The most efficient way to compute the wavelet transform is to use a function like

Figure 18.1 Five wavelet scales reveal different spatial structures in the classic Lena image. Scale 1 shows small spatial structures with a kernel radius of 1; the radius of the Scale 2 is 2, that of Scale 3 is 4, and so on—each wavelet scale doubling the radius of the preceding scale and showing large structures.

Figure 18.1 Five wavelet scales reveal different spatial structures in the classic Lena image. Scale 1 shows small spatial structures with a kernel radius of 1; the radius of the Scale 2 is 2, that of Scale 3 is 4, and so on—each wavelet scale doubling the radius of the preceding scale and showing large structures.

the Mexican Hat transform to isolate each wavelet scale (i.e., each band of spatial frequencies) from the original image in turn. For astronomy, the a trous algorithm is particularly effective, and it has the added advantage of being computationally fast and efficient. The name of the algorithm—a trous—means "with gaps," since, as we shall see, at the higher wavelet scales, the wavelet convolution kernel has big holes in it.

The a trous wavelet transform builds wavelet scales iteratively. It begins by applying a 3 x 3 high-pass filter to the original image to form Scale 1. This level contains all of the small-scale high-frequency components in the image, and because the sum of elements in the high-pass filter is zero, the mean value of Scale 1 is also zero. The a trous procedure then subtracts Scale 1 from the original image, leaving a residual image free of small-scale high-frequency components. The algorithm now iterates by applying a high-pass filter "with gaps" to the residual image to create Scale 2. This filter is diluted by placing its outer elements two pixels from the center. The procedure then subtracts Scale 2 from the residual image, creating a new residual free of both scale 1 and Scale 2 components.

The algorithm iterates again, applying an even more dilute high-pass filter in which the outer elements lie four pixels from the center, creating Scale 3. It then subtracts Scale 3 from the residual to make a new residual.

At each following step, the radius of the high-pass filter doubles: 8, 16, 32, 64, 128, 256, subtracting the next larger structure from the image. Successive wavelet scales contain progressively coarser structures, as shown by the example in Figure 18.1. By Scale 8, the residual contains very little structure. However, when the residual and all the wavelet scales are added together, they recreate the original image exactly.

Because the mean of each scale removed from the original is zero, the total pixel value of the residual remains the same as the original image. This has the interesting consequence that—without making the image darker or lighter—we can multiply, divide, or threshold wavelet scales to enhance, control, or remove image features based on their spatial frequency content.

The Mexican Hat function is a continuous function equal to the difference between a positive Gaussian function and a negative Gaussian function of twice the width and half the value. The smallest equivalent one-dimensional convolution kernel is:

The smallest equivalent two-dimensional wavelet kernel can be formed as the convolution of the row kernel with the column kernel:

Because this is a separable kernel, convolution of the image to produce wavelet scales is fast and efficient.

18.1.2 Properties of the A Trous Wavelet Transform

Understanding the properties of the wavelet transform is the key to using it effectively. Here is the nomenclature:

• The original image is S0, and a pixel in image S0 at location (x,y) is S0(x,y).

• The wavelet kernel is cp, containing elements q>(i,j).

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