In this paper, we propose relative projective differential invariants (RPDIs) which are invariant to general projective transformations. By using RPDIs and the structural frame of integral invariant, projective weighted moment invariants (PIs) can be constructed very easily. It is first proved that a kind of projective invariants exists in terms of weighted integration of images, with relative differential invariants as the weight functions. Then, some simple instances of PIs are given. In order to ensure the stability and discriminability of PIs, we discuss how to calculate partial derivatives of discrete images more accurately. Since the number of pixels in discrete images before and after the geometric transformation may be different, we design the method to normalize the number of pixels. These ways enhance the performance of PIs. Finally, we carry out some experiments based on synthetic and real image datasets. We choose commonly used moment invariants for comparison. The results indicate that PIs have better performance than other moment invariants in image retrieval and classification. With PIs, one can compare the similarity between images under the projective transformation without knowing the parameters of the transformation, which provides a good tool to shape analysis in image processing, computer vision and pattern recognition.
This paper presents a novel method to derive invariants of 2D grayscale images under projective transformation. Invariants of images are good features for object recognition and have attracted extensive attention. Although geometric invariants of point locations such as cross ratios are well known for centuries, we have found no reported invariants for grayscale images that remain the same under projective transformation. It has even been proven that projective invariants of images cannot be derived from the standard geometric moments of images. However, this does not mean that there is no projective invariant of images in other forms. We will prove in this paper that projective invariants of images do exist as functions of the generalized moments of images. We first derive some projective invariant relations between an image function and its derivative functions. Next, we extend the traditional definition of moments by considering both the image function and its derivative functions. Then we derive a set of functions of the generalized moments that are projective invariant. Experimental results indicate that the proposed invariants have certain discriminating power for object recognition.