I have a halfling who is wearing a black hood covering his head in a moving carriage.
Can I cast Silent Image and present an image of myself with the hood on, but actually take the hood off while the carriage is moving from location to location?
Or is the Image fixed in a geographical location as opposed to following a moving target?
Using an illusion to alter a creature's appearance is the province of spells of the illusion subschool glamer like disguise self, hallucinatory terrain, and veil. A spell like illusory wall or silent image is an illusion spell of the subschool figment, and those spells typically add information to the environment rather than change already-present information. This GM would rule that the spell silent image can't be used to change a creature's appearance, even if the caster adds to the environment a floating, stationary hood beneath which the caster stands. Seriously, illusions are complicated enough without trying to force one illusion spell's effect into a different subschool. (Instead of trying to use the silent image spell this way, this GM suggests buying a hat of disguise or using the Disguise skill: this latter is free, and others can help you… or someone else who's better at the skill Disguise than you can put a disguise on you!)
The spell silent image possesses the entry Range long (400 ft. + 40 ft./level) and and the entry Effect visual figment that cannot extend beyond four 10-ft. cubes + one 10-ft. cube/level (S). The spell's range—as per Aiming a Spell—indicates that the caster
must designate the location where [its effect is] to appear, either by seeing it or defining it. Range determines how far away an effect can appear, but if the effect is mobile, after it appears it can move regardless of the spell’s range.
…But the effect created by the silent image spell does not say it's mobile in the same way, for example, that the sphere created by the spell flaming sphere is or that the creatures brought forth by a summon monster effect are. That is, the silent image spell says that the caster "can move the image within the limits of the size of the effect," yet that means the created image is limited to moving within the spell's shapeable area. (Again see Aiming a Spell for an explanation of a shapeable area). In short, a spell's point of origin is fixed when it's cast unless otherwise stated, and the spell silent image does not say otherwise, this despite the fact that the image can roam freely within the designated area around the spell's point of origin.
Whether existence is composed of unchanging, stationary-in-space 5-ft. squares over which everything moves or if 5-ft. squares travel with the map that they're attached to is something for the GM to decide. (Also see this question—while for 3.5e same rules apply to Pathfinder.)
While the first ruling is elegant, the second ruling allows many spells to be cast on, for example, moving ships (although Pathfinder sometimes creates unique rules spells when they're cast on boats) and, similarly, carriages of size sufficient to have their own maps—often in such a way that makes spells comprehensible, manageable, and playable. Thus, under this second ruling—which this GM supports—, a carriage big enough to warrant its own unique map should be able to support the point of origin of a silent image spell's effect and that effect would move with that big carriage, but a carriage that's too little to warrant its own map will see it leave a silent image effect in its dust.
