Bhatia and Wolf have shown [15] that a polynomial that is invariant in form for any rotation of axes about the origin must be of the form

where is a radial polynomial in of degree. We now defined a new set of orthogonal moments, Pseudo-Jacobi -Fourier Moments (PJFM’s), based on Jacobi polynomials. In the polar coordinates, radial Jacobi polynomials are expressed as

where the function is orthogonal over the range [0,1]

where is Kronecker symbol, and

The radial polynomials of OFMM’s, CHM’s and ZM’s belong to Jacobi polynomials with and respectively. When , radial Jacobi polynomials become, so Pseudo-Jacobi polynomials are defined as

So a new set of orthogonal polynomial function, which consists of radial function and angular function, is obtained as

obviously, the is orthogonal over the range [0, 1].

According to the orthogonal theory, the image function can be written as an infinite series expansion in terms of

where

then is defined as Pseudo-Jacobi-Fourier Moments (PJFM’s), and is the maximum size of the objects that can be encountered in a particular application.

Actually, most of the images are defined in the Cartesian coordinate system. When calculating PJFM’s, the images need to be converted to polar coordinates, so may cause some problems: 1) Quantization error is introduced into the calculation of PJFM’s, which increases calculating amount and additional noise; 2) In polar coordinates, the discrete points near from the origin are more than those in the Cartesian coordinate system, which lead to information redundancy; 3) Far from the origin, the scattered points in the polar coordinate system are less than those in the Cartesian coordinate system, which lead to information loss. Therefore, an improved algorithm for computing PJFM’s Cartesian coordinate system is developed in this paper.

When calculating PJFM’s in the Cartesian coordinates, the image should first be normalized into the unit circle, and the integral region of Equation (11) as shown in Figure 1.

Substitute Equation (6) into Equation (8)

where

and (13)

Substitute Equation (13) and into Equation (12)

Figure 1 shows that s1 - s2 is not the part of the image and there is no pixel over it, Equation (15) can be written as

So Equation (16) is the general formula for calculating PJFM’s in Cartesian coordinate system.

Image reconstruction can be used as an effective means to evaluate the quality of feature extraction. The more PJFM’s used to reconstruct images, the closer to the original image. An image of capital E, shown in Figure 2, is reconstructed by Equation (17).

The four corners of the reconstructed image are black and some pixels have been lost in the Figure 3. But the black area of the four corners of the reconstructed image is shrinking with the increase of N and M, and the edge information is well preserved in the Figure 4. Figure 4 also shows that E is differentiated well when N = M = 6 instead of N = M = 10 as in Figure 3.

As for computation time, reconstruction time in the Cartesian coordinate system is much shorter than the polar coordinates system, shown in Table 1.

In order to test the feature extraction performance of the improved PJFM’s, a recognition experiment was done by using microscopic image of 8 kinds of helminth eggs, such as Fasciola hepatica (a), moniezia (b), Hairy ail nematode (c), paramphistomum (d), Nematodirus (e), Dicrocoelium chinensis (f), coccidium (g), Pancreatic Eurytrema (h).

The training sets consist of 20 different versions from each kind of helminth eggs, including 160 images. Testing sets consist of 307 untrained images from different version of 8 kinds of helminth eggs. Figure 5 gives a multi-distorted image of the microscopic image of some helminth eggs in a testing set.

Choosing N = M = 8, the image feature extracted respectively by using Equation (16) in Cartesian coordinate system and Equation (11) in polar coordinate system, then the target objects were recognized by the minimum average distance rules . Euclidean distances are calculated by Equation (18), and the result is shown in Table 2.

where is the PJFM of the testing object, and is the PJFM of the reference object of class i.

As can be seen from Table 2, the total recognition rate of the microscopic images of 8 helminth eggs was 92.2% with improved PJFM’s, 7.2% higher than PJFM’s in polar coordinate system.