Shannon information storage in noisy phasemodulated fringes and fringe-data
compression by phase-shifting algorithms
MANUEL SERVIN,* AND MOISES PADILLA
Centro de Investigaciones en Optica A. C., Loma del Bosque 115, 37150 Leon Guanajuato, Mexico.
*mservin@cio.mx
www.cio.mx

Abstract: Optical phase-modulated fringe-patterns are usually digitized with XxY pixels and
8 bits/pixel (or higher) gray-levels. The digitized 8 bits/pixel are raw-data bits, not Shannon
information bits. Here we show that noisy fringe-patterns store much less Shannon
information than the capacity of the digitizing camera. This means that high signal-to-noise
ratio (S/N) cameras may waste to noise most bits/pixel. For example one would not use
smartphone cameras for high quality phase-metrology, because of their lower (S/N) images.
However smartphones digitize high-resolution (12 megapixel) images, and as we show here,
the information storage of an image depends on its bandwidth and its (S/N). The standard
formalism for measuring information are the Shannon-entropy H, and the Shannon capacity
theorem (SCT). According to SCT, low (S/N) images may be compensated with a larger
fringe-bandwidth to obtain high-information phase measurements. So broad bandwidth
fringes may give high quality phase, in spite of digitizing low (S/N) fringe images. Most reallife images are redundant, they have smooth zones where the pixel-value do not change much,
and data compression algorithms are paramount for image transmission/storage. Shannon's
capacity theorem is used to gauge competing image compression algorithms. Here we show
that phase-modulated phase-shifted fringes are highly correlated, and as a consequence,
phase-shifting algorithms (PSAs) may be used as fringe-data compressors. Therefore a PSA
may compress a large number of phase-shifted fringes into a single complex-valued image.
This is important in spaceborne optical/RADAR phase-telemetry where downlink is severely
limited by huge distance and low-power downlink. That is, instead of transmitting M phaseshifted fringes, one only transmit the phase-demodulated signal as compressed sensing data.
2017-11 Centro de Investgaciones en Optica A. C.
OCIS Codes: (120.0120) Instrumentation, measurement, and metrology; (100.2650) Fringe analysis; (120.3180)
Interferometry; (120.5050) Phase measurement.

References and Links
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.

K. J. Gasvik, Optical Metrology, 3rd ed. (John Wiley & Sons, 2003).
D. Malacara, Optical Shop Testing, 3rd ed. (Wiley-Interscience, 2007).
M. Servin, J. A. Quiroga, and M. Padilla, Fringe Pattern Analysis for Optical Metrology, Theory, Algorithms,
and applications (Wiley-VCH, 2014).
M. Schwartz, Information Transmission Modulation and Noise, 4th ed. (McGraw-Hill, 1990).
A. Papoulis, Probability, Random Variables and Stochastic Processes, 4th ed. (McGraw Hill, 2000).
C. E. Shannon and W. Weaver, The Mathematical Theory of Communication. (University of Illinois Press.
Urbana. IL, 1949).
H. Nyquist, “Certain factors affecting telegraph speed,” Bell Syst. Tech. J., 3, 412-422, (1924)
R. V. L. Hartley, “Transmission of information,” Bell Syst. Tech. J., 3, 535–564 (1928).
F. T. S. Yu, Optics and Information Theory, (John Wiley & Sons, 1976).
T. M. Cover and J. A. Thomas, Elements of Information Theory, 2th ed. (Wiley-Interscience, 2006).
A. N. Kolmogorov, " On the Shannon Theory of Information Transmission in the Case of Continuous Signals,"
IRE Trans. Inf. Theory, 102-108 (1956).
T. J. Lim and M. Franceschetti, "Information without rolling dice," IEEE Trans. Inf. Theory, 63(3), 1349-1363
(2017).
Y. C. Eldar, Compressed Sensing: Theory and Applications, (Cambridge Univ. Press, 2012).

1

1. Introduction
Optical phase-metrology has productively incorporated well known results from digital signal
processing, stochastic processes and telecommunications [1-5]. In particular, we have applied
the frequency transfer function (FTF) paradigm to the theory of phase-shifting algorithms
(PSAs) [3]. The FTF is now widely used in phase-shifting algorithm (PSA) theory to
investigate their signal-to-noise-ratio (S/N), their harmonic and detuning robustness [3]. Also
stochastic processes theory applied to linear system has been used in PSAs to study the
signal-to-noise-ratio (S/N) of the phase demodulation processes [1,3,4,5]. By plotting the
absolute value of the FTF associated to a PSA one may gauge graphically its measuring noise,
detuning and harmonic rejection [3].
Shannon information theory has not been applied to wavefront phase-metrology [4-11].
Many scientific and engineering disciplines, including optics, use Shannon theory to study the
information processing behavior of their systems [9,10]. Shannon built on Nyquist [7] and
Hartley [8] to develop a general theory for reliable information transmission/storage over
noisy channels [6]. Nowadays this is simply known as Shannon information theory [4,10]. In
this work we use Shannon information theory [6] to quantify the information storage of noisy
fringe-patterns and fringe-data compression by phase-shifting algorithms (PSAs) [1-3].
In digital phase-metrology one wrongly equals the amount of raw fringe data-bits with
information. One normally think that a fringe-image with XxY pixels, and 8 bit/pixel, has
8(XxY) bits of information; however these are raw-data bits, not Shannon information bits.
We show that noisy fringe patterns have much lower Shannon information than the capacity
of their digitizing cameras. Quantifying the Shannon information storage of noisy fringeimages is important for fringe data compression for phase-measuring telemetry. For example,
a spaceborne interferometer may take many phase-shifted fringe-images and send them back
to earth for phase-demodulation. Or phase demodulate them within the spacecraft, and send
just a single complex-valued demodulated signal as compressive sensing wave-data [13].
Knowing the Shannon information storage of noisy-fringes is also important because
costly, high (S/N) CCD-cameras may not be necessary to digitize phase-modulated fringes for
precision phase-metrology. That is because noisy fringe-images contains only a fraction of
the information capacity of the CCD cameras which digitize them. Most bits/pixel
information are wasted to speckle/electronic fringe-noise; for example an electronic speckle
interferometry (ESPI) fringe-image, probably waste 7 out of 12 bits/pixel to noise. So a much
cheaper, high resolution, low (S/N) camera, may be enough for ESPI metrology. In this work
we show that poor (S/N) fringe-images may be compensated by large bandwidth B fringes.
For example, low (S/N) smartphone cameras, compensate this with high resolution images.
The SCT allows us to think seriously about low (S/N), high resolution, mass-produced
smartphone cameras for high-quality wave phase-metrology.
2. Mathematical theory of communication
Here we outline two important contributions on the mathematical theory of information by
Claude E. Shannon: information-source entropy and channel-capacity [6]. Figure 1 shows the
Shannon's abstraction of a telecommunication process [6]. The information source (the
message) may be continuous or discrete data [4]. The message may then be amplitude or
phase-modulated, digitally encoded/compressed, and finally transmitted. The radio signal
then propagate towards the receiver. The communication channel is assumed bandlimited and
noisy. The received signal is then decoded to obtain a reliable estimation of the original
message. Signal decoding may consist on phase-demodulation, decompressing and error
correction-algorithms [4]. We finally produce a recovered information which is a reliable
copy the original message.

2

Fig. 1. Shannon abstraction of a memoryless noisy digital communication process in electrical
engineering. The information source generate r symbols/second [6].

Claude Shannon proved a formula for the channel capacity (the SCT) needed to transmit a
message over a memoryless, noisy-channel with negligible information degradation [6]. In
this paper [6], Shannon introduced the information-entropy and channel-capacity formulas for
reliable transmission over a noisy additive white Gaussian noise (AWGN) channel [6].
2.1 Shannon information entropy of a discrete information source
Shannon information entropy H of a discrete message consisting of a large sequence
 xm (t  m r ) of symbols, drawn from {x0 , x1,..., xK 1} , each sampled with probability

p( xk )  0 is [6],
K 1

H    p ( xk ) log 2  p ( xk ) 
k 0

In the average, a large data-sequence

bits
;
symbol

 x  (t  m r )
m

K 1

 p( x )  1.0 . .
k 0

k

(1)

have an information entropy of H

bits/symbol. If these symbols {xk } are transmitted at a rate of r symbols/second then, the
information-rate is,
 K 1
 bits
.
R  rH   r   p ( xk ) log 2  p ( xk )
 k 0
 second

(2)

The prototypical example of a discrete source is the English language where the data are
{ A, B, C ,..., Z } , with p ( A)  ...  p ( Z )  1.0 , being { p ( A),..., p ( Z )} the probability of
occurrence of a single letter in a text, then H= [ p ( A) log 2 p( A)  ...  p ( Z ) log 2 p ( Z )] .
2.2 Shannon capacity theorem (SCT) for a bandlimited AWGN-channel
The message

 x  (t  m r ),
m

is then transmitted over a AGWN-channel (see Fig. 1). The

received signal is,
X (t )   ( axm  nm ) (t  m r );

t  [0, T ]; T  1.

(3)

m

Where a<1.0 is the channel attenuation. Shannon proved that the rate of information that a
bandlimited AWGN-channel can transmit with negligible errors is [4-6,10],
S  bits

C  B log 2  1  
; S  a 2 E xm2 ; N  E nm2  . .
N
second



(4)

Where B is the channel's bandwidth in cycles/second, S is the received signal-power, N is the
channel noise-power, and E 
 is the ensemble average. This is the famous Shannon capacity
theorem (SCT) for a bandlimited AWGN-channel [4-6,10-11]. The relation between the
source information-rate R and channel capacity C for reliable communication is,

3

R  C ; Unreliable Communication ;

.

(5)

R  C ; Negligible Error Communication .

Therefore in order to transmit reliably through an AWGN-channel the information-rate R
must be below C. For example, a typical AWGN telephone line with B  3kHz and
(S/N)=103 has a capacity of C  3000 log 2 (1  103 )  30,000 bits/second.
The SCT also tell us that we can interchange (S/N) with bandwidth B while keeping the
channel capacity C constant, as Table 1 shows,
Table 1. Channel capacity C=log2[1+(S/N)] trade between (S/N) and bandwidth B
Channel capacity C (bits/sec)

Channel Bandwidth B (Hz).

Signal-to-noise ratio (S/N)

30,000

1,500

1,100,000

30,000

3,000

1,000

30,000

6,000

31

30,000

9,000

9

30,000

12,000

5

To keep C constant, B increases linearly while (S/N) decreases exponentially. That is, for low
(S/N), one may keep C constant by increasing B, and still have reliable communication; this is
why spread-spectrum communication is so robust against thermal-noise, radio-jamming and
low-power spacecraft communications [4]. A daily example of the use of the SCT is when
one walks away from a Wi-Fi transmitter. Given that the hardware have not changed, the
channel bandwidth B remains the same. However at a large distance from the Wi-Fi
transmitter the signal-power decreases. This reduces the data transfer rate R towards our
laptop because the channel encoding must be more redundant, less information-rate efficient.
2.3 Shannon's capacity is a mathematical existence theorem
Finally keep in mind that Shannon capacity theorem (SCT) is a theoretical upper-bound limit
for reliable classical communication/storage of information independently of any engineering
apparatus. The SCT is an existence theorem, it say nothing on how to implement it physically
or algorithmically. That is, the SCT is a mathematical abstraction disembodied of any
implementation, such as Einstein's E=mc2 is not a receipt for building a nuclear reactor. In
fact the SCT may be seen as a sphere-packing theorem in a K-dimensional Euclidian space
[6,10,12]. In other words, no matter how advanced (non-quantum) digital-modem one may
built, it cannot reliably transmit more classical information than C. This is like trying to
obtain more energy than E=mc2 from a rest mass, or trying to build a perpetual moving
machine bypassing the second law of thermodynamics. Modern digital communication
modems using efficient block error-correcting codes approaches the SCT limit, i.e. R  C .
3. Shannon information storage of noisy fringe-patterns
Many signals that we communicate/store are continuous such as voice, video, or telemetry,
where the discrete-data entropy H cannot be used. The source information-rate R=R(H,r) and
channel-capacity C=C(B,S,N) are both given in bits/second, however they use different
parameters [6]. The entropy H, the rate r, and signal-power S are information-source variables
[6]. On the other hand, the bandwidth B, and AWGN-power N are communication channel
variables [6,4,10]. To estimate the source information-rate of a continuous-signal, one would
need a continuous-density entropy. However a continuous-density entropy is not always welldefined [10]. In contrast, the bandwidth, the noise, and the signal-power of a noisy
continuous-source are always well-defined. So it is better to have an information-rate formula
for continuous-signals based on its bandwidth and signal-to-noise ratio, rather than on its

4

continuous-density entropy [10,11]. Fortunately mathematician Andrei Kolmogorov proposed
an information-rate formula for continuous, noisy information-sources, the   entropy
[11,12]. That is, Kolmogorov   entropy does not depend on a continuous-density entropy
which is good, because it avoids some theoretical problems [10,11,12].
In wave phase-metrology the message is the continuous phase-field which modulate the
fringes; it could be an optical wavefront W ( x, y ) , or a solid z  h ( x , y ) in fringe-projection
profilometry [1,2]. The model for a continuous fringe-pattern is,
 2

I ( x, y )  a ( x, y )  b( x, y ) cos  W ( x, y )   n ( x, y ).



(6)

Here a ( x, y ) , b( x, y ) are smooth functions, and  the illumination wavelength. The
probability density p ( I ) of these fringes is continuous. Shannon defined the differential
entropy of a continuous random variable X with density p(x) as [6,10],

H    p( x ) log 2  p( x ) dx;

 p( x) dx  1.0.

(7)

As in every example involving an integral, or even a density, we should include the statement
if it exist [10]. It is easy to construct examples of random variables for which a density
function does not exist or for which the above integral does not exist [10]. Instead of trying to
estimate a continuous-density entropy H of a fringe-image, we use the Kolmogorov's
  entropy H  ( ) . In 1956 Kolmogorov wrote [11]: According to the proper meaning of the
word, the entropy of a signal with a continuous distribution is always infinite. If the
continuous signals can, nevertheless, serve to transmit finite information, then it is only
because they are always observed with bounded accuracy. Consequently, it is natural to
define the appropriate "   entropy " H  ( ) of the object  by giving the accuracy of
observation  . Shannon did this under the designation "rate of creating information with
respect to a fidelity criterion" [11]. In our notation, the noiseless object is   a  b cos[ ] ,
which is observed with bounded-accuracy   n . Kolmogorov proved that H  ( I )
approximates Shannon's capacity C [11, 12]. The information-rate H  ( I ) for I ( x , y ) is [11],
H  ( I )  BI log 2 1  SNR K 

bits
;
pixel

BI in

fringes
.
pixel

(8)

; s( x, y )  b cos   .

(9)

Being SNRK the signal-to-noise ratio of I ( x , y ) , given by,

SNR K 



s 2 ( x , y ) dxdy



n 2 ( x, y ) dxdy

( x , y )

I ( x, y )





2

s (u, v ) dudv

( u , v ) B

  
 

2

n (u, v ) dudv

[  , ]x[  , ]

Where s(u, v ) , n(u, v ) are the spectra of s( x, y ) , n( x, y ) respectively; ( x, y )   is the
region with well defined fringes; (u, v)  B the fringes' spectral lobe; BI is the average
bandwidth (see Fig. 2). Note that formally H  ( I ) equals C; but the difference is that now B
and SNRK refer to the noisy signal, not the noisy channel. Kolmogorov H  ( I ) cannot exceed
the storage-capacity of the digitizing CCD-camera [11]. For example, for a 256 gray-levels
per pixel camera, then H  ( I ) <8 bits/pixel.

5

Fig. 2. A 8 bits/pixel quantized, noiseless simulated fringes I(x,y). The spectrum |I(u,v)| has a
bandwidth .

Let us consider the limits for H  ( I ) when the fringe's noise n( x, y ) tends to zero or infinite,
lim  BI log 2 1  SNR K     ;

lim  BI log 2 1  SNR K    0.
n 

n 0

(10)

The information-rate is infinite H  ( I )   for noiseless fringes, and zero H  ( I )  0 for
infinite noisy fringes; regardless of BI. One might think that a constant-signal would hold
little information; however if that signal is coded into a large number with say, 1x108
significant digits (above the noise n ), then one may encode a large amount of information
within that single number.
As mentioned, Shannon capacity C (Eq. (3)) and Kolmogorov entropy-rate H  ( I ) (Eq.
(7)) look identical, just substituting (S/N) for SNRK. However in Shannon's capacity N is a
Gaussian stochastic processes [6], while the observation accuracy  , in H  ( I ) , is a
deterministic error-signal [11,12]. This seems as an irrelevant theoretical nuance, but it has
however practical consequences [6,10,11,12]. Lim and Franceschetti proved that C and
H  ( I ) , are indeed "identical" under appropriate mathematical conditions [12].
4. Simulated Shannon information storage of fringe-images
Here we give two simple but illustrative examples of noisy fringe information storage. We
use low resolution fringes to show the fringes at their Nyquist sampling rate of BI=0.5
fringes/pixel. Figure 3 shows three 200x200-pixel, computer-generated noisy fringes phasemodulated by defocus and increasing n(x,y).

Fig. 3. Information storage-rate for fringes with BI= fringes/pixel. The 8 bit/pixel fringes
have SNRK=70,000 (about 8-bit quantizing noise). The noise is obvious by 1.0 bit/pixel
(SNRK=3.0). The noisier fringes have 0.5 bits/pixel (SNRK=1.0).

6

We then decrease the SNRK, until reaching H  ( I ) =0.5-bits/pixel. Figure 3 and Fig. 4 show
the interplay between BI and SNRK stated by H  ( I ) . Figure 4 shows a similar sequence of
fringes but with reduced bandwidth BI=0.125 fringes/pixel. In Fig. 4 the maximum fringefrequency is one fourth of the Nyquist rate: BI=0.25(0.5)=0.125 fringes/pixel, therefore the
average information-rate H  ( I ) is reduced one fourth with respect to Fig. 3 for the same
SNRk.

Fig. 4. Average information-rate of fringe-images with BI=0.125 fringes/pixel. The 2-bit/pixel
fringes have SNRK=70,000 (about 8-bits quantizing noise). The noisier fringes have 0.2
bits/pixel (SNRK=2.03). The reduced spectrum of the noisier fringes is at far right.

The information-rate H  ( I ) is reduced by decreasing the SNRk until the noisier fringes at the
far-right with 0.2-bit/pixel is obtained. Finally the spectrum of the noisiest-fringes with 0.2
bits/pixel is shown at the far right.
Remember that we are finding the low information-rate H  ( I ) of noisy fringes with
respect to the CCD camera used to digitize them. Kolmogorov's H  ( I ) tell us nothing about
how to program a compressing algorithm to store/transmit efficiently a large number of noisy
fringe images.
5. Shannon information storage of experimental fringes
Here we are estimating the Shannon information content of experimental fringe-projection
profilometry fringes. To estimate H  ( I ) for profilometry fringes, one possibility is to take two
images: the noisy fringes, and the noisy background. The formalism for both images are [1],
 2

I ( x , y )  a ( x, y )  b( x, y ) cos 
tan( )h ( x , y )   n ( x, y );
p


I n ( x, y )  a ( x, y )  n ( x, y ).

(11)

Where h( x, y ) is the object being digitized; p the period of the projected fringes and  is
the sensitivity angle between the projector and the (8 bits/pixel) CCD-camera [1].

7

Fig. 5. High spatial frequency fringe-images (512x640, 8 bits/pixel) for obtaining an
estimation of the information storage-rate for fringe-projection profilometry carrier-fringes.

White-light fringe-projection patterns have relatively high SNRK. We then find the spectrum
of the fringes I (u, v )  F {I ( x, y )} , and define the indicator function for the signal-plus-noise
region as (u, v )  SN , and for the noise-only region as (u, v )  NR .

Fig. 6. Fourier spectrum corresponding to the fringes and noise (512x640 pixels).

Figure 7 shows the Fourier spectrum of the profilometry fringes and the Fourier spectrum of
the background-image; both images defined in Eq. (11). Figure 7 shows the spatial filters to
keep just the signal-lobes spectra within the region (u, v )  SN .

Fig. 7. Fourier filtered fringe-lobes (BI=0.11), and high-pass filtered noise.

Next we show the demodulated wrapped-phase obtained by taken the right lobe of the signalplus-noise region (u, v )  SN in Fig. 7. This was made to be sure that we have included all
the phase signal bandwidth. This is shown in Fig. 8,

8

Fig. 8. The fringe-pattern and its wrapped demodulated phase (512x640 pixels). At the fringe
self-occluding shadow the fringe-amplitude drops and the wrapped phase becomes noisier.

We then find the noise-power density within the noise-only region (u, v )  NR as,



1
ANR



a ( u, v )  n (u , v ) .
2

(12)

( u , v )NR

Where a (u, v )  n(u, v )  F{a ( x, y )  n( x, y )} and ANR the area of (u, v )  NR . We then
assume that the noise-density within (u, v )  SN is also  (Eq. (15)). Finding (S/N) as,

SNR K 



( u , v )SN

 I ( u, v )

2

512(640)





 11.2 .

(13)

We finally estimate the information-rate (with BI=0.11, Fig. 7) of this fringe-pattern as,
H  ( I )  BI log 2 1  SNR K   0.41

bits
.
pixel

(14)

The average bandwidth BI (Fig. 7) of the fringes may be estimated by looking at the spectrum
the fringes I (u, v )  F {I ( x, y )} . This profilometry fringe-image has in the average 0.41
bits/pixel, much less information than the 8 bits/pixel capacity of the CCD-camera. The
information storage of most fringe patterns is usually well below the capacity of the digitizing
CCD-camera. This means that if we want to transmit/store fringe-images one may effectively
use compression image algorithms.
6. Real-life versus fringe-pattern images data compression
Real life images are highly redundant; have low information entropy. A real life photograph
usually have smooth image regions, along with sparse noisy-like regions with detailed image
features. Data compression algorithms may use local space-frequency wavelets expansion to
reduce redundancy [13]. Image compression algorithms are broadly classified as lossy and
lossless. In lossy compression the encoding/decoding algorithm decompress the original
image not 100% accurately. However, subjective quality perception may tolerate large
compression ratios without subjective degradation [10]. In lossless algorithms, decompressed
images remains 100% accurate to the original. Therefore the most compressible images are
usually smooth piece-wise continuous. On the other hand, one cannot compress a purelyrandom image, if we want to preserve exactly every pixel value; a white noise image is not
compressible. Common image storage formats are: 8-bits/pixel Bit Maps (BMP); Joint
Photographic Experts Group (JPEG); 8-bits/pixel Graphic Interchange Format (GIF); Tagged
Image File Format (TIFF), and lossless Portable Network Graphics (PNG). These file
formats have been designed to store real-life color images. These formats store noisy fringepatterns as images containing more "information" than noiseless fringes. The noise is

9

interpreted as fine image-details that the user wants to "preserve". On the contrary, noiseless
fringes occupy less storage bytes; they are interpreted as smoother images (see Fig. 9).

Fig. 9. Noiseless and noisy fringes with 200x200 pixel resolution and 8 bits/pixel. At right, the
PNG file size is 40-Kbytes, while the noisy fringe image has a PNG file size of 180-Kbytes.

We have simulated two fringe patterns shown in Fig. 9. We have store them using lossless
PNG format, and Fig. 9 shows the resulting PNG file sizes in kilobytes. The noisy image file
is about four times bigger because the noise is interpreted as high frequency image details.
The PNG format is designed to compress color real-life images for good visual perception,
not phase-modulated noisy fringes. Fringe patterns have more mathematical structure than
real-life images. So noisy fringe images are far more redundant and as a consequence, more
compressible. The mathematical model for fringe patterns (Eq. (6)) allow us to distinguish
between phase information and noise. Due to these considerations, real-life image
compressors are not suitable for efficient transmission/storage of noisy fringe patterns; they
were simply not designed for that purpose.
7. Fringe-data compression by phase-shifting algorithms (PSAs)
We have seen that noisy fringe patterns store much less Shannon information than the
capacity of the CCD used to digitize them. In this section we show that phase-shifted fringes
are even more information redundant. In remote optical/RADAR phase-metrology one may
digitize M phase-shifted fringes at a spacecraft (see Fig. 10). Then, one may downlink these
M fringes sequentially, and phase-demodulate them at the receiver. Or we may use a PSA to
phase-demodulate them at the spacecraft, and downlink just the demodulated analytic-signal.
A PSA may then be regarded as compressive-sensing, where we are "sensing" not real-valued
fringes, but less noisy complex-valued wave-data [13]. So PSA compression of phase-shifted
fringes is paramount in remote phase-shifting metrology where communications are limited
due to large distances and few watts for radio transmission. A PSA then packs the information
contained in M phase-shifted images into a single phase-demodulated complex-valued image.

Fig. 10. Schematic of a spacecraft transmitting high-information, complex-valued wave-data
from a set of M phase-shifted fringe-patterns.

To quantify the increase of information-rate by phase-shifting algorithms (PSAs),
consider a sequence of M phase-shifted noisy interferograms,

10

I ( x, y, t ) 

M 1



 2

 a  b cos  

m 0

2


W  m 0   nm   (t  m ); 0 
.
M



(15)

Here some (x,y) coordinates were omitted for clarity. We may phase-demodulate these fringes
by a PSA whose impulse response is [3],
h (t ) 

M 1

c

m 0

m

 (t  m ).

(16)

Being cm complex-valued. The Fourier transform of h(t ) ( H ( )  F [h(t )] ) must have at
least the following zeroes [3],
H ( 0 )  H (0)  0;

and

H (0 )  0.

(17)

Obtaining the estimated quadrature analytic signal as [3],
Aei  ( x , y )  I ( x, y , t )  h (t ) t  M 1 

M 1

c
m 1

m

I ( x, y , m ).

(18)

A PSA increases the (S/N) of the analytic-signal with respect to the fringes (Eq. (15)) by [3],
GS / N 

H (0 )
1
2





2

(19)

.

H ( ) d 
2



The highest GS / N is obtained by the least-squares PSA, where GS / N  M [3]. Then the SNRK
of the noisy fringes increases to M(SNRK) for the complex-valued phase-demodulated signal
Aei  ( x , y ) . Then the information-rate contained in the analytic signal Aei  ( x , y ) is,
H  ( A; M )  BI log 2 1  M SNR K 

bits
.
pixel

(20)

The information of Aei  ( x , y ) increases logarithmically with M. An interesting question is: how
many phase-shifted fringes double the information content of H  ( A; M ) , or,
BI log 2 1  M SNR K   2  BI log 2 1  SNR K  .

(21)

Solving for M one obtains,
M  2  SNR K .

This is shown in Fig. 11,

Fig. 11. Information storage as a function of SNRK, with BI=0.5 fringes/pixel. At the
green-zone, the spatial information-rate increases almost linearly from 1 to 2
bits/pixel. In contrast in the blue-zone, this rate increases logarithmically from 2 to 4
bits/pixel.

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(22)

For example with SNR K  1 one would need M=3 phase-shifted fringes to double the
information-rate of Aei  ( x , y ) . In contrast for SNR K  20 , one would need M=22 phaseshifted fringes to double the information-rate of Aei  ( x , y ) from 2 to 4 bits/pixel. An example
of this is shown in Fig. 12 and Fig. 13 (for M=12).

Fig. 12. Four out-of 12 phase-shifted fringes degraded with AWGN. These fringes have an
average information-rate of 0.4bits/pixel (BI=0.125fringes/pixel, SNRK=8.1).

The M=12 phase-shifted fringes in Fig. 12 were phase-demodulated using a least-squares
PSA and the resulting real and imaginary images of the analytic signal are shown in Fig. 13,

Fig. 13. Real and imaginary parts of the phase-demodulated signal from the 12 noisy phaseshifted fringes in Fig. 12. The information-rate has increased about twice, from 0.4 bits/pixel to
0.82 bits/pixel (BI=0.12fringes/pixel, SNRK=97.5, M=12).

As conclusion, a PSA regarded as fringe-image compressor packs into a single analyticsignal Aei ( x , y ) the phase information stored in M phase-shifted fringes. In other words,
instead of transmitting/storing M phase-shifted fringes, one may send/store a single, higherinformation signal Aei ( x , y ) , or wrapped-phase arg[ Aei  ( x , y ) ] . Therefore a PSA may be
regarded as an efficient compression algorithm for transmission or storage of large blocks of
phase-shifted fringe-data. This is paramount for efficient transmission of optical/RADAR
interference fringes from remote phase-metrology sites where the channel is severely limited
by large distances (high attenuation) and low power radio link. Also a PSA may be regarded
as a compressive-sensing algorithm where the collected data is complex-valued with lowernoise and as consequence higher information rate [13].
8. Summary
We have shown that noisy fringe-images in wave phase-metrology contain much less
Shannon information than the capacity of the CCD camera used to digitize them. For
example, it does not make sense to store electronic speckle-pattern interferometry (ESPI)
fringes using a costly 14 bits/pixel cooled CCD-camera. That is because most data bits (not
information bits) are wasted to fringe-noise. ESPI noisy fringes have very-low spatial
information-rate. Shannon information may be used as cost-efficiency criteria for choosing a
CCD camera according to the class of fringes that we are dealing with. For example, mass-

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produced smartphone cameras may do just fine in many optical-metrology cases. Smartphone
cameras have low signal-to-noise ratio, but this is compensated with high image-pixel
resolution; 12 megapixels or more is nowadays available. So smartphone cameras with
moderate (S/N) and very high resolution images, may be efficiently used for remote phasewave telemetry applications.
We showed that low-noise fringe-images may be obtained from white-light, fringeprojection profilometry. Here we have estimated that an experimental carrier-fringe
profilometry image may contain less than 0.5 bits/pixel of Shannon information (Figs. 5-6).
With these small information-rates one realizes that there is plenty of room for noisy fringedata compression for efficient transmission or storage. We have seen that PSAs may be
regarded as fringe-data compression algorithms. Using PSAs, all Shannon information
contained in M phase-shifted fringes may be compressed into a single analytic signal. In
spaceborne phase-metrology for example, one may use PSAs to compress large blocks of M
phase-shifted fringes for efficient downlink. Finally we saw that PSAs compress noisy fringedata information by increasing the SNR K of the phase-demodulated complex-signal.

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