You may achieve it using Fisheye Equisolid type of the Panoramic camera (with some manual vertices tweaking).
Add a camera and position it verticaly as pictured below. Add a plane and place it in front of it. Add a background image (display axis type: Camera), decrease its opacity and push the Front button.
Set the render resolution so it is the same as image's resolution.
Enter the camera view (Ctrl+Num 0) and rotate it along the Z axis to fit the plane to the reference. Extrude the plane until you you build the rough representation of the object from the picture.
Select the extruded object in Object Mode and give it a material (Emission e.g.). Also give it a Wireframe Modifier with some thickness.
In Cycles go to Camera Data header--> Lens panel and set the camera type to Panoramic (Fisheye Equisolid). Experiment with Lens and Field of View values. In this case I found Lens=14 and FoV=180 optimal.
Divide the 3d view into two windows. Set the Solid viewport shading mode for the first and Rendered for the second one.
Snap the cursor to the origin point of the plane. Grab the camera and translate it along the Z axis. Next rotate it along the local Y axis (press R, then double tap Y) to fit the image as precisely as you can.
Controling two views at the same time you can now easily make the subdivisions to the mesh and model the details.
To make the model exactly fit the reference's perspective you may tweak some vertices.
Here's the comparision between original and tweaked perspectives.