The drift-diffusion model (DDM) is a common approach to understanding human decision making. It considers decision making as accumulation of evidence about visual stimuli until sufficient evidence is reached to make a decision (decision boundary). Recently, Smith and colleagues proposed an extension of DDM, the time-varying DDM (TV-DDM). Here, the standard simplification that evidence accumulation operates on a fully formed representation of perceptual information is replaced with a perceptual integration stage modulating evidence accumulation. They suggested that this model particularly captures decision making regarding stimuli with dynamic noise. We tested this new model in two studies by using Bayesian parameter estimation and model comparison with marginal likelihoods. The first study replicated Smith and colleagues' findings by utilizing the classical random-dot kinomatogram (RDK) task, which requires judging the motion direction of randomly moving dots (motion discrimination task). In the second study, we used a novel type of stimulus designed to be like RDKs but with randomized hue of stationary dots (color discrimination task). This study also found TV-DDM to be superior, suggesting that perceptual integration is also relevant for static noise possibly where integration over space is required. We also found support for within-trial changes in decision boundaries ("collapsing boundaries"). Interestingly, and in contrast to most studies, the boundaries increased with increasing task difficulty (amount of noise). Future studies will need to test this finding in a formal model.