Requirements: Sufficient knowledge of PhotoShop and AfterEffects, understanding the principles of animation and different animation techniques. A Bear!

This method is based on the simple observation that objects which are positioned at a different distance from the observer move with different amplitudes while he/she is approaching them. The nearer the objects, the bigger the amplitude. This is called Parallax.

This technique was used since the 1930s in the classic cartoons to simulate depth in the scenes, especially in esatablishing shots, panoramas, etc.

Below is an picture, illustrating the method:

Having the camera zoom into the main background, while sliding the foreground layers to the sides, produces a convincing depth effect on screen. This was later modified into the so-called multi-plane camera, where the different levels of the image were really placed at different distances from the camera (on transparent glass shelves) and it gave additional sense of realism with the in-focus / out-of-focus effects it produced.

A digital version of the classic multi-plane camera is built into AfterEffects. Based on a Z or depth coordinate it automatically calculates the parallax and renders the images as if positioned in 3D space.

1. Prepare your file in PhotoShop.
You need a file prepared properly, according to common sense and your understanding on how you need to cut the separate facial slices. Think about it the same way the geographical maps display areas with the same altitudes - joining them with curved lines. In fronton view of a generic human face we have several planes:
-ears / back of head,
-cheeks / eyes,
-forehead / chin,
-hair,
-nose, etc.
It depends much on the particular design and your desire to go deeper into individual details. For low resolution animations, such as an avatar-sized movie, four or five layers should be absolutely enough.

Name your layers and position them in Photoshop the way you want them to appear in the AfterEffects timeline. If you plan to have more states for some of the layers, for example blinking eyes, smiling mouths, etc., position them where you need them and turn the layers off.

Later you can edit your source file in PhotoShop, as AfterEffects actually reads the PSD (and all the other footages) externally. But remember that if you change the dimensions of any of the layers it will loose its coordinates and will appear at a wrong spot on screen. What you can do to go round this problem and be sure that it wouldn't cause you troubles on later stages of your work is to add markers at the top left and bottom right corners of every layer. Thus AE will register layers as full-sized and you can do whatever you like to them without losing the right coordinates. You can record yourself an action that adds a pixel in the layers' corners and run this action every time you need.

2. Import in AfterEffects and prepare the layers for animation
Import your PSD in AfterEffects AS A COMPOSITION.
Extend the layers to the end of the composition. Your Timeline should look like this:

Then turn all the layers into 3D layers - only 3D layers interact with the camera and behave the way we need them in order to achieve the desired multi-plane effect. You do this by activating the 3D check-box for each layer.

Turning a layer into a 3D layer gives you access to some additional properties, such as Z-position coordinate (indicating depth), orientation, and more. Basically the layer starts to behave as a sheet of paper that can be moved freely in space.

Next we need to link the layers in a logical hierarchy. This is done either by choosing parent's name from the corresponding pop-up menu in Timeline or by dragging the lasso and pointing to the parental layer (which is my preferred way).
Below is an image of the linked layers:

As this animation will be a pretty basic demonstration of the multi-plane method you will notice that most of the layers are children to the face and the face will be a child to the body. That's because we will not move ears or other parts separately. The only extra thing will be the blinking eyes.

Now go to LAYER>NEW>CAMERA… and insert a new camera layer and extend it to the end of the Timeline.
If you play with the camera tools, you will notice how the whole image now behaves like cut out of cardboard.

Finally we need to adjust the Z position of the different facial parts.
Select the following layers and by pressing P open the position property for each of them.

Select the View window (the Stage) and from VIEW>NEW VIEW open a second instance of it. And from the pop-up choose TOP.

Now we have to position the parts of the image in depth and then correct the scaling. This can be done by sliding the numerical values for the Z-coordinate or by dragging the blue arrow in the top view. When done the Timeline and the Stage look like this: [click on image below to open larger version]

OK. Select the same layers and press S on the keyboard to activate the scaling property. Turn all the layers off, except for the Face, which was left in zero position. Then turn them on one by one and scale them up or down as necessary, so that the bounding boxes match.

When you're done the windows should look something like this:

If you play now with the orbit camera tool you will notice the difference!

Finally move the Pivot Point of layer Face a bit lower using the pivot repositioning tool and then link the Face to the Body. I would also suggest to move body's pivot point quite low.

Hooray!
The difficult part is over.

3. Animate
We animate the character by inserting key-frames and timing them according to our initial idea, storyboard, mockup or internal sense.

The animation techniques are similar to the ones explained in the first tutorial on cut-outs.
For the final example I have added paws and some background objects that accentuate the depth of the space.

Below are three animations I did at Konami, using this technique. And the bear is Barney Bear, of course!

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