ j , k 2tdt=1)( ψ j , k ( t ) 2 =1) t ψ j , k t 2 1 ψ j , k ( t ) 2 1
(2)

Cualesquiera dos funciones de la base son ortogonales.

Figura 9: Integral del producto = 0
(a) Misma escala(b) Diferente escala
Figura 9(a) (fig9a.png)Figura 9(b) (fig9b.png)

También, ψ j , k φ ψ j , k φ generan L 2 0 T L 2 0 T

Transformada de la Ondoleta de Haar

Usando lo que conocemos sobre espacios de Hilbert : Para cualquier ft L 2 0 T f t L 2 0 T , podemos escribir

Sintesis

ft=jk w j , k ψ j , k t+ c 0 φt f t j j k k w j , k ψ j , k t c 0 φ t
(3)

Análisis

w j , k =0Tft ψ j , k tdt w j , k t 0 T f t ψ j , k t
(4)
c 0 =0Tftφtdt c 0 t 0 T f t φ t
(5)

nota:

los w j , k w j , k son reales
La transformación de Haar es muy útil especialemte en compresión de imagenes.

Ejemplo 1

Esta demostración nos permite crear una señal por combinación de sus funciones de la base de Haar, ilustrando la ecuación de sistesis de la ecuación de la Transformada de la Ondoleta de Haar. Veámos aquí para las instrucciones de como usar el demo.

LabVIEW Example: (run) (source)

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