why is an image non-stationary?

Hi all,

I am not sure I completely understood...

but I've heard people say that an image is non-stationary, blah blah blah,

what does that mean?

and what does that imply?

I vaguely heard that people say since an image is not-stationary, so Fourier 
Transform should not be applied, etc...

Can anybody throw some lights to me?

thanks a lot!

Hello,
               4) It should be BIBO stable( Bounded Input and Bounded
Output) says, for finte input the systme output should be finite.

Regards,
Athreya

"kiki" <lunaliu3@yahoo.com> wrote in message
news:co1gqv$4hr$1@news.Stanford.EDU...
> Hi all,
>
> I am not sure I completely understood...
>
> but I've heard people say that an image is non-stationary, blah blah blah,
>
> what does that mean?
>
> and what does that imply?
>
> I vaguely heard that people say since an image is not-stationary, so
Fourier
> Transform should not be applied, etc...
>
> Can anybody throw some lights to me?
>
> thanks a lot!
>
>
>

Hello,
             Earlier posted for wrong question.
             Image is always non stationary, as the expectation of pixel
matrix is non zero one.

Regards,

Athreya
"kiki" <lunaliu3@yahoo.com> wrote in message
news:co1gqv$4hr$1@news.Stanford.EDU...
> Hi all,
>
> I am not sure I completely understood...
>
> but I've heard people say that an image is non-stationary, blah blah blah,
>
> what does that mean?
>
> and what does that imply?
>
> I vaguely heard that people say since an image is not-stationary, so
Fourier
> Transform should not be applied, etc...
>
> Can anybody throw some lights to me?
>
> thanks a lot!
>
>
>

kiki wrote:
> Hi all,
> 
> I am not sure I completely understood...
> 
> but I've heard people say that an image is non-stationary, blah blah blah,
> 
> what does that mean?
> 
> and what does that imply?
> 
> I vaguely heard that people say since an image is not-stationary, so Fourier 
> Transform should not be applied, etc...

Fourier analysis is based on two assumptions. One is that the system is
superposable, usually just called linear. The other is that the system
is time invariant in that the origin of time does not matter for the
analysis. So sound filters work the same at 9:00AM (before your coffee)
as they do at 10:30AM (after your coffee).

For images one uses position rather than time. Superposable is OK.
Position invariance would mean that a seascape would have the same
properties as an urban image. If you think not then you have given
up the Fourier assumptions. The things folks will agree on tend to
suggest Haar analysis or perhaps wavelets.

> Can anybody throw some lights to me?
> 
> thanks a lot!
> 
> 
> 

In article <tg0pd.205586$9b.150943@edtnps84>,
Gordon Sande  <g.sande@worldnet.att.net> wrote:

>kiki wrote:
>> 
>> I am not sure I completely understood...
>> 
>> but I've heard people say that an image is non-stationary, blah blah blah,
>> 
>> what does that mean?

[I was sort of hoping that a mathematician would chime in here, because
I had noticed a spate of sci.math threads about these signal-processing
applications, all very confusing to me: they appear to use mathematicians' 
terminology ("linear", etc.) to mean something else.]

>Fourier analysis is based on two assumptions. One is that the system is
>superposable, usually just called linear. The other is that the system
>is time invariant in that the origin of time does not matter for the
>analysis. So sound filters work the same at 9:00AM (before your coffee)
>as they do at 10:30AM (after your coffee).
>
>For images one uses position rather than time. Superposable is OK.
>Position invariance would mean that a seascape would have the same
>properties as an urban image. If you think not then you have given
>up the Fourier assumptions. The things folks will agree on tend to
>suggest Haar analysis or perhaps wavelets.

Um, right. THere has to be a more rigorous way to put this! Sure,
sound filters work the same at morning and night. And image filters
work the same in London as in Hong Kong. So what's the point? 

I think what you _meant_ to say was that the Fourier series can be computed
over any length of time (if it's a multiple of the period of the signal);
you don't need to know what the starting point of a period is.
But the same could be said of an image IF it's periodic. You could
for example compute a fourier series for the background images on many
computer screens --- the ones that tile endlessly. Just as with the
sound filter, you could start your computations anywhere within the
tile ("fundamental domain", in math parlance) and get the same answers.

What prompts the use of wavelets and other things is precisely the
lack of periodicity. That would be true of sound filters too: it would
be pointless to try to view the sound wave of, say, a one-hour conversation
as if it were a complex superposition of sound waves with periods which
evenly divided 1 hour!

dave
(writing from sci.math)

Dave Rusin wrote:

(someone wrote)

>>Fourier analysis is based on two assumptions. One is that the system is
>>superposable, usually just called linear. The other is that the system
>>is time invariant in that the origin of time does not matter for the
>>analysis. So sound filters work the same at 9:00AM (before your coffee)
>>as they do at 10:30AM (after your coffee).

>>For images one uses position rather than time. Superposable is OK.
>>Position invariance would mean that a seascape would have the same
>>properties as an urban image. If you think not then you have given
>>up the Fourier assumptions. The things folks will agree on tend to
>>suggest Haar analysis or perhaps wavelets.

> Um, right. THere has to be a more rigorous way to put this! Sure,
> sound filters work the same at morning and night. And image filters
> work the same in London as in Hong Kong. So what's the point? 

Time and position invariance.  There is a close relation between
symmetry and conservation laws.  Conservation of energy is related
to time invariance, conservation of momentum to position invariance.
Conservation of angular momentum to rotation invariance.

Note also that energy*time, momentum*distance, and angular position
all have the same dimensions.

(snip)

-- glen

glen herrmannsfeldt wrote:

>
> Dave Rusin wrote:
>
> (someone wrote)
>
>>> Fourier analysis is based on two assumptions. One is that the system is
>>> superposable, usually just called linear. The other is that the system
>>> is time invariant in that the origin of time does not matter for the
>>> analysis. So sound filters work the same at 9:00AM (before your coffee)
>>> as they do at 10:30AM (after your coffee).
>>
>
>>> For images one uses position rather than time. Superposable is OK.
>>> Position invariance would mean that a seascape would have the same
>>> properties as an urban image. If you think not then you have given
>>> up the Fourier assumptions. The things folks will agree on tend to
>>> suggest Haar analysis or perhaps wavelets.
>>
>
>> Um, right. THere has to be a more rigorous way to put this! Sure,
>> sound filters work the same at morning and night. And image filters
>> work the same in London as in Hong Kong. So what's the point? 
>
>
> Time and position invariance.  There is a close relation between
> symmetry and conservation laws.  Conservation of energy is related
> to time invariance, conservation of momentum to position invariance.
> Conservation of angular momentum to rotation invariance.
>
> Note also that energy*time, momentum*distance, and angular position
> all have the same dimensions.
>
> (snip)
>
> -- glen 


Hi all,

Just wanted to chip in with an electrical engineer's viewpoint.

First of all, describing a system as LSI - linear and shift-invariant - 
only describes the system itself. It says nothing at all about the 
characteristics of the input signal.  Shift invariance thus does NOT 
imply that "...a seascape would have the same properties as an urban image."

If a system is shift invariant, it just means that the system response 
does not vary with time or position. Mathematically, if a given input 
f(t) produces an output g(t), then if the system is LSI, a time-delayed 
input f(t + T) produces a time-delayed output g(t + T).

For an LSI imaging system, shift invariance means that the optical 
system's impulse response (point spread function) is constant over the 
field of view of the system.

Modeling image formation as an LSI process is the basis of Fourier 
optics - see the excellent text by J. Goodman.

Cheers,
H

PS Regarding the OP's question on image stationarity:  If the intensity 
distribution of the object being imaged is time-varying, then a time 
sequence of images of that object will also display a time-varying 
intensity, and hence the image intensity is non-stationary.

Dave Rusin wrote:
> In article <tg0pd.205586$9b.150943@edtnps84>,
> Gordon Sande  <g.sande@worldnet.att.net> wrote:
> 
> 
>>kiki wrote:
>>
>>>I am not sure I completely understood...
>>>
>>>but I've heard people say that an image is non-stationary, blah blah blah,
>>>
>>>what does that mean?
> 
> 
> [I was sort of hoping that a mathematician would chime in here, because
> I had noticed a spate of sci.math threads about these signal-processing
> applications, all very confusing to me: they appear to use mathematicians' 
> terminology ("linear", etc.) to mean something else.]

Good mathematicians are able to use the terminology of their
applications if that helps communications. Linear is so pervasive
that it has many detailed technical meanings when it is not
embedded in much longer unambiguous technical phrases.

>>Fourier analysis is based on two assumptions. One is that the system is
>>superposable, usually just called linear. The other is that the system
>>is time invariant in that the origin of time does not matter for the
>>analysis. So sound filters work the same at 9:00AM (before your coffee)
>>as they do at 10:30AM (after your coffee).
>>
>>For images one uses position rather than time. Superposable is OK.
>>Position invariance would mean that a seascape would have the same
>>properties as an urban image. If you think not then you have given
>>up the Fourier assumptions. The things folks will agree on tend to
>>suggest Haar analysis or perhaps wavelets.
> 
> 
> Um, right. THere has to be a more rigorous way to put this! Sure,
> sound filters work the same at morning and night. And image filters
> work the same in London as in Hong Kong. So what's the point? 

Sound filters work the same if the time origin is relabelled, otherwise
known as time invariance. Images do not have position invariance even
if the same processing might be applied in both NYC and HK. Depending
upon the invariance conditions you get differing sorts of a diagonalizing
transformation for the operators.

> I think what you _meant_ to say was that the Fourier series can be computed
> over any length of time (if it's a multiple of the period of the signal);

Gee! I was not aware that the usual integral Fourier transform defined
on the real line had any requirements that the function under analysis be
periodic. If the indexing group is periodic then you get what is usually
called Fourier Series with its discrete frequencies. What was meant is
that the Fourier transform is a diagonalizing operation for operators
which are superposable, linear in the common jargon, and time invariant
with respect to their time index whether it be unbounded or periodic,
continuous or discrete. There being slightly differing FTs for the
various underlying indexing groups.

> you don't need to know what the starting point of a period is.
> But the same could be said of an image IF it's periodic. You could

But images are not periodic. Compression schemes based on periodic
assumptions tend to particular styles of artifacts. Often they are made
periodic by first reflecting and then repeating to generate only even
Fourier coefficients, i.e. cosines.

> for example compute a fourier series for the background images on many
> computer screens --- the ones that tile endlessly. Just as with the
> sound filter, you could start your computations anywhere within the
> tile ("fundamental domain", in math parlance) and get the same answers.
> 
> What prompts the use of wavelets and other things is precisely the
> lack of periodicity. That would be true of sound filters too: it would
> be pointless to try to view the sound wave of, say, a one-hour conversation
> as if it were a complex superposition of sound waves with periods which
> evenly divided 1 hour!
> 
> dave
> (writing from sci.math)

Dave Rusin wrote:

> [massive SNIP] ...
> lack of periodicity. That would be true of sound filters too: it would
> be pointless to try to view the sound wave of, say, a one-hour conversation
> as if it were a complex superposition of sound waves with periods which
> evenly divided 1 hour!
> 

I have amateur/hobbyist/??? interest in characterizing speech.

My question was "What frequencies are significant?"

As a first pass I plotted the magnitude of the FFT of ~1 minute of a 
speaker.

 From that plot I "discovered" formants. I think my underlying 
assumption was that I was plotting a 'probability' of a frequency being 
present.

How far off the mark am I?

Thanks.

Dave Rusin wrote:

> In article <tg0pd.205586$9b.150943@edtnps84>,
> Gordon Sande  <g.sande@worldnet.att.net> wrote:
> 
> 
>>kiki wrote:
>>
>>>I am not sure I completely understood...
>>>
>>>but I've heard people say that an image is non-stationary, blah blah blah,
>>>
>>>what does that mean?
> 
> 
> [I was sort of hoping that a mathematician would chime in here, because
> I had noticed a spate of sci.math threads about these signal-processing
> applications, all very confusing to me: they appear to use mathematicians' 
> terminology ("linear", etc.) to mean something else.]
> 
Partially there needs to be a translation guide, partially some of the 
terms are used loosely.  "Linear System", specifically, causes confusion.

I think that when a mathematician sees the phrase he thinks "linear 
system of equations".  The signal & systems person, however, sees this 
and thinks "linear dynamical system", in which a system is a thingie 
that transforms one signal into another, a dynamical system is one where 
the current value of the output signal is a function of the history of 
the input signal (possibly only the past history, possibly past, present 
& future), and a linear dynamical system is one that obeys 
superposition, i.e. the output signal that results from the sum of two 
input signals is exactly equal to the sum of the output signals that 
would have resulted from the two input signals processed individually.
> 
>>Fourier analysis is based on two assumptions. One is that the system is
>>superposable, usually just called linear. The other is that the system
>>is time invariant in that the origin of time does not matter for the
>>analysis. So sound filters work the same at 9:00AM (before your coffee)
>>as they do at 10:30AM (after your coffee).
>>
>>For images one uses position rather than time. Superposable is OK.
>>Position invariance would mean that a seascape would have the same
>>properties as an urban image. If you think not then you have given
>>up the Fourier assumptions. The things folks will agree on tend to
>>suggest Haar analysis or perhaps wavelets.
> 
> 
> Um, right. THere has to be a more rigorous way to put this! Sure,
> sound filters work the same at morning and night. And image filters
> work the same in London as in Hong Kong. So what's the point? 
> 
> I think what you _meant_ to say was that the Fourier series can be computed
> over any length of time (if it's a multiple of the period of the signal);
> you don't need to know what the starting point of a period is.
> But the same could be said of an image IF it's periodic. You could
> for example compute a fourier series for the background images on many
> computer screens --- the ones that tile endlessly. Just as with the
> sound filter, you could start your computations anywhere within the
> tile ("fundamental domain", in math parlance) and get the same answers.
> 
> What prompts the use of wavelets and other things is precisely the
> lack of periodicity. That would be true of sound filters too: it would
> be pointless to try to view the sound wave of, say, a one-hour conversation
> as if it were a complex superposition of sound waves with periods which
> evenly divided 1 hour!
> 
> dave
> (writing from sci.math)

I have a question myself about the use of the word "stationary" that I'm 
going to post separately.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com