After a few decades of video, there would have to be more than enough data to do the reconstruction.
Sandberg & Bostrom are skeptical. Page 109:
Again, it is sometimes suggested that recording enough of the sensory experiences and actions would be enough to produce brain emulation. This is unlikely to work simply because of the discrepancy in the number of degrees of freedom between the brain (at least 10^14 synaptic strengths distributed in a 10^22 element connectivity matrix) and the number of bits recorded across a lifetime (less than 2 * 10^14 bits (Gemmell, Bell et al., 2006)).
Also, Sandberg in 2010:
I would really like to develop a good argument about when reconstructing a mind from its inputs and outputs is possible. Being a slice-and-dice favoring WBE thinker, I am suspicious of the feasibility. But am I wrong?
It is not too hard to construct "minds" that cannot be reconstructed easily from outputs. Consider a cryptographically secure pseudorandom number generator: watching the first k bits will not allow you to predict the k+1 bit with more than 50% probability, until you have run through the complete statespace (requires up to ~2^(number of state bits) output bits). This "mind" is not reconstructible from its output in any useful way.
However, this cryptographic analogy also suggests that some cryptographic approaches might be relevant. Browsing a paper like Cryptanalytic Attacks on Pseudorandom Number Generators by Kelsey, Schneier, Wagner and Hall (PDF) shows a few possibilities: input-based attacks would involve sending various inputs to the mind, and cryptoanalyzing the outputs. State compromise extension attacks make use of partially known states (maybe we have some partial brainscans). But it also describes ways the attacks might be made harder, and many of these seem to apply to minds: outputs are hashed (there are nontrivial transformations between the mindstate and the observable behavior), inputs are combined with a timestamp (there might be timekeeping or awareness that makes the same experience experienced twice feel different), occasionally generate new starting state (brain states might change due to random factors such as neuron misfiring, metabolism or death, sleep and comas, local brain temperature, head motion, cell growth etc). While the analogy is limited (PRNGs are very discrete systems where the update rule is simple rather than messy, more or less continuous systems with complex update rules - much harder to neatly cryptoanalyze) I think these caveats do carry over.
But this is not a conclusive argument. Some minds are likely nonreconstructible (imagine the "mind" that just stores a list of its future actions is another example: it can be reconstructed up until the point where the data runs out, and then becomes completely opaque), but other minds are likely trivially reconstructible (like the "mind" that just outputs 1 at every opportunity). A better kind of argument is to what extent our behavioural output constrains possible brain states. I think the answer is hidden in the theory of figuring out hidden Markov models.
Well, if I'm doing morningstar rhetoric, I'd best get my game face on.
First, in the paper below, their estimate of the information density of the brain is somewhat wrong. What you actually need is the number of neurons in the brain (10^11), squared, times two bytes for half-precision floating point storage of the strength of the synaptic connection, plus another two bytes to specify the class of neuron, times two, to add fudge factor for white matter doing whatever it is that white matter does, which all works out to 6.4 * 10^23.
Now that we've actually se...
Paul Christiano recently suggested that we can use neuroimaging to form a complete mathematical characterization of a human brain, which a sufficiently powerful superintelligence would be able to reconstruct into a working mind, and the neuroimaging part is already possible today, or close to being possible.
Paul was using this idea as part of an FAI design proposal, but I'm highlighting it here since it seems to have independent value as an alternative or supplement to cryonics. That is, instead of (or in addition to) trying to get your body to be frozen and then preserved in liquid nitrogen after you die, you periodically take neuroimaging scans of your brain and save them to multiple backup locations (1010 bits is only about 1 gigabyte), in the hope that a friendly AI or posthuman will eventually use the scans to reconstruct your mind.
Are there any neuroimaging experts around who can tell us how feasible this really is, and how much such a scan might cost, now or in the near future?
ETA: Given the presence of thermal noise and the fact that a set of neuroimaging data may contain redundant or irrelevant information, 1010 bits ought to be regarded as just a rough lower bound on how much data needs to be collected and stored. Thanks to commenters who pointed this out.