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As photons bounce around and finally find their way into our eyes, are they continually relinquishing the information of the previous thing they "bounced off of"? Is this the reason why we receive a coherent, continuous image and not a chronologically chaotic image of the past meshed with the present?

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It depends on what the photon is bouncing off of. For most materials, it's true that a reflected photon carries no information about what it interacted with prior to the reflection, but that's not the case for all materials.

To be precise, when a photon - actually an electromagnetic wave - is reflected, what really happens is that it gets absorbed by the reflecting material, and then a new EM wave is emitted to "replace" it. (Actually, it's more complicated than that; see this other question for example, but the re-emission explanation will suffice for now.) There are two properties of interest that could apply to the re-emitted wave:

  • For most materials, the re-emitted wave's frequency is determined by the electronic properties of the material itself; these materials can only emit certain specific frequencies of light, which determines their color. Other materials, though, can emit any frequency of light, which means the frequencies (and thus colors) of the re-emitted EM waves will match those of the absorbed waves. These materials are either reflective or white.
  • For most materials, the absorption and emission are separate events, which means that the emitted EM wave can be in a random direction, regardless of which direction the absorbed wave came from. This is called diffuse reflection and it means that the appearance of the reflecting material will be determined by the material itself. But for some materials, due to their electronic structure the absorption and emission are "linked" in a sense, such that the emitted wave comes out at an angle which depends on the direction of the absorbed wave and the orientation of the surface. This is called specular reflection and it is the kind of thing that happens in mirrors. In this case, the "appearance" of the material is determined by the absorbed light. If you think about it, when you look at a mirror, you don't see the mirror itself, but rather some other thing whose reflection appears in the mirror.

In fact, for both these effects, the possibilities listed are only extremes of a continuum. A material can be partially reflective, for example, meaning that part of the absorbed EM wave is re-emitted in a specific direction and part of it is re-emitted randomly in any direction, as in window glass. In a case like this, you could say that the reflected light waves carry partial information about what they had previously been emitted from.

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