Kirill Serkh

Philip Greengard, Kirill Serkh

Zernike polynomials are a basis of orthogonal polynomials on the unit disk that are a natural basis for representing smooth functions. They arise in a number of applications including optics and atmospheric sciences. In this paper, we provide a self-contained reference on Zernike polynomials, algorithms for evaluating them, and what appear to be new numerical schemes for quadrature and interpolation. We also introduce new properties of Zernike polynomials in higher dimensions. The quadrature rule and interpolation scheme use a tensor product of equispaced nodes in the angular direction and roots of certain Jacobi polynomials in the radial direction. An algorithm for finding the roots of these Jacobi polynomials is also described. The performance of the interpolation and quadrature schemes is illustrated through numerical experiments. Discussions of higher dimensional Zernike polynomials are included in appendices.

Zakaria Patel, Kirill Serkh

Diffusion models are a powerful class of generative models capable of producing high-quality images from pure noise using a simple text prompt. While most methods which introduce additional spatial constraints into the generated images (e.g., bounding boxes) require fine-tuning, a smaller and more recent subset of these methods take advantage of the models' attention mechanism, and are training-free. These methods generally fall into one of two categories. The first entails modifying the cross-attention maps of specific tokens directly to enhance the signal in certain regions of the image. The second works by defining a loss function over the cross-attention maps, and using the gradient of this loss to guide the latent. While previous work explores these as alternative strategies, we provide an interpretation for these methods which highlights their complimentary features, and demonstrate that it is possible to obtain superior performance when both methods are used in concert.