This paper presents a spherical measure based spherical image representation(SMSIR) and sphere-based resampling methods for generating our representation. On this basis, a spherical wavelet transform is also proposed. We first propose a formal recursive definition of the spherical triangle elements of SMSIR and a dyadic index scheme. The index scheme, which supports global random access and needs not to be pre-computed and stored, can efficiently index the elements of SMSIR like planar images. Two resampling methods to generate SMSIR from the most commonly used ERP(Equirectangular Projection) representation are presented. Notably, the spherical measure based resampling, which exploits the mapping between the spherical and the parameter domain, achieves higher computational efficiency than the spherical RBF(Radial Basis Function) based resampling. Finally, we design high-pass and low-pass filters with lifting schemes based on the dyadic index to further verify the efficiency of our index and deal with the spherical isotropy. It provides novel Multi-Resolution Analysis(MRA) for spherical images. Experiments on continuous synthetic spherical images indicate that our representation can recover the original image signals with higher accuracy than the ERP and CMP(Cubemap) representations at the same sampling rate. Besides, the resampling experiments on natural spherical images show that our resampling methods outperform the bilinear and bicubic interpolations concerning the subjective and objective quality. Particularly, as high as 2dB gain in terms of S-PSNR is achieved. Experiments also show that our spherical image transform can capture more geometric features of spherical images than traditional wavelet transform.