Michael Dessauer and Sumeet Dua

µ30 = M30 − 3xM¯ 20 + 2x¯2M10, and

µ03 = M03 − 3yM¯ 02 + 2y¯ 2M01.

From these moments, ηij can be made invariant to translation and scale by dividing by the scaled 00th moment, written as:

Finally, the seven Hu moments, ϕi

ϕ1 = η20 + η02,

ϕ2 = (η20 + η02)2 + (2η11)2,

ϕ3 = (η30 + 3η12)2 + (3η21 + η03)2, ϕ4 = (η30 + η12)2 + (η21 + η03)2,

In Fig. 2.24, we show that Hu moments produce almost identical values for rotated, scaled, and translated versions of a segmented region. Invariant moments work well in situations in which both the orientation of a segment can be unreliable, and characteristics of the region’s spatial intensity provide important information for classification.

2.4.2. Data Transformations

These next feature descriptor sets use methods to transform the data from the spatial domain, or Euclidean space, to coordinate systems that provide different types of characterization for an ROI that is not obvious or directly calculable from the image matrix. We will briefly describe and discuss

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Fig. 2.24. (Top: left–right): an original segmented optic disk, scaled 2x, rotated 45◦, rotated 45◦ and translated, (bottom) table of Hu moments for each image.

two methods that can both contribute useful discriminative information to a classification algorithm and reduce data dimensionality by retaining only data of interest. As you will see, computational methods for image transformations use classic linear algebraic operations to produce refined sets of data that can better represent image regions in a reduced, descriptive domain.

2.4.2.1. Fourier descriptors

The frequency domain describes an image not by its intensities at specific (x, y) locations on a 2D matrix, but as magnitudes of periodic functions of varying wavelengths. Although the functions’ coefficients no longer retain any spatial information, they are divided into low, middle, and high frequencies with low frequencies providing overall structural information and high frequencies containing the detail. Feature descriptors can be localized from this set of Fourier coefficients, with low frequency coefficients retaining region structure (Fig. 2.25). To obtain the 2D discrete Fourier transform coefficients of a region f(x, y) of size M × N, we use the equation26

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Michael Dessauer and Sumeet Dua

Fig. 2.25. (a) Original ROI, (b) visualization of a center-shifted 2D discrete Fourier transform,

(c) reduced set of Fourier coefficients, and (d) 2D inverse transform of the reduced coefficients.

descriptors, or we can set values of F to zero and transform back to Euclidean space using the inverse transform equation:

for and x = 0, 1, 2, . . . , M − 1 and y = 0, 1, 2, . . . , N − 1. In Fig. 2.25, we demonstrate how a Fourier transform can retain the structural information of a 166×133 region using a reduced set of 13×13 Fourier descriptors.A subset of frequency-domain descriptors has previously been used to extract blood vessels details from retinal images successfully.27 Although not rotation, scale, or translation invariant, a linear operation with constant values in the frequency domain can correct for the identified spatial changes.8

2.4.2.2. Principal component analysis (PCA)

Another linear transformation that creates a useful feature set is principal component analysis, PCA. This method is defined as an orthogonal linear transformation of data projected onto a new coordinate system, where the greatest variance of any projection of the data lies on the line that passes through the first coordinate (principal component), with the second greatest variance on the second coordinate, and so on.28 This method reduces a set of variables by calculating the eigenvalues of the covariance matrix, after a mean normalization occurs for each attribute.

We show how to use PCA with image regions by first vectorizing an M × N image region into a vector, v, with M N dimensions. We then take a set, K, of vectorized images, which creates M N sets of 1D vectors, x,

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of size k:

x1

x2

x = . , i = 1, 2, . . . , M N. (2.52)

..

xk

We then construct a vector of mean values, mx , calculated for each i as

K

mx = 1 xk , (2.53)

K k=1

which we can then use to form a covariance matrix, Cx , which is written as

Next, let A be a 2D matrix with rows that are filled with eigenvectors of Cx sorted so that eigenvector rows correspond to eigenvalues of descending size. We use A to map x values into vectors, y, using the following:

which has diagonal terms that are the eigenvalues of Cx . We can then recover any x from a corresponding y without using the entire matrix A. We can use only the k large eigenvectors, using a transform matrix, Ak , of size k × n by using the following:

This method is used in optic disk localization by normalizing and registering a set of well-defined M × N optic disk regions, then vectoring each image to form a set of 1 − D M N vectors. The PCA transform is performed first on a set of training image vectors, and the top six eigenvalues are kept as features to represent the training set. Tested ROIs are projected onto these eigenvectors, and a Euclidean distance metric is used to measure optic disk “likeness.” We provide an example of how these lower dimensional eigenvectors reconstruct an optic disk using a training image convolved with differing Gaussian filters in Fig. 2.26.

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