Michael Dessauer and Sumeet Dua

Fig. 2.26. (a) Three optic disk images smoothed with Gaussian filters of increasing scale and (b) first component from PCA transform, containing the largest variability in the image.

2.4.3. Multiscale Features

We will next discuss representing a 2D image with varying degrees of scale through computational operations that reduce the resolution, yet retain multiple levels of detail that can be used as feature descriptors. The texture or shape of an image region at its native resolution can provide an excessive amount of unnecessary detail or noise artifacts, which can reduce classification accuracy. In addition, discriminative characteristics of an image region may occur only at certain reduced scales, which would go unnoticed at a lower scale. We will describe two methods that reduce an image’s dimensionality at multiple levels (scales), representing the image as 3D pyramid structure of linearly decreasing resolutions to more easily extract discriminative features. We will give examples of how such methods can be used to extract features in retinal images.

2.4.3.1. Wavelet transform

We will discuss an image transformation that creates a multi-scale pyramid of coefficients that can each represent the ROI at different resolutions. Unlike the Fourier transform, wavelet transformations retain spatial and texture information, which can then be used as input in other feature extraction methods, such as those described above. Wavelets retain this information

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because their basis functions (or wavelets) are localized in the image, where a Fourier basis function spans the entire image. One such method is the discrete wavelet transform.

Although a full explanation of wavelet theory will not be provided here, we will give the mathematical formulation of the 2D discrete wavelet transform,29 which is typically the wavelet transform used with images. We first need to chose basis functions, which includes a wavelet, ψ(x, y), and a scaling function, ϕ(x, y). Both functions are linearly separable, so that the 2D combinations give information for a horizontal (H), vertical (V), and diagonal (D) direction:

Each of these values describes the variations at a particular point along the specified direction. The linear combination of the scaling function produces:

ϕ(x, y) = ϕ(x)ϕ(y).

We use scaling and translation variables with the functions above to position the function correctly before convolving with the image of size M × N using the following formula:

The 2D discrete wavelet transform of the function f(x, y) of an image of size M × N is

j0 is the starting scale, which is typically set to j0 = 0, then set to N = M = 2j , j = 0, 1, 2, . . . , J − 1, and m, n = 0, 1, 2, . . . , 2j − 1. We can

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Michael Dessauer and Sumeet Dua

Fig. 2.27. (a) Original ROI, (b) 2D discrete wavelet-transform image pyramid, and (c) third-level reconstruction from approximation coefficients.

then perform the inverse discrete wavelet transform to find f(x, y) using

We use the simplest wavelet function, the Haar wavelet (Fig. 2.27), to construct a wavelet decomposition pyramid, containing details in the horizontal, vertical, and diagonal directions at different sizes. We can use these details to extract shape, texture, and moments at multiple scales, thus, increasing available ways to describe a ROI. The approximation of the image at varying resolutions also provides a compact representation of a region, reducing dimensionality in cases where fine details of are not necessary.

A method similar to that illustrated in Fig. 2.26 is used to localize the optic disc by performing a fourand five-level Haar wavelet decomposition, with the optic disc reduced to a small cluster of coefficients.30

2.4.3.2. Scale-space methods for feature extraction

As discussed in the previous section, we are interested in finding discriminative characteristics of a ROI, many of which occur over varying spatial scales. Several methods other than wavelet decomposition can provide a scale-space representation by convolving an image with kernels of varying size, typically suppressing details of varying scales. We will discuss two sets of scale space representations that can be used to extract multi-scale

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features from the retinal anatomy: difference of Gaussians and Hessian determinants.

As discussed in Sec. 2.2.2.2, Gaussian kernels smooth an image by varying amounts based upon the overall size and magnitude of the discrete 2D Gaussian curve. We can use subsequently smoothed images, L(x, y; t), convolved with increasing kernel sizes (t = σ2) to create a set of difference images, written as

which reveal scale-specific features of the ROI (Fig. 2.28). This method can also be used to approximate the Laplacian of the Gaussian (LoG as discussed in Sec. 2.3.1.3) for edge detection. We can now perform shape and texture feature extraction operations to the new set of multi-scale images to find a scale-specific features.

We can derive a set of 2D matrices that find corners for each smoothed image, L(x, y; t) by first calculating the Hessian matrix, which is the square

Fig. 2.28. Top-left: Gaussian smoothed image with increasing t; top-right: difference of Gaussian images (DoG); bottom-left: determinant of Hessian matrices with increasing t; and bottom-right: feature location (markers) and scale (radius of circle) from Hessian scale space.

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