The nonsubsampled Laplacian pyramid (NSLP) is widely used as a common multiresolution decomposition method in various nonsubsampled image transforms. However, the NSLP has a fixed spectrum partition, and thus cannot represent images accurately, and flexibly. We propose a new adaptive arbitrary multiresolution decomposition to solve these problems. First, using the affine characteristics of a pseudopolar Fourier transform (PPFT), we apply a 1-D nonuniform filter bank to the modulated PPFT to obtain a 2-D arbitrary resolution filter bank. This filter surpasses the limitation of fixed spectrum partitioning of the traditional tree structure. We then demonstrate that the proposed method satisfies the compact frame condition, and has translation invariance, and a linear phase. Furthermore, we propose an adaptive spectrum division approach at various scales based on image spectrum information based on a 2-D filter bank of arbitrary multiresolution, so our method can capture important visual information more accurately. Finally, we combine our method with a nonsubsampled directional filter bank of the nonsubsampled contourlet transform to create a new multiscale geometric analysis (MGA) method, and verify that the new method can have perfect reconstruction properties. The new MGA also performs better in image denoising, and recognition experiments than state-of-the-art MGA methods with translation invariance.