Visual Anagrams: Generating Multi-View Optical Illusions with Diffusion Models

Our method is conceptually simple. We take an off-the-shelf diffusion model and use it to estimate the noise in different views or transformations, \(v_i\), of an image. The noise estimates are then aligned by applying the inverse view, \(v_i^{-1}\), and averaged together. This averaged noise estimate is then used to take a diffusion step.

We find that not every view function works with the above method. Of course, \(v_i\) must be invertible, but we discuss two additional constraints.

A diffusion model is trained to estimate the noise in noisy data \(\mathbf{x}_t\) conditioned on time step \(t\). The noisy data \(\mathbf{x}_t\) is expected to have the form \[\mathbf{x}_t = w_t^{\text{signal}}\underbrace{\mathbf{x}_0}_{\text{signal}} + w_t^{\text{noise}}\underbrace{\epsilon\vphantom{\mathbf{x}_0}}_{\text{noise}}.\] That is, \(\mathbf{x}_t\) is a weighted average of pure signal \(\mathbf{x_0}\) and pure noise \(\epsilon\), specifically with weights \(w_t^{\text{signal}}\) and \(w_t^{\text{noise}}\). Therefore, our view, \(v\) must maintain this weighting between signal and noise. This can be achieved by making \(v\) linear, which we represent by the square matrix \(\mathbf{A}\). By linearity \[\begin{aligned} v(\mathbf{x}_t) &= \mathbf{A}(w_t^{\text{signal}} \mathbf{x}_0+w_t^{\text{noise}} \epsilon)\\[7pt] &= w_t^{\text{signal}} \underbrace{\mathbf{A}\mathbf{x}_0}_{\text{new signal}} + w_t^{\text{noise}} \underbrace{\mathbf{A}\epsilon}_{\text{new noise}}. \end{aligned}\] Effectively, \(v\) acts on the signal and the noise independently, and combines the result with the correct weighting.

Diffusion models are trained with the assumption that the noise is drawn iid from a standard normal. Therefore we must ensure that the transformed noise also follows these statistics. That is, we need \[\mathbf{A}\epsilon \sim \mathcal{N}(0, I).\] For linear transformations, this is equivalent to the condition that \(\mathbf{A}\) is orthogonal. Intuitively, orthogonal matrices respect the spherical symmetry of the standard multivariate Gaussian distribution.

Therefore, for a transformation to work with our method, it is sufficient for it to be orthogonal.

Most orthogonal transformations on images are meaningless, visually. For example, we transform the image below with a randomly sampled orthogonal matrix.

However, permutations matrices are a subset of orthogonal matrices, and are quite interpretable. They are just rearrangements of pixels in an image. This is where the idea of a visual anagram comes from. The majority of illusions here can be interpreted this way—as specific rearrangements of pixels—such as rotations, flips, skews, "inner rotations," jigsaw rearrangements, and patch permutations. Finally, color inversions are not permutations, but are orthogonal as they are a negation of pixel values.