Optics 039 · Imaging, instruments, and visual systems

Ghost Image, Pepper’s Ghost & HUD

An exact half-mirror workbench links a real flat display to its same-size virtual image. A computed fixed-exposure view and finite-aperture admission map expose additive mixing, background washout and viewpoint clipping.

Virtual center and observer distance

y_V,z_V,D_{EV}

$0.50$

Admitted reference-patch radiance and contrast

L_g,L_b,C_M

$50\%$

Source-area admission, size ratio and energy closure

f_{\mathrm{area}},w_V/w_S,R+T

$0.00\pi$

Model regime $\text{valid model regime}$

Ideal zero-thickness lossless combiner, not a volumetric hologram or a complete automotive HUD. The virtual construction is not an emitting surface; reference contrast is conditional on geometric admission. No rear-surface plate ghost or calibrated luminance is claimed.

Physics tutorial

Audit the physics of Ghost Image, Pepper’s Ghost & HUD

BackgroundA tilted half-mirror reflects a flat display into a virtual image. For a unit normal and a mirror through the origin, the image position is $\mathbf V=\mathbf S-2\mathbf n(\mathbf n\cdot\mathbf S)$ . This exact reflection preserves every distance and therefore image size. Physical light turns at the sheet and enters the observer; only its backward extensions reach the virtual plane.

Why it mattersWhere is a half-mirror virtual image, why does a bright background wash it out, and when does the finite mirror clip it?

Start with the essentials

Focus question: Where is a half-mirror virtual image, why does a bright background wash it out, and when does the finite mirror clip it?
One-sentence intuition: At an admitted ray, display light adds to the transmitted real background: $L_{\mathrm{out}}=R L_d+(1-R)L_s$ . Black display pixels contribute no light but do not remove scene light. This is why a bright background washes out an overlay. The virtual glyph is flat, not a volumetric hologram; a real automotive HUD also needs image-forming and often collimating optics absent here.

Core mathematical model

Reflection is an isometry, not an image-size slider

H=I-2\mathbf n\mathbf n^{\mathsf T},\quad H^{\mathsf T}H=I,\quad \|H(\mathbf S_1-\mathbf S_2)\|=\|\mathbf S_1-\mathbf S_2\|

The real emitter and virtual plane have identical 240 by 180 mm bounds. At a forty-five-degree sheet tilt a horizontal display becomes an upright virtual plane. Its changing perspective size is not optical magnification. The overview shows the virtual construction from arbitrary orbit angles; only the computed card represents the selected observer.

Every displayed reflection has a real intersection

\mathbf M=\mathbf E+t(\mathbf V-\mathbf E),\quad t=\frac{\mathbf n\cdot\mathbf E}{\mathbf n\cdot(\mathbf E-\mathbf V)},\quad \mathbf d_r=\mathbf d_i-2\mathbf n(\mathbf n\cdot\mathbf d_i)

Intersect the observer-to-virtual-point line with the actual tilted sheet. The source-to-intersection ray obeys specular reflection into the observer. The dashed intersection-to-image segment is a backward construction, never a light ray passing through an opaque mirror. At full reflectance the transmitted background ray disappears.

Finite sheet, finite viewing window

|M_x|\le a,\quad |\mathbf M\cdot\mathbf t|\le b,\quad f_{\mathrm{area}}=\frac{\operatorname{area}(P_{\mathrm{admitted}})}{240\cdot180}

The four aperture inequalities clip a polygon in source coordinates exactly. Green observer positions admit the whole source; amber admits part; black admits none. The map uses a fixed 700 mm observer-depth slice, horizontal positions from minus 700 to 700 mm and vertical positions from minus 100 to 500 mm. Its white ring follows the selected point. This geometric source-area measure is independent of brightness and is not finite-pupil power or a three-dimensional eye-box.

A fixed view is not an aperture-admission map

\mathbf d=(q_x,q_y,-1),\quad |q_x|\le0.30,\quad |q_y|\le0.225

Every observer pixel launches this backward ray from the selected point. The sheet either splits it into reflected and transmitted branches, the opaque border blocks it, or it passes outside the finite sheet through open space. First-hit handling includes the one-sided source. The angular window and exposure never zoom to fit a moving image. A geometrically admitted virtual point may still lie outside this view; the area map deliberately measures aperture admission, not camera framing.

Radiance mixing and an explicitly defined contrast

T=1-R,\quad L_{\mathrm{on}}=R L_d+T L_s,\quad L_{\mathrm{off}}=T L_s,\quad C_M=\frac{L_{\mathrm{on}}-L_{\mathrm{off}}}{L_{\mathrm{on}}+L_{\mathrm{off}}}=\frac{R L_d}{R L_d+2T L_s}

This is reference Michelson contrast for white and black display patches over a uniform unit background patch, conditional on an admitted ray. It is not an object-visibility score for the patterned city image or a human-perception prediction. With no light the contrast is undefined. Free propagation preserves radiance; no inverse-square multiplier is applied to a resolved surface pixel. A fixed Reinhard exposure and sRGB encoding display all relative linear-radiance images without auto-normalization.

Common difficulties

A virtual black pixel can cover the real scene

Typical misconceptionA dark pixel on the display should erase whatever is behind the ghost.

Better mental modelA passive combiner adds radiance. A black source pixel adds zero while the background remains transmitted. Full reflectance can block the background across the physical sheet, but cannot create arbitrary per-pixel scene occlusion. Actual occlusion requires additional hardware.

Run the experiment

01
Scene 1: Dark-stage phantom
Start with the dark-stage phantom. Change the real display distance and sheet tilt: compare its actual motion, the same-size virtual plane and the observer image.
What to observe: The overview is explanatory; observer pixels are independently backward-traced through the same finite aperture. A directly visible source is included if it enters the fixed view. Sheet borders are opaque, while empty space outside the finite sheet is open.
02
Scene 2: Bright-background washout
Choose bright-background washout. Raise background radiance, then display radiance. Finally set reflectance to zero and one: reflection and transmission disappear at their respective limits.
What to observe: Reference contrast may stay high while the ghost is entirely clipped. Read geometric admission separately; brightness cannot fix a blocked path.
03
Scene 3: Finite viewing window
Choose finite viewing window. Translate the observer through green, amber and black; enlarge the actual sheet to recover the ghost. Geometric admission is unchanged by brightness.
What to observe: The classical tilted half-mirror is not the holographic mirror in the cited research chapter. Plate thickness, double-surface ghosts and commercial HUD collimation are deliberately not claimed.