Moments are important features used in pattern recognition, image analysis and computer vision. Geometric moments are calculated using window functions with great discontinuities at window boundary and the bases are not orthogonal. In order to better characterize noisy images, one should use the orthogonal moments with a smoothing window function. In the present paper, by use of the well-known Gaussian functions as smoothing widow function, we first introduce Gaussian-Hermite moments which are shown to be orthogonal smoothed moments. Then we present the recursive calculation of Gaussian-Hermite moments. The recursive calculation not only gives an efficient scheme for Gaussian-Hermite moment calculation, but also reveals the essential relation between orthogonal Gaussian- Hermite moments and Gaussian-filtered signal derivatives. From this relation, we see that the Gaussian-Hermite moments are good local features for noisy signal. Moreover, because Gaussian derivatives satisfy the conditions for mother wavelets, Gaussian-Hermite moments defined from Gaussian functions of different (sigma) correspond to wavelet development of the input signal. In fact, the n-order Gaussian-Hermite moment is a linear combination of different Gaussian-derivative wavelets of the input signal. So the use of Gaussian-Hermite moments gives also an efficient approach to representing the input signal in an orthogonal functional space from wavelet analysis. All the results above obtained for 1-D signals are generalized to 2-D image analysis.