Optics 070 · Diffraction, Fourier optics, and computational imaging

Computational Deblurring Lab

Restore the actual noisy camera record with truncated inverse, Wiener-type ridge and RL. Expose wrong-kernel ghosts and irrecoverable frequencies.

Simulation-only blur / restoration error

\varepsilon_{\rm blur},\varepsilon_{\rm restore}

$0.50$

Data fit / small-transfer area

\varepsilon_D,m

$50\%$

Image-side NA / actual sensor distance

{\rm NA},d_{\rm sensor}

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Computational Deblurring Lab

Blur type (named buttons below)

j_h

0

Pupil-edge defocus relative to centre

w_d

0.55\lambda

Total image travel during shutter

\ell

12\mathrm{px}

Travel direction clockwise from image right

\theta_m

25^{\circ}

Gaussian seeing kernel standard deviation

\sigma_s

2\mathrm{px}

Astigmatism phase coefficient (not RMS)

w_a

0.75\lambda

Synthetic target (named buttons below)

j_f

0

Expected photoelectrons per unit intensity

E

100\mathrm{e}^{-}

Independent Gaussian read-noise SD

\sigma_r

0\mathrm{e}^{-}

Zero: expectation; one: noisy acquisition

n

0

Restoration (named buttons below)

j_{\rm restore}

1

Assumed constant noise / signal spectral ratio

K

0.002

RL iteration count

k

30

Assumed kernel scale / actual scale

s_h

1

Horizontal probe (positive right)

q_x

12\mathrm{px}

Vertical probe (positive down)

q_y

-14\mathrm{px}

One camera record, three ways back. Move the sensor out of focus, smear the image during exposure, add Gaussian seeing or astigmatism. Compare the acquired image with restoration, then deliberately supply a wrong PSF. Drag the answer-key probe to inspect a cross-section. Noise, ringing and lost frequencies are visible costs: a sharp-looking image is not evidence that missing detail returned.

Synthetic incoherent periodic test field and known/perturbed PSF; no real camera data or blind recovery. Gaussian seeing is approximate. RL with read noise is not exact Poisson–Gaussian likelihood.

Start with the essentials

Focus question: When does deconvolution recover detail, and when does it only amplify noise or model error?
One-sentence intuition: No algorithm gets an answer key. At an optical null, $H(\nu)=0\ \Longrightarrow\ \text{lost mode}$ . Priors may select an answer, not establish a measurement of missing detail. Try the beyond-cutoff target.

Core mathematical model

Count light, then read electronics

\begin{aligned}\mu_i&=E(h*f)_i\\P_i&\sim\operatorname{Poisson}(\mu_i)\\D_i&=P_i+\epsilon_i\\\epsilon_i&\sim\mathcal N(0,\sigma_r^2)\end{aligned}

Independent shot noise and electronic read noise; expectation mode records the mean instead. Negative electronic samples are retained for spectral filters and scoring, not called negative photons.

A physical intensity kernel

\begin{aligned}P&=\mathbf1_{\rho\le1}e^{i\phi},\quad \rho^2=u^2+v^2\\\frac{\phi}{2\pi}&=-w_d\rho^2+w_a(u^2-v^2)\\h&=\frac{|\mathcal F^{-1}P|^2}{\sum|\mathcal F^{-1}P|^2}\end{aligned}

Sampled circular pupil and scalar diffraction. Defocus or astigmatism changes phase, not aperture transmission. Motion convolves a shutter-time path; seeing convolves a Gaussian surrogate. All convolution is periodic on the declared grid.

Sensor travel matches defocus

\Delta z=\frac{2\lambda w_d}{{\rm NA}^2},\quad d_{\rm sensor}=f+\Delta z

First-order paraxial defocus magnitude. Positive control moves the sensor behind nominal focus; negative before it. Pure defocus intensity is invariant under sign reversal. No claim of an exact large-displacement spherical-wave calculation.

Truncated inverse

\widehat F_{\rm inv}(\nu)=\begin{cases}G(\nu)/\widetilde H(\nu),&|\widetilde H|\ge10^{-3}\\0,&\text{otherwise}\end{cases}

Data are normalized by the known electron budget. The assumed transfer can differ from the true one. The threshold avoids division by numerical zeros but still permits severe noise gain.

Constant-ratio Wiener-type ridge

\widehat F_K=\frac{\widetilde H^*}{|\widetilde H|^2+K}\,G

A user-assumed constant spectral ratio, equivalent here to identity-penalty ridge. Not a calibrated frequency-dependent optimal Wiener filter; Poisson noise is signal-dependent. Raw signed estimates are scored without display clipping.

RL updates only the acquired record

f^{(k+1)}=f^{(k)}\odot\widetilde H^T\!\left(\frac{D/E}{\widetilde H f^{(k)}}\right)

A positive uniform start and the normalized-kernel adjoint. Poisson-only likelihood, maximum eighty steps; early stopping is an implicit prior. With Gaussian read noise, negative data are clipped for this approximate update and a warning is shown. No automatic truth-based tuning.

Common difficulties

Sharper means more correct?

Typical misconceptionA sharp reconstruction with a small fit residual proves fine detail has returned.

Better mental modelInverse filtering can amplify noise; regularization adds bias; RL can fit noise after too many steps. Compare the held-out synthetic truth score and the transfer band, not sharpness alone.

Run the experiment

01
Scene 1: Defocus · a recoverable band
Start with defocus and inspect the sensor front. Compare acquired and restored images using their common scale.
What to observe: Defocus moves the actual camera but does not change the clear pupil. The normalized kernel conserves the declared photoelectron budget.
02
Scene 2: Motion with read noise
Keep one record fixed and switch restoration algorithms. Add noise, increase RL steps, or deliberately perturb the supplied PSF.
What to observe: The same raw record is used by all algorithms. Changing the assumed kernel alters processing, not acquisition.
03
Scene 3: Poisson stars · RL iteration
Drag the answer-key probe for a shared cross-section. Select beyond-cutoff stripes to see information absent from the acquisition.
What to observe: Noise gain, bias, wrong-kernel ghosts and null bands remain visible. A better fit need not mean a more faithful reconstruction.