Wheel lacing patterns
The main picture shows a bike with each wheel having spokes in a different pattern. The rear wheel is the normal pattern, with the spokes crossing over one another - they don't follow the radius of the wheel, but are angled between the hub and the rim. This is because the spokes of the rear wheel perform 2 functions. Firstly, they support the weight of the bike and the rider from the ground. Secondly, they have to transmit the drive torque from the hub to the wheel rim. If they were arranged radially they would simply flex and bend under the force.
A radially laced front wheel will be fractionally lighter than a cross laced one because the spokes will be shorter. this is not the reason for doing it though, the main reason is that the wheel is stiffer and therefore gives a more responsive ride.
If the front wheel has disk brakes (attached to the hub) then it should not be radially laced.This is because the braking force will be the same, or greater than the drive force, but in reverse. Rim braking is very different in terms of how the force is applied as most of the force is taken by the mechanism holding the brakes to the frame.
Each wheel has spokes in a different pattern
A close up view, showing the spoke pattern in more detail
The lacing pattern of the rear wheel can be clearly seen. Pull spokes are in blue, and push spokes are in red.