Effros and Chou (see Proceedings of the IEEE International Conference on Image Processing, Washington, DC, 1995) introduce a two-stage universal transform code called the weighted universal transform code (WUTC). By replacing JPEG's single, non-optimal transform code with a collection of optimal transform codes, the WUTC achieves significant performance gains over JPEG. The computational and storage costs of that performance gain are effectively the computation and storage required to operate and store a collection of transform codes rather than a single transform code. We consider two complexity- and storage-constrained variations of the WUTC. The complexity and storage of the algorithm are controlled by constraining the order of the bases. In the first algorithm, called the fast WUTC (FWUTC), complexity is controlled by controlling the maximum order of each transform. On a sequence of combined text and gray-scale images, the FWUTC achieves performance comparable to the WUTC. In the second algorithm, called the jointly optimized fast WUTC (JWUTC), the complexity is controlled by controlling the average order of the transforms. On the same data set and for the same complexity, the performance of the JWUTC always exceeds the performance of the FWUTC. The JWUTC and FWUTC algorithm are interesting both for their complexity and storage savings in data compression and for the insights that they lend into the choice of appropriate fixed- and variable-order bases for image representation.