Summary: For \(L_2(\mathbb{R}^2)\), one can easily construct an orthonormal basis from an orthonormal basis of \(L_2(\mathbb{R})\) using the tensor product; and that orthonormal basis is in fact separable orthonormal basis. The separable basis have a number of disadvantages, as they have very little design freedom. Furthermore, the separability imposes an unnecessary product structure on \(\mathbb{R}^2\), that is artificial for natural images. In this paper, we construct a compactly supported nonseparable smooth orthonormal basis for \(L_2({\mathbb{R}}^2)\) by constructing smooth projection \(P_{I\times M}\) associated with a rectangle \(I\times M=[a,b]\times [c,d]\subset \mathbb{R}^2\) on \(L_2({\mathbb{R}}^2)\) without using tensor product.