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Of course you can - and tens of thousands of thoughts have gone through various government's bureaux over the last centuries to be assigned to owners - who have paid for the privilege.
After that, a fair number of them have vanished into obscurity - having been published as possible but ending as improbable. The ones that have made it into commercial life have not had it easy, either. No end of truly good thoughts make truly bad products. Along the way, some of them have stripped fortunes from the high and low alike.
Well, how about scenes? Can you patent a photograph of a scene - and prevent anyone else from ever making that same image? A little harder to do - as the photo world operates upon the nursery-rhyme idea that a cat may look at a king. The cat can't sell someone else's view of royalty to the illustrated papers as it may be covered by copyright, but if Tiddles can formulate his own view, he's entitled to hold it.
It has been tried - and in some venal jurisdictions the attempt has succeeded. Possessiveness'll only last as long as the money does - when that runs out no-one cares.
What does this mean to the average photographer? Not a lot. You can go down to Canal Rocks or up to the Pinnacles and take as many shots as you please despite the fact that thousands of tourists have done so before you, and thousands will after you're gone. If you succeed in selling a view of either attraction, good on you. There will be enough variation in your photo to remove any copyright objection.
Can you patent an idea in the studio? or on a wedding shoot? Well, try, and see if it works. You will have a difficult time stopping others from copying your position or lighting if you depend upon natural light and the actions of the day. It is tough enough to stop the rellies popping into the garden shots at a wedding with their phones as it is. Don't expect a lot of sympathy from people who are related to the wedding couple - they have a feeling of entitlement stronger than any copyright act.
In the end, the defence of your images sometimes comes down to an act - you act as if you are entitled to take the only picture of a sunset on the west coast, and then when you discover someone else has done one you demand recompense for copyright, time, sunshine, and Magna Carta. Bluster or blister, you might end up with the same response.
Note: Taking pictures straight out of someone else's website, book, or lounge room and selling them as your own is always considered bad form. There really are laws agin it, and people can belong to organisations that will vigorously defend their ownership. You can still steal, but you'd better be prepared to steal something big to pay for the legal battle that ensues. Even then, it pays to be selective in what you plagiarise...some originals are going to make you look as bad as they did the original creator. If people push the images on you in a dark alley and run away laughing, you have a right to be suspicious.