We propose a new class of generalized moment functions (GMFs) that scan the object with different probing functions. Using the GMF, it is possible to extract a unique geometric point within the object, called the generalized centroid (G-centroid). We can obtain a set of discrete G-centroids from the same object by using different GMFs. The GMFs, which are similar to traditional moment functions, can also be used to describe the global shape of the object, including symmetry and fullness. However, the GMFs, along with the G-centroids, can further serve to construct a feature vector of the object, which is critical to the process of image registration and pattern recognition. Conformal transforms (C-transforms) are another tool used to probe the object by rearranging the latter's mass distribution, without distorting its shape. Using the C-transformed object, it is possible to detect a new mass centroid and G-centroid. More distinguishing feature points can be extracted from the same object by changing the combination of different GMFs and C-transforms. As GMF and centroid detection can be performed by convolution, the centroid and G-centroid can be detected optically in real time. It is also possible to optically implement some of the C-transforms. We present the results of GMF and C-transform applications, including image registration and pattern recognition.