WAVELET TRANSFORM IN TWO DIMENSIONS

Presented By:Ayushi Gagneja

Wavelet TransformThe wavelet transform is a tool that cuts up data, functions or operators into different frequency components, and then studies each component with a resolution matched to its scale

Uses a variable length window, e.g.:Narrower windows are more appropriate at high frequencies Wider windows are more appropriate at low frequencies

Wavelet Functions

■ Different purposes, different wavelets

Two-Dimensional Wavelets• For image processing applications we need

wavelets that are two-dimensional.• This problem reduces down to designing 2D filters.• We will focus on a particular class of 2D filters: separable filters (can be directly designed from their 1D counterparts)

2-D Separable WT

•For images we use separable Wavelet Transform

First we apply a 1-D filter bank to the rows of the image

Then we apply same transform to the columns of each channel of the result

Therefore, we obtain 3 high-pass channels corresponding to vertical, horizontal, and diagonal, and one approximation image

■In two dimensions, a two-dimensional scaling function, ϕ(x, y) , and three two-dimensional wavelet , and are required.

■Excluding products that produce one dimensional results, like ϕ(x) (x) , the four remaining products produce the separable scaling function

ϕ(x, y) = ϕ(x)ϕ(y)And separable, “directionally sensitive” wavelets

𝛹𝐻 (𝑥 , 𝑦 ) 𝛹𝑉 (𝑥 , 𝑦 )𝛹𝐷 (𝑥 , 𝑦 )

(x)ϕ(y) ϕ(x)(y) (x)(y)

corresponds to variations along diagonals

measures variations along columns (like, horizontal edges)

responds to variations along rows (like vertical edges)

The scaled and translated basis functions:

j / 2 j j j,m,n (x, y) 2 (2 x m,2 y n)

i H ,V, D i j ,m,n (x, y) 2 j / 2 (2 j x m,2 j y n),

where i is the subscript that assumes the values H, V and D

The Discrete Wavelet Transform of function f(x,y) of size M*N is then

1 M 1 N 1

x0 y 0MNW j0 , m, n f (x, y) j0 ,m,n (x, y)

1 M 1 N 1

x0 y 0

i i H ,V , DMN

W j, m, n f (x, y) i

j ,m,n (x, y)

Here, j0 is an arbitrary starting scaleW j0, m, n coefficients define an approximation of of f(x,y) at scale j0

coefficients add horizontal, vertical and diagonal details for the scales j>=j0

iW j, m, n

, f(x,y) is obtained viathe inverse discrete wavelet transform

iGiven the W and W

i

m n

MN i H ,V , D j jm n

ij ,m,n0

(x, y)

MN

1

1

W ( j, m, n)

f (x, y) W ( j0 , m, n) j0 ,m ,n (x, y)

Inverse Wavelet Transforms in Two Dimensions

10

2-D Analysis Filter Bank

1h

0h

1h

1h

0h

0h

x diagonal

vertical

horizontal

approximation

Like 1-D DWT, the 2-D DWT can be implemented using digital filters and downsamplers.

■The 2-D FWT filters the scale j+1 approximation coefficients to construct the scale j approximation and detail coefficients.

■The single scale filter can be “iterated” to produce P scale transforms in which scale j is equal to J-1, J-2,……., J-P.

■ Image f(x,y) is used as the W j0, m, n input.

■High pass or detail component characterizes the image’s high frequency information with vertical orientation.

■ Lowpass approximation component contains its low frequency vertical information.

■Both images are then filtered columnwise and downsampled to yield four quarter size output subimages ,.

Resultant Decomposition

2D Discrete Wavelet Transformation

d2 h2

v2 a2

a1

h1d1

v1

Original imageNxN

a3

d3 h3

v3

Level/Band/Scale 1

Level/Band/Scale 2Level/Band/Scale 3

d = diagonal detail (LOW/LOW)h = horizontal detail (HIGH/LOW)v = vertical detail (LOW/HIGH)a = approximation (HIGH/HIGH)

2D Discrete Wavelet Transformation (cont.)

d2

h2

v2h1

d1v1

a3d3h3

v3

Original imageNxN

Wavelet coefficientsNxN

2-D Synthesis Filter Bank

x̂diagonal

vertical

horizontal

approximation

1g

1g

1g

0g

0g

0g

■Synthesis Filter bank reverses the process.■At each iteration, four scale j approximation and detail

subimages are upsampled and convolved with the 2-D filters, one operating on subimages columns and other on its rows.

■Addition of the results yields the scale j+1 approximation, and process is repeated until the original image is reconstructed.

computer-generated image consisting of 2-D sine-like pulses on a black background

Each pass through the filter bank produced four quarter-size output images that were substituted for the input from which they were derived.

Example:

Next example • The decomposition filters used in the preceding example are

part of a well known family of wavelets called symlets, short for "symmetrical wavelets.“

• Figure 7.26(a) and (d), shows the corresponding decomposition and reconstruction filters.

• The coefficients of lowpass reconstruction filter go(n) = hφ(n) for 0 < n < 7 .

• Figure 7.26(g), a low-resolution graphic depiction of wavelet , is provided as an illustration of how a one-dimensional scaling and wavelet function can combine to form a separable, two-dimensional wavelet .

Fig. 7.26 (Con’t)

As in the Fourier domain, the basic approach is to – Step 1. Compute a 2-D wavelet transform of an image. – Step 2. Alter the transform.– Step 3. Compute the inverse transform.

How??■ In Fig 7.27(a) the lowest scale approximation component of the

discrete wavelet transform shown in Fig 7.25(c) has been eliminated by setting its values to zero.

■ Fig 7.27(b) shows, the net effect of computing the inverse wavelet transform using these modified coefficients is edge enhancement.

Note how well the transitions between signal and background are delineated, despite the fact that they are relatively soft, sinusoidal transitions.By zeroing the horizontal details as well—see Figs. 7.27(c) and (d)—we can isolate the vertical edges .

Next Example

Thresholding

hard thresholding, means setting to zero the elements whose absolute

values are lower than the threshold

soft thresholding, involves first setting to zero the elements whose absolute values are lower than the threshold and then scaling the nonzero coefficients toward zero

General wavelet-based procedure for denoising the image :Step 1. Choose a wavelet (Haar, symlet) and number of levels (scales), P, for the decomposition. Then compute the FWT of the noisy image. Step 2. Threshold the detail coefficients. That is, select and apply a threshold to the detail coefficients. Soft thresholding eliminates the discontinuity(at the threshold) that is inherent in hard thresholding. Step 3. Compute the inverse wavelet transform (i.e., perform a wavelet reconstruction) using the original approximation coefficients at level J - P and the modified detail coefficients for levels J — 1 to J — P.

shows the information that is lost.note the increase in edge informationin(f) and the corresponding decrease in edge detail in (e).

generated by simply zeroing the highest-resolution detail coefficients and reconstructing

shows the information that is lost in the process which was generated by computing the inverse FWT of the two-scale transform with all but the highest-resolution detail coefficients zeroed

Reconstruction of the DWT in which the details at both levels of the two-scale transform have been zeroed;

Wavelets Applications

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