Eyes (and cameras) form images by bending light so rays from each direction meet at the right spot on a curved receptor. The front surfaces — in the eye, the cornea and lens — refract (bend) incoming rays. Because the retina is roughly spherical, rays from different parts of the visual field are focused onto corresponding locations across that curved surface. This produces a spatial map of the scene, but inverted (top becomes bottom, left becomes right) because the lens flips the incoming light. Photoreceptors on the retina then convert the focused light into neural signals the brain interprets as the upright image we perceive.

For cameras, the same principles apply: a (usually) curved or flat sensor receives light focused by a lens system; optics compensate for curvature and field-of-view to keep images sharp across the sensor. (See basic optics texts, e.g., Hecht, Optics.)

• Basic imaging by a lens: A thin lens forms an image by bending (refracting) rays from each point of an object so they converge to a corresponding point on an image plane. The lens equation 1/f = 1/s + 1/s’ determines where the image of an object at distance s will form (s’ is the image distance, f the focal length). Aperture size controls how much light and which bundle of rays contribute to each image point; smaller apertures increase depth of field and reduce blur (Hecht, Ch. on lens imaging).

• Field of view and wide-angle optics: Field of view (FOV) is the angular extent of the scene that the lens can image onto the sensor/film. For a given sensor size, a shorter focal length (wide-angle lens) increases the FOV. Wide-angle lenses collect rays from larger angles relative to the optical axis, which requires special design to control aberrations and maintain sharpness across the image (Hecht, section on wide-angle optics).

• Projection and distortion: Lenses map directions in object space to positions on the image plane. Wide-angle lenses cause more oblique rays to hit the edges of the image, producing characteristic geometric distortion (e.g., barrel distortion) and perspective exaggeration: near objects appear larger relative to background. Catadioptric or specially designed wide-angle lenses use curved image surfaces or corrective elements to reduce distortion (Hecht discusses projection geometry and aberration correction).

• Vignetting and illumination falloff: At large angles, less light reaches the image plane per unit area (cosine^4 falloff and mechanical vignetting), so image corners can be darker. Lens design and aperture placement mitigate this (Hecht, optics of illumination).

• Practical considerations: Designing wide-angle optics balances focal length, sensor size, aberration correction, aperture, and desired FOV. Hecht explains mathematical ray-tracing, imaging equations, and trade-offs used to analyze and design such systems.

References:

• Hecht, E. Optics (relevant chapters: lens imaging, wide-angle optics, aberrations and illumination).

Each eye sees a very wide horizontal span — roughly 120°–200°. Where the two eyes’ views overlap (about 120° centrally) the brain uses differences between the images to judge depth (binocular stereopsis).

To capture similarly wide angles with a camera or sensor you need an imaging geometry that “compresses” peripheral directions onto the flat (or curved) imaging surface. Fisheye lenses, panoramic cameras, and spherical projections do this by bending incoming rays so large angular ranges correspond to positions on the sensor. Near the center of the image angular changes map to larger positional shifts, while toward the edges many incoming directions are squashed into a smaller sensor area. The result is that a single image records a much wider field of view, at the cost of peripheral distortion—just as the eye’s optics and neural processing trade spatial resolution in the periphery for a broad visual field.

References: basic optics and imaging texts on fisheye/spherical projection (e.g., Horn & Burns on panoramic imaging; standard vision science summaries of binocular overlap and stereopsis).

Wide-angle or “round” images map a wide field of view from a spherical scene onto a flat image using non-linear projection formulas. Common mappings include:

• Equirectangular: maps latitude and longitude linearly to x and y. It preserves neither angles nor areas uniformly, so shapes near the poles stretch vertically.
• Stereographic: projects the sphere from one pole onto a plane. It is conformal (preserves angles and small shapes locally), but it enlarges areas far from the projection point and bends straight lines that do not pass through the projection center.
• Fisheye (equidistant, equisolid-angle, etc.): maps angle from the optical axis to radius in the image. Different fisheye formulas preserve different properties (e.g., equal angles or equal solid angles) but all produce strong scale variation with viewing angle.

Trade-offs and visible effects

• Angle vs. area: No flat-map projection can preserve both everywhere. Conformal projections keep local angles intact but distort relative sizes; equal-area projections keep areas correct but distort shapes.
• Straight lines: Only lines through the projection center remain straight. Other great circles on the sphere generally appear as curves in the image.
• Scale variation: Objects near the image edge are stretched or magnified compared with those near the center; the amount depends on the chosen projection.

In short: wide-angle views require choosing a projection that preserves some geometric property while necessarily distorting others, producing curved lines and nonuniform scaling across the image.

(For technical references, see Snyder, J.P., “Map Projections — A Working Manual” (USGS), and Gentile et al., “Fisheye Lens Projection Models and Calibration”.)

The visual cortex combines the separate, slightly different images from each eye into one continuous spatial model. It aligns and “stitches” these inputs over time, correcting for lens and perspective distortions so objects appear at stable sizes and shapes. Depth is inferred from binocular disparity (the small positional differences between the two eyes’ views), motion parallax (relative movement of objects as we move), and monocular cues like perspective and texture gradients. The cortex also fills in missing information—such as the physiological blind spot—using nearby patterns and prior expectations so we perceive an uninterrupted scene. Together, these processes produce a single, stable, 3-D representation of the world.

Suggested sources: seminal work on binocular vision and depth perception (Gibson, 1950s–1970s), textbooks on visual neuroscience (e.g., Kandel et al., Principles of Neural Science), and reviews of predictive coding in vision (Friston).

David H. Hubel’s Eye, Brain, and Vision explains how visual perception arises from interactions between the eye’s optics and the brain’s neural processing. Key points:

• Optics and receptors: Light is focused by the cornea and lens onto the retina, where photoreceptors (rods and cones) transduce light into electrical signals. Cones mediate high-acuity, color vision in bright light; rods mediate low-light vision.

• Retinal preprocessing: Retinal neurons (horizontal, bipolar, amacrine, and ganglion cells) transform receptor signals. Ganglion cells have center–surround receptive fields that encode contrast and edges rather than uniform brightness.

• Pathways to cortex: Ganglion cell axons form the optic nerve and project to subcortical structures (especially the lateral geniculate nucleus, LGN) and then to primary visual cortex (V1). Parallel pathways (magnocellular and parvocellular) carry different information (motion/temporal vs. detail/color).

• Cortical feature extraction: In V1, neurons are selective for simple features such as orientation, spatial frequency, location, and motion direction. Hubel and Wiesel’s discovery of simple and complex cells showed hierarchical processing: simple cells respond to oriented bars in specific positions; complex cells respond to orientation moving across a region.

• Hierarchical and modular organization: Successive cortical stages combine simpler features into more complex ones (e.g., edges → contours → shapes → object recognition), with increasing receptive-field size and invariance.

• Plasticity and development: Visual circuits are shaped by experience during critical periods; deprivation or abnormal input can permanently alter cortical organization.

Hubel’s work emphasizes that vision is not a passive photograph but an active, hierarchical neural construction extracting features (edges, orientations, motion) from retinal signals to build perceptual representations. (See Hubel, D. H. Eye, Brain, and Vision, 1988.)

Short explanation: Gonzalez & Woods treat wide-angle or “round” vision as the result of applying a non‑linear geometric projection that maps 3D scene directions onto a 2D image plane (or sensor) with substantial angular extent. The key idea is that ordinary perspective projection (pinhole camera) maps scene points along straight rays into an image plane using a linear relation in homogeneous coordinates; wide‑angle lenses and imagers instead use alternative projection mappings that preserve different properties and produce the characteristic distortions near the edges.

Common projection models discussed include:

• Perspective (central) projection: standard pinhole mapping; straight lines through the center remain straight, but large fields of view produce extreme stretching near the image periphery.
• Stereographic, equidistant, equisolid‑angle, orthographic projections: these are radial, central projections that map the polar angle θ (angle between optical axis and incoming ray) to an image radius r by different functions r = f(θ). Examples:
  • Equidistant: r = f·θ (angles map linearly to radius — useful for some fisheye lenses).
  • Equisolid‑angle: r = f·2·sin(θ/2) (preserves solid angle increments).
  • Stereographic: r = f·2·tan(θ/2) (conformal: preserves angles locally).
  • Orthographic: r = f·sin(θ) (projects onto a plane by dropping depth). Each choice yields different radial distortion patterns and tradeoffs (angle preservation, area preservation, straight‑line behavior).

Practical use in Gonzalez & Woods:

• They explain these transforms to model fisheye and panoramic imaging and to correct or simulate wide‑angle distortions.
• Projection transforms are implemented as forward mapping (scene → image) or inverse mapping (image → scene) and are frequently used with resampling/interpolation to build rectified images or to reproject images between coordinate systems.
• For tasks like panoramic stitching, one commonly remaps images from camera coordinates onto a chosen projection (cylindrical, spherical, or planar) to align and blend multiple views.

Why it matters: Understanding which projection governs your imaging device lets you:

• Correct distortions (undistort).
• Reproject images onto different surfaces (spherical panoramas, equirectangular maps).
• Preserve desired properties (angles, area, or line straightness) depending on application.

Reference:

• Gonzalez, R. C., & Woods, R. E. (2008). Digital Image Processing (3rd ed.). Chapters on geometric transformations and projection models (see sections on wide‑angle/fisheye and projection transforms).

“Round vision” means seeing rays coming from a wide range of directions around a point—not just a narrow forward cone. Optical and imaging technologies replicate this by capturing light over large solid angles (wide fields of view) and then mathematically transforming those incoming directions into a flat image that we can view.

• Fisheye lenses: These lenses use extreme wide-angle optics to collect light across very large angular extents (up to 180° or more). They map incoming ray directions to image coordinates with nonlinear projection formulas (e.g., equidistant, equisolid-angle, or stereographic projections), producing the characteristic curved, wide-field image.

• Panoramic stitching: Multiple normal or wide-angle images are taken from a single viewpoint covering different directions. Software identifies matching features and warps each image into a common projection (often cylindrical or equirectangular) and blends them so the assembled picture represents a continuous wide-angle, near-“round” view.

• Spherical (360°) cameras: These cameras use multiple lenses/sensors or special optics to capture the full sphere of incoming directions around a point. The raw directional data are mapped into a 2D representation (commonly equirectangular projection) using spherical-to-planar projection formulas so viewers can navigate the full surrounding scene.

In all cases, the core idea is the same: collect light from many directions around a point and remap ray directions into 2D coordinates using projection mathematics so a “round” field of view can be displayed on a flat image.

References: basic optics and imaging texts; see discussions of fisheye projections and equirectangular mapping in imaging literature (e.g., Gonzalez & Woods, Digital Image Processing).