What is the relation of visual quality and distance from the central field of view? I have read somewhere that 


  Our visual perception degrades non-linearly with distance from the
  fovea.


However, it was not referenced. I was wondering if there is a reference talking about this? 

Also, is there a scientific reference stating the degradation of vision as a function of degrees in the field of view?
Assuming you are talking about the ability of the visual system to resolve spatial detail (resolution, if you wish), then your question is on visual acuity as a function of eccentricity on the retina (Fig. 1). 

Due to the densely packed cones at the fovea (corresponding to the central field of view (FOV), see Fig. 1), visual acuity (VA) is the greatest at the center of fixation. More eccentrically, VA declines markedly. E.g., at 10 minutes of arc (1/6 of a degree) from fixation, there is a 25 % loss of visual acuity (Fig. 1.) This dependence of VA on eccentricity is indeed nonlinear (cf. Fig. 1) and correlates with the cone packing density and photopic resolution in the region up to a few degrees eccentricity. 

However, at larger eccentricities, visual acuity is worse than that predicted by cone spacing alone, indicating that other post-receptoral retinal elements are the limiting factor. One of these factors is the increased density of rods in eccentric regions; rods are built for speed and sensitivity. The latter is accomplished in part by funneling the inputs of multiple rods onto a single retinal ganglion cell (RGC). RGCs transmit the signals via the optic nerve to the visual centers in the brain stem, ultimately ending up in the brain. Foveal cones, on the other hand, are connected 1:1 and hence the spatial resolution and thus SA is much better in foveal regions. The foveal region contains high densities of cones (good VA), while more eccentrically more rods are present (worse VA); see Fig. 2. This is exactly the reason that we read by focusing our central foveal view on the characters. 


Fig. 1. VA as a functin of eccentricity. Source: Kalloniatis &amp; Luu (2016)


Fig. 2. Rod and cone densities as a function of eccentricity. Source: Kolb (2016)

References
- Kalloniatis &amp; Luu. Visual acuity. In: Webvision. The Organization of the Retina and Visual System. Kolb et al. (eds). Utah University (2016)
- Kolb. Photoreceptors. In: Webvision. The Organization of the Retina and Visual System. Kolb et al. (eds). Utah University (2016)