Lecture Notes 9 Spatial Resolution - Stanford University

Lecture Notes 9 Spatial Resolution

? Sensor Modulation Transfer Function (MTF) ? MTF Calculation ? Aliasing

EE 392B: Spatial Resolution

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Preliminaries

? The image sensor is a spatial (as well as temporal) sampling device (of the incident photon flux image) -- the sampling theorem sets the limits for the reproducibility in space (and time) of the input spatial (and temporal) frequencies

? So spatial (or temporal) frequency components higher than the respective Nyquist rate cannot be reproduced and cause aliasing

? The image sensor, however, is not a point sampling device in space (or time), and cannot be approximated as such

Photocurrent is integrated over the photodetector area (and in time) before sampling

Photogenerated carriers in quasi-neutral regions of a pixel may diffuse and be collected by its neighboring pixels

These effects (in addition to the optics) result in low pass filtering and crosstalk before spatial (and temporal) sampling

EE 392B: Spatial Resolution

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? We focus here on spatial sampling

p

p-sub

Assuming a square pixel with width (pitch) p, the spatial Nyquist

rate

in

each

dimension

is

fNyquist

=

1 2p

and

is

typically

reported

in

line

pairs per millimeter (lp/mm)

Signals (photon flux images) with spatial frequencies higher than fNyquist cannot be faithfully reproduced, and cause aliasing

The low pass filtering caused by integration and diffusion degrades the reproduction of frequencies below fNyquist -- degradation measured by the Modulation Transfer Function (MTF)

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Modulation Transfer Function (MTF)

? The contrast in an image can be characterized by the modulation

M

=

Smax Smax

- +

Smin Smin

,

where Smax and Smin are the maximum and minimum pixel values over

the image

Note that 0 M 1

? Let the input to an image sensor be a 1-D sinusoidal monochromatic photon flux

F (x, f ) = Fo(1 + cos(2f x)), for 0 f fNyquist

The sensor modulation transfer function is defined as

MTF(f )

=

Mout(f ) Min(f )

From the definition of the input signal, Min = 1

? MTF is in general difficult to model and analyze for a real sensor and is determined experimentally

EE 392B: Spatial Resolution

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? By making several simplifying assumptions (as we shall see), the sensor can be modeled as a 1-D linear space-invariant system with impulse response h(x) that is real, nonnegative, and even

In this case the transfer function H(f ) = F[h(x)] is real and even, and the signal at x

S(x) = F (x, f ) h(x) = Fo(1 + cos(2f x)) h(x) = Fo (H(0) + H(f ) cos(2f x))

Therefore

Smax = Fo(H(0) + |H(f )|) Smin = Fo(H(0) - |H(f )|),

and the sensor MTF is given by

MTF(f

)

=

|H(f )| H (0)

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