Spatial coding: scaling, magnification & sampling

Spatial coding: scaling, magnification & sampling

Snellen Chart Snellen fraction: 20/20, 20/40, etc. 100 40 20 10

Visual Axis Visual angle and MAR A B C Dots just resolvable F 20 f 40 Visual angle Minimal angle of resolution 60 f MAR can be expressed in degrees or minutes or seconds of arc. 1 o = 60, 1 = 60 (fovea)

T N N T Visual Field Eye t n OS Oculus Sinister n t OD Oculus Dexter

Minimal Angle of Resolution (degrees) Spatial Resolution as a Function of Distance from the Fovea (Eccentricity) 0.25 0.2 Under optimal lighting and contrast conditions, human foveal MAR is about 1/120 deg. or 0.5. 0.15 0.1 0.05 0 0 10 20 30 40 50 60 Eccentricity (degrees)

Innervation density Spatial resolution Skin resolution 0.6 Two-point discrimination units/sq. mm. 1/mm 0.3 0 120 80 Meissner s corpuscles 40 0 Palm Finger Finger tip

Left Eye t n

Size and Spacing of Retinal Beta (X) Cells in Cat (Correspond roughly to P(arvo) cells of the monkey) Increasing distance from area centralis ~5 o ~8 o ~25 o 250 mm

Optic chiasm Optic nerve Optic tract LGN Optic radiation Striate cortex (V1)

Optic chiasm Optic nerve Optic tract Lateral geniculate nucleus Optic radiation Primary visual cortex (striate cortex, V1)

Left Visual Field Superior Right Occipital Lobe Medial Surface Fixation Point Inferior Calcarine Sulcus Inferior visual field Superior visual field Anterior

S1 Differential Cortical Magnification of the Receptor Surface Regions of high spatial acuity have: Small receptive fields High receptive field density Large representations in sensory cortex

Critical separation of activity in cortex

Deriving a central map by single cell recordings Vertical meridian Fixation point 30 20 10 Horizontal meridian 5 15 10 5 10 20 30 40 0 10 20 30 40 0-10 -20-30 Visual Space (tangent screen) -5-10 -15 Visual structure (V1, LGN, SC)

(Cortical) Magnification Factors 0 5 15 10 A 2 mm Retinotopic map (not in the retina!) 5 10 20 30 40 B 1 mm Magnification Factor in Vision M = mm cortex/degree visual angle Region A M = 2 mm/5 deg = 0.4 mm/deg. Region B M = 1 mm/10 deg = 0.1 mm/deg -5-10 -15

Minimal Angle of Resolution (degrees) Spatial Resolution as a Function of Distance from the Fovea (Eccentricity) 0.25 0.2 0.15 M 0.1 0.05 0 0 10 20 30 40 50 60 Eccentricity (degrees)

MAR and Magnification Eccentricity 2.5 o 5 o 10 o 20 o 30 o 40 o 50 o M (mm/deg)* 3.87 2.35 1.31 0.70 0.48 0.36 0.29 MAR (deg)**.018.028.044.080.116.163.210 MAR x M (mm).070.066.058.056.056.059.061 * Foster et al. 1981 ** Wertheim, 1894

30 Up Visual Space 15 10 Retinotopic map 5 10 20 30 40 20 5 10 MAR Fovea 10 20 30 40 0-10 -5-20 -10-15 -30 Down (deg * mm/deg = mm) MAR x M = equivalent cortical distance of the MAR MAR x M is translationally invariant in cortical coordinates.

Critical separation of activity in cortex

30 Up Visual Space 15 Retinotopic map 5 10 20 30 40 10 20 5 10 Fovea 10 20 30 40 0-10 Minimum resolvable separation of visual points -5-20 Minimal resolvable separation of patches of neural activity -10-15 -30 Down

Analysis in the Frequency Domain Hi Lo Luminance Profile of a Sinusoidal Grating

Parameters of Sinusoidal Gratings Increasing spatial frequency (c/deg) High contrast Low contrast Contrast Sensitivity Function

Contrast Sensitivity (1/C) Contrast Sensitivity Function Cat Human Cutoff Frequency (highest frequency that grating can be perceived) Spatial frequency (cycles/degree)

A Square-wave grating B MAR Hi Grating period Lo Luminance Profile P = 2 x MAR F = 1/(2 x MAR) Period: Frequency: degrees/cycle (or minutes/cycle) cycles/degree

If the MAR = 0.5, what is the cut-off frequency of the CSF? uare-wave grating B MAR 0.5 minutes Grating period 1 minute/cycle (2 MAR) Luminance Profile P = 2 x MAR F = 1/(2 x MAR) The period of the grating corresponding to the MAR is 2 0.5 = 1 The frequency of this grating is 1 cycle per minute Because there are 60 minutes in a degree: 1 cycle/minute = 60 cycles/degree The cut-off frequency of the CSF should be 60 cycles/deg (CPD).

What factors limit our ability to resolve gratings of high frequency?

Aliasing A B C x Photodetectors Photodetectors (or picture elements pixels )

Aliasing A B C x Photodetectors

Given a particular receptor spacing (sampling interval), what is the highest frequency that can be unambiguously sampled? Nyquist frequency 2 samples/period

A If the Nyquist frequency of a sampling array is 120 CPD, what is the interval, X, between detectors? BA BC C x Freq = 120CPD X =? The Nyquist frequency is 120CPD or 2 cycles/minute The Nyquist period is 1/Freq = 1/(2 cycles/minute) or 1/2 minute (30 ) At the Nyquist frequency, the detectors must sample twice every cycle, so there are 2 intervals / 0.5 minutes The interval X must be 0.25 Photodetectors x Photodetectors X =.25 minute of arc (15 seconds)

Avoiding Aliasing Increase sampling frequency More detectors, smaller spacing Remove high frequencies from stimulus Low pass filter input before sampling Optically Have detectors sample a larger area

% Transmission Filters Low pass High pass Band Pass 100 0 Frequency Frequency Frequency

Filter Effect of Aperture Size Luminance profile of moving stimulus Detector Output Narrow-aperture detector Wide-aperture detector

Size and Spacing of Retinal Beta (X) Cells in Cat Increasing distance from area centralis ~5 o ~8 o ~25 o 250 mm

Two distinct problems require converting back and forth between frequency units and interval units Frequency units Interval units Behavior Physics Cut-off frequency of the CSF Nyquist frequency of an array Minimal angle of resolution (MAR) Sampling interval of an array Grating at CSF Cut-off Frequency Grating at Nyquist Frequency of an Array MAR 1 cycle 1 cycle 1 cycle or period Frequency = # cycles (or periods)/degree Period = 2 x MAR Frequency = 1/(2 x MAR) Watch your units! Sampling interval Nyquist period = 2 x Sampling interval Nyquist frequency = 1/(2 x sampling interval) Sampling array