SLIDE 1
Convolu'on
NEU 466M Instructor: Professor Ila R. Fiete Spring 2016
SLIDE 2 Convolu'on
g: a 'me-varying signal sampled at discrete intervals h: another 'me-varying signal, not necessarily of same length
{· · · gt−1, gt, gt+1 · · · }
{· · · ht−1, ht, ht+1 · · · }
(g ∗ h)(n) =
∞
X
m=−∞
g(n − m)h(m)
Finite-length g, h: g*h has length N+M-1, where N=length(g), M=length(h).
SLIDE 3 Proper'es of convolu'on
- Commuta've/symmetric (unlike cross-
correla'on):
(g ∗ h)(n) =
∞
X
m=−∞
g(n − m)h(m)
g ∗ h = h ∗ g f ∗ (g ∗ h) = (f ∗ g) ∗ h f ∗ (g + h) = f ∗ g + f ∗ h
SLIDE 4 Convolu'on
(g ∗ h)(n) =
∞
X
m=−∞
g(n − m)h(m)
- Typically, one short series, one long.
- Long series called “signal”; the other called the “kernel”.
- The convolu'on is viewed as a weighted version/moving average of the by
the kernel.
SLIDE 5 [· · · gn−3 gn−2 gn−1 gn gn+1 gn+2 gn+3 · · · ] [· · · h2 h1 h0 h−1 h−2 · · · ]
Convolu'on (g ∗ h)(n) =
∞
X
m=−∞
g(n − m)h(m)
(g ∗ h)n
Say h: kernel (short/”finite support”), g: signal (long).
SLIDE 6
Convolu'on (g ∗ h)(n) =
∞
X
m=−∞
g(n − m)h(m)
[· · · gn−2 gn−1 gn gn+1 gn+2 gn+3 gn+4 · · · ] [· · · h2 h1 h0 h−1 h−2 · · · ]
(g ∗ h)n+1
Flip h (kernel) and keep it in one place, move g-tape (signal) leZ.
SLIDE 7 Convolu'on g h g ∗ h
“signal” “kernel”
(g ∗ h)(n) =
∞
X
m=−∞
g(n − m)h(m)
Flip h (kernel), keep g-tape (signal) fixed, sweep h (kernel) rightward.
· · · · · · · · · · · ·
SLIDE 8 − − −
−
Convolu'on g h g ∗ h · · · · · ·
−
SLIDE 9 − − −
−
Convolu'on g h g ∗ h · · · · · ·
−
SLIDE 10 − − −
−
Convolu'on g h g ∗ h · · · · · ·
−
SLIDE 11 − − −
−
Convolu'on g h g ∗ h · · · · · ·
−
SLIDE 12 − − −
−
Convolu'on g h g ∗ h · · · · · ·
−
SLIDE 13 − − −
−
Convolu'on g h g ∗ h · · · · · ·
−
SLIDE 14 − − −
−
Convolu'on g h g ∗ h · · · · · ·
−
SLIDE 15 − − −
−
Convolu'on g h g ∗ h · · · · · ·
−
SLIDE 16 − − −
−
Convolu'on g h g ∗ h · · · · · ·
−
SLIDE 17 − − −
−
Convolu'on g h g ∗ h · · · · · ·
−
SLIDE 18 − − −
−
Convolu'on g h g ∗ h · · · · · ·
−
SLIDE 19 Common convolu'on kernels
(e.g. rates from spikes)
(e.g. EPSPs from spikes). Called a linear low-pass filter.
(e.g. smoothing) h = 1/N for N samples, 0 elsewhere h = 1 τ e−t/τ for t > 0, 0 otherwise h = 1 √ 2πσ e−t2/2σ2
SLIDE 20 MATLAB DEMOS
Spikes to rate, smoothing, EPSPs
SLIDE 21 RETINA AS A CONVOLUTIONAL FILTER
Edge detec'on, HDR imaging
SLIDE 22
Mach bands (Ernst Mach 1860’s)
Eight bars of stepped grayscale intensity. Each bar: constant intensity.
SLIDE 23 lighter
Interes'ng perceptual effect in Mach bands
Illumina'on at a point on the re'na is not perceived objec'vely, but only in reference to its neighbours. Why/how does this happen?
darker
SLIDE 24
Difference-of-Gaussians or center-surround recep've field.
Electrophysiology of a re'nal ganglion cell (RGC)
SLIDE 25 Anatomy of RGC circuit
Off-center (surround) s'mulus has reverse effect of on-center s'mulus because of inhibitory horizontal cells.
SLIDE 26 RETINAL KERNEL SENSES CHANGES IN ILLUMINATION (MATLAB)
Reproducing the Mach band illusion
SLIDE 27
Ganglion cells code contrast: difference in brightness between center and surround
SLIDE 28
Re'nal filter performs edge detec'on
SLIDE 29 RETINAL KERNEL AS EDGE DETECTOR (MATLAB)
Demo
SLIDE 30 How edge detec'on works: theory
difference-of-Gaussians Smoothing filter:
Hretinal(x) = 1 p 2πσ2
1
e−x2/2σ2
1 − α
1 p 2πσ2
2
e−x2/2σ2
2
= Hσ1(x) − Hσ2(x)
Hσ(x) = 1 √ 2πσ2 e−x2/2σ2
Re'nal filter model: Gaussian
How to interpret Re'nal/difference-of-Gaussians filter?
σ1 < σ2
SLIDE 31 How edge detec'on works: theory
- D. Marr, E. Hildreth (1980) "Theory of edge detec'on."
- Proc. R. Soc. Lond. B, 207:187-217.
difference-of-Gaussians with some
− d2 dx2 Hσ(x) = − d2 dx2 1 √ 2πσ2 e−x2/2σ2 = 1 σ2 ✓ Hσ(x) − x2 σ2 Hσ(x) ◆ ≈ Hσ1(x) − Hσ2(x)
σ2 < σ1
SLIDE 32 How edge detec'on works: theory
− d2 dx2 Hσ(x) = − d2 dx2 1 √ 2πσ2 e−x2/2σ2 = 1 σ2 ✓ Hσ(x) − x2 σ2 Hσ(x) ◆ ≈ Hσ1(x) − Hσ2(x)
− − − −
σ2 = 3 2σ1
≈
SLIDE 33 How edge detec'on works: theory
Re'nal filter (difference-of-Gaussians) is like smoothing filter followed by a 2nd deriva've filter:
Hretinal ≈ (Hσ1(x) − Hσ2(x)) ≈ − d2 dx2 Hσ(x)
Hretinal ≈ H2nd diff ∗ Hsmooth
2nd deriva've smoothing
SLIDE 34 Re'nex-based adap've filter: global compression, local processing
High dynamic-range imaging
SLIDE 35 Comparison: convolu'on, cross- correla'on, autocorrela'on
Image: wikimedia commons hjps://en.wikipedia.org/wiki/Convolu'on
SLIDE 36 Summary
- Convolu'on: a kernel (short) acts on a signal (long), to
produce a locally reweighted version of the signal.
- Useful in engineering sense: smooth signals, extract rates
from spikes, template matching, other processing.
- Opera'ons of re'na on visual s'mulus may be interpreted as
convolu'on.
- Re'nal difference-of-Gaussians convolu'on: edge
enhancement, edge detec'on, contrast normaliza'on.
