*AI Summary*
*# 1. Analyze and Adopt*
*Domain:* Computer Vision / Computational Geometry / Pattern Recognition
*Persona:* Senior Research Scientist in Computer Vision and Mathematical Optimization
---
### 2. Summarize (Strict Objectivity)
*Abstract:*
This seminal paper introduces a computationally efficient, non-iterative method for fitting ellipses to scattered 2D data. While traditional least-squares methods for conic fitting often produce non-elliptical results (hyperbolae or parabolae) under high noise or occlusion, the proposed algorithm incorporates a specific ellipticity constraint—$4ac - b^2 = 1$—directly into the normalization factor. Mathematically, the problem is formulated as a constrained minimization of algebraic distance, which is solved via a generalized eigenvalue system ($Sa = \lambda Ca$). The authors provide a theoretical proof establishing that the system yields exactly one positive generalized eigenvalue, which corresponds to the unique elliptical solution. Experimental results demonstrate that the method is affine-invariant, robust against noise, and exhibits a graceful degradation under occlusion compared to iterative or general conic-fitting approaches.
*Technical Summary: Direct Least Square Fitting of Ellipses*
* *[Section 1] Foundational Problem:* The paper addresses the requirement in pattern recognition to fit ellipses—critical as perspective projections of circles—to image data. Previous methods were either computationally expensive iterative procedures or general conic fitters that frequently failed by returning unbounded hyperbolae when data was noisy or occluded.
* *[Section 2] Constraint Limitations:* Existing least-squares techniques minimize the sum of squared algebraic distances. However, the choice of parameter constraint (e.g., $|a|^2=1$ or $a+c=1$) determines the behavior of the fit. Prior to this work, a direct, ellipse-specific constraint that could be solved linearly had not been identified.
* *[Section 3] The Ellipticity Constraint:* The authors propose the quadratic constraint $4ac - b^2 = 1$. This specific constraint is represented by a $6 \times 6$ constraint matrix $C$. Because the constraint $b^2 - 4ac < 0$ defines an ellipse, forcing the discriminant to a negative constant ensures the resulting conic is always an ellipse, regardless of data quality.
* *[Section 3.1] Generalized Eigensystem:* The minimization of algebraic distance $E = \|Da\|^2$ subject to $a^T Ca = 1$ is solved using Lagrange multipliers, resulting in the generalized eigenvalue problem $Sa = \lambda Ca$, where $S$ is the scatter matrix.
* *[Section 3.2] Uniqueness Proof:* Through Lemma 1 and Theorem 1, the authors prove that the system $(S, C)$ possesses exactly one positive eigenvalue. This ensures the algorithm always identifies a single, unique elliptical solution rather than multiple local minima or non-elliptical conics.
* *[Section 3.3] Affine Invariance:* The method is proven to be invariant to affine transformations. Because the error measure is a scalar multiple of the original under transformation, the minimizer scales accordingly, ensuring consistency across different coordinate mappings.
* *[Section 4] Bias and Noise Performance:* The algorithm exhibits a "low-eccentricity bias," pulling solutions away from the parabolic singularity ($b^2 - 4ac = 0$). Comparative testing against Bookstein, Taubin, and Gander methods shows superior stability in center-position estimation and shape retention as Gaussian noise levels increase.
* *[Section 4.3] Resilience to Occlusion:* In experiments with partial elliptical arcs, the "B2AC" (new) method demonstrates graceful degradation. While all methods show "shrinking" bias under high noise, the proposed method remains more stable and predictable than iterative alternatives.
* *[Section 5] Implementation Efficiency:* The algorithm is highly accessible for industrial applications, capable of being implemented in six lines of Matlab code. It functions as a standalone solution or a robust initial estimate for more complex geometric distance minimizations.
---
### 3. Reviewer Recommendations
This topic would be best reviewed by *Senior Computer Vision Engineers,* *Computational Mathematicians,* and *Robotics Researchers specializing in SLAM (Simultaneous Localization and Mapping).*
These professionals are concerned with the trade-offs between algorithmic robustness and real-time execution. A Computer Vision Engineer would prioritize the "ellipse-specificity" (guaranteeing a valid shape), while a Mathematician would focus on the "generalized eigensystem" solution for its numerical stability and the "uniqueness proof" which eliminates the need for heuristic initialization.
AI-generated summary created with gemini-3-flash-preview for free via RocketRecap-dot-com. (Input: 17,472 tokens, Output: 967 tokens, Est. cost: $0.0116).
Below, I will provide input for an example video (comprising of title, description, and transcript, in this order) and the corresponding abstract and summary I expect. Afterward, I will provide a new transcript that I want a summarization in the same format.
**Please give an abstract of the transcript and then summarize the transcript in a self-contained bullet list format.** Include starting timestamps, important details and key takeaways.
Example Input:
Fluidigm Polaris Part 2- illuminator and camera
mikeselectricstuff
131K subscribers
Subscribed
369
Share
Download
Clip
Save
5,857 views Aug 26, 2024
Fluidigm Polaris part 1 : • Fluidigm Polaris (Part 1) - Biotech g...
Ebay listings: https://www.ebay.co.uk/usr/mikeselect...
Merch https://mikeselectricstuff.creator-sp...
Transcript
Follow along using the transcript.
Show transcript
mikeselectricstuff
131K subscribers
Videos
About
Support on Patreon
40 Comments
@robertwatsonbath
6 hours ago
Thanks Mike. Ooof! - with the level of bodgery going on around 15:48 I think shame would have made me do a board re spin, out of my own pocket if I had to.
1
Reply
@Muonium1
9 hours ago
The green LED looks different from the others and uses phosphor conversion because of the "green gap" problem where green InGaN emitters suffer efficiency droop at high currents. Phosphide based emitters don't start becoming efficient until around 600nm so also can't be used for high power green emitters. See the paper and plot by Matthias Auf der Maur in his 2015 paper on alloy fluctuations in InGaN as the cause of reduced external quantum efficiency at longer (green) wavelengths.
4
Reply
1 reply
@tafsirnahian669
10 hours ago (edited)
Can this be used as an astrophotography camera?
Reply
mikeselectricstuff
·
1 reply
@mikeselectricstuff
6 hours ago
Yes, but may need a shutter to avoid light during readout
Reply
@2010craggy
11 hours ago
Narrowband filters we use in Astronomy (Astrophotography) are sided- they work best passing light in one direction so I guess the arrows on the filter frames indicate which way round to install them in the filter wheel.
1
Reply
@vitukz
12 hours ago
A mate with Channel @extractions&ire could use it
2
Reply
@RobertGallop
19 hours ago
That LED module says it can go up to 28 amps!!! 21 amps for 100%. You should see what it does at 20 amps!
Reply
@Prophes0r
19 hours ago
I had an "Oh SHIT!" moment when I realized that the weird trapezoidal shape of that light guide was for keystone correction of the light source.
Very clever.
6
Reply
@OneBiOzZ
20 hours ago
given the cost of the CCD you think they could have run another PCB for it
9
Reply
@tekvax01
21 hours ago
$20 thousand dollars per minute of run time!
1
Reply
@tekvax01
22 hours ago
"We spared no expense!" John Hammond Jurassic Park.
*(that's why this thing costs the same as a 50-seat Greyhound Bus coach!)
Reply
@florianf4257
22 hours ago
The smearing on the image could be due to the fact that you don't use a shutter, so you see brighter stripes under bright areas of the image as you still iluminate these pixels while the sensor data ist shifted out towards the top. I experienced this effect back at university with a LN-Cooled CCD for Spectroscopy. The stripes disapeared as soon as you used the shutter instead of disabling it in the open position (but fokussing at 100ms integration time and continuous readout with a focal plane shutter isn't much fun).
12
Reply
mikeselectricstuff
·
1 reply
@mikeselectricstuff
12 hours ago
I didn't think of that, but makes sense
2
Reply
@douro20
22 hours ago (edited)
The red LED reminds me of one from Roithner Lasertechnik. I have a Symbol 2D scanner which uses two very bright LEDs from that company, one red and one red-orange. The red-orange is behind a lens which focuses it into an extremely narrow beam.
1
Reply
@RicoElectrico
23 hours ago
PFG is Pulse Flush Gate according to the datasheet.
Reply
@dcallan812
23 hours ago
Very interesting. 2x
Reply
@littleboot_
1 day ago
Cool interesting device
Reply
@dav1dbone
1 day ago
I've stripped large projectors, looks similar, wonder if some of those castings are a magnesium alloy?
Reply
@kevywevvy8833
1 day ago
ironic that some of those Phlatlight modules are used in some of the cheapest disco lights.
1
Reply
1 reply
@bill6255
1 day ago
Great vid - gets right into subject in title, its packed with information, wraps up quickly. Should get a YT award! imho
3
Reply
@JAKOB1977
1 day ago (edited)
The whole sensor module incl. a 5 grand 50mpix sensor for 49 £.. highest bid atm
Though also a limited CCD sensor, but for the right buyer its a steal at these relative low sums.
Architecture Full Frame CCD (Square Pixels)
Total Number of Pixels 8304 (H) × 6220 (V) = 51.6 Mp
Number of Effective Pixels 8208 (H) × 6164 (V) = 50.5 Mp
Number of Active Pixels 8176 (H) × 6132 (V) = 50.1 Mp
Pixel Size 6.0 m (H) × 6.0 m (V)
Active Image Size 49.1 mm (H) × 36.8 mm (V)
61.3 mm (Diagonal),
645 1.1x Optical Format
Aspect Ratio 4:3
Horizontal Outputs 4
Saturation Signal 40.3 ke−
Output Sensitivity 31 V/e−
Quantum Efficiency
KAF−50100−CAA
KAF−50100−AAA
KAF−50100−ABA (with Lens)
22%, 22%, 16% (Peak R, G, B)
25%
62%
Read Noise (f = 18 MHz) 12.5 e−
Dark Signal (T = 60°C) 42 pA/cm2
Dark Current Doubling Temperature 5.7°C
Dynamic Range (f = 18 MHz) 70.2 dB
Estimated Linear Dynamic Range
(f = 18 MHz)
69.3 dB
Charge Transfer Efficiency
Horizontal
Vertical
0.999995
0.999999
Blooming Protection
(4 ms Exposure Time)
800X Saturation Exposure
Maximum Date Rate 18 MHz
Package Ceramic PGA
Cover Glass MAR Coated, 2 Sides or
Clear Glass
Features
• TRUESENSE Transparent Gate Electrode
for High Sensitivity
• Ultra-High Resolution
• Board Dynamic Range
• Low Noise Architecture
• Large Active Imaging Area
Applications
• Digitization
• Mapping/Aerial
• Photography
• Scientific
Thx for the tear down Mike, always a joy
Reply
@martinalooksatthings
1 day ago
15:49 that is some great bodging on of caps, they really didn't want to respin that PCB huh
8
Reply
@RhythmGamer
1 day ago
Was depressed today and then a new mike video dropped and now I’m genuinely happy to get my tear down fix
1
Reply
@dine9093
1 day ago (edited)
Did you transfrom into Mr Blobby for a moment there?
2
Reply
@NickNorton
1 day ago
Thanks Mike. Your videos are always interesting.
5
Reply
@KeritechElectronics
1 day ago
Heavy optics indeed... Spare no expense, cost no object. Splendid build quality. The CCD is a thing of beauty!
1
Reply
@YSoreil
1 day ago
The pricing on that sensor is about right, I looked in to these many years ago when they were still in production since it's the only large sensor you could actually buy. Really cool to see one in the wild.
2
Reply
@snik2pl
1 day ago
That leds look like from led projector
Reply
@vincei4252
1 day ago
TDI = Time Domain Integration ?
1
Reply
@wolpumba4099
1 day ago (edited)
Maybe the camera should not be illuminated during readout.
From the datasheet of the sensor (Onsemi): saturation 40300 electrons, read noise 12.5 electrons per pixel @ 18MHz (quite bad). quantum efficiency 62% (if it has micro lenses), frame rate 1 Hz. lateral overflow drain to prevent blooming protects against 800x (factor increases linearly with exposure time) saturation exposure (32e6 electrons per pixel at 4ms exposure time), microlens has +/- 20 degree acceptance angle
i guess it would be good for astrophotography
4
Reply
@txm100
1 day ago (edited)
Babe wake up a new mikeselectricstuff has dropped!
9
Reply
@vincei4252
1 day ago
That looks like a finger-lakes filter wheel, however, for astronomy they'd never use such a large stepper.
1
Reply
@MRooodddvvv
1 day ago
yaaaaay ! more overcomplicated optical stuff !
4
Reply
1 reply
@NoPegs
1 day ago
He lives!
11
Reply
1 reply
Transcript
0:00
so I've stripped all the bits of the
0:01
optical system so basically we've got
0:03
the uh the camera
0:05
itself which is mounted on this uh very
0:09
complex
0:10
adjustment thing which obviously to set
0:13
you the various tilt and uh alignment
0:15
stuff then there's two of these massive
0:18
lenses I've taken one of these apart I
0:20
think there's something like about eight
0:22
or nine Optical elements in here these
0:25
don't seem to do a great deal in terms
0:26
of electr magnification they're obiously
0:28
just about getting the image to where it
0:29
uh where it needs to be just so that
0:33
goes like that then this Optical block I
0:36
originally thought this was made of some
0:37
s crazy heavy material but it's just
0:39
really the sum of all these Optical bits
0:41
are just ridiculously heavy those lenses
0:43
are about 4 kilos each and then there's
0:45
this very heavy very solid um piece that
0:47
goes in the middle and this is so this
0:49
is the filter wheel assembly with a
0:51
hilariously oversized steper
0:53
motor driving this wheel with these very
0:57
large narrow band filters so we've got
1:00
various different shades of uh
1:03
filters there five Al together that
1:06
one's actually just showing up a silver
1:07
that's actually a a red but fairly low
1:10
transmission orangey red blue green
1:15
there's an excess cover on this side so
1:16
the filters can be accessed and changed
1:19
without taking anything else apart even
1:21
this is like ridiculous it's like solid
1:23
aluminium this is just basically a cover
1:25
the actual wavelengths of these are um
1:27
488 525 570 630 and 700 NM not sure what
1:32
the suffix on that perhaps that's the uh
1:34
the width of the spectral line say these
1:37
are very narrow band filters most of
1:39
them are you very little light through
1:41
so it's still very tight narrow band to
1:43
match the um fluoresence of the dies
1:45
they're using in the biochemical process
1:48
and obviously to reject the light that's
1:49
being fired at it from that Illuminator
1:51
box and then there's a there's a second
1:53
one of these lenses then the actual sort
1:55
of samples below that so uh very serious
1:58
amount of very uh chunky heavy Optics
2:01
okay let's take a look at this light
2:02
source made by company Lumen Dynamics
2:04
who are now part of
2:06
excelitas self-contained unit power
2:08
connector USB and this which one of the
2:11
Cable Bundle said was a TTL interface
2:14
USB wasn't used in uh the fluid
2:17
application output here and I think this
2:19
is an input for um light feedback I
2:21
don't if it's regulated or just a measur
2:23
measurement facility and the uh fiber
2:27
assembly
2:29
Square Inlet there and then there's two
2:32
outputs which have uh lens assemblies
2:35
and this small one which goes back into
2:37
that small Port just Loops out of here
2:40
straight back in So on this side we've
2:42
got the electronics which look pretty
2:44
straightforward we've got a bit of power
2:45
supply stuff over here and we've got
2:48
separate drivers for each wavelength now
2:50
interesting this is clearly been very
2:52
specifically made for this application
2:54
you I was half expecting like say some
2:56
generic drivers that could be used for a
2:58
number of different things but actually
3:00
literally specified the exact wavelength
3:02
on the PCB there is provision here for
3:04
385 NM which isn't populated but this is
3:07
clearly been designed very specifically
3:09
so these four drivers look the same but
3:10
then there's two higher power ones for
3:12
575 and
3:14
520 a slightly bigger heat sink on this
3:16
575 section there a p 24 which is
3:20
providing USB interface USB isolator the
3:23
USB interface just presents as a comport
3:26
I did have a quick look but I didn't
3:27
actually get anything sensible um I did
3:29
dump the Pi code out and there's a few
3:31
you a few sort of commands that you
3:32
could see in text but I didn't actually
3:34
manage to get it working properly I
3:36
found some software for related version
3:38
but it didn't seem to want to talk to it
3:39
but um I say that wasn't used for the
3:41
original application it might be quite
3:42
interesting to get try and get the Run
3:44
hours count out of it and the TTL
3:46
interface looks fairly straightforward
3:48
we've got positions for six opto
3:50
isolators but only five five are
3:52
installed so that corresponds with the
3:54
unused thing so I think this hopefully
3:56
should be as simple as just providing a
3:57
ttrl signal for each color to uh enable
4:00
it a big heat sink here which is there I
4:03
think there's like a big S of metal
4:04
plate through the middle of this that
4:05
all the leads are mounted on the other
4:07
side so this is heat sinking it with a
4:09
air flow from a uh just a fan in here
4:13
obviously don't have the air flow
4:14
anywhere near the Optics so conduction
4:17
cool through to this plate that's then
4:18
uh air cooled got some pots which are
4:21
presumably power
4:22
adjustments okay let's take a look at
4:24
the other side which is uh much more
4:27
interesting see we've got some uh very
4:31
uh neatly Twisted cable assemblies there
4:35
a bunch of leads so we've got one here
4:37
475 up here 430 NM 630 575 and 520
4:44
filters and dcro mirrors a quick way to
4:48
see what's white is if we just shine
4:49
some white light through
4:51
here not sure how it is is to see on the
4:54
camera but shining white light we do
4:55
actually get a bit of red a bit of blue
4:57
some yellow here so the obstacle path
5:00
575 it goes sort of here bounces off
5:03
this mirror and goes out the 520 goes
5:07
sort of down here across here and up
5:09
there 630 goes basically straight
5:13
through
5:15
430 goes across there down there along
5:17
there and the 475 goes down here and
5:20
left this is the light sensing thing
5:22
think here there's just a um I think
5:24
there a photo diode or other sensor
5:26
haven't actually taken that off and
5:28
everything's fixed down to this chunk of
5:31
aluminium which acts as the heat
5:32
spreader that then conducts the heat to
5:33
the back side for the heat
5:35
sink and the actual lead packages all
5:38
look fairly similar except for this one
5:41
on the 575 which looks quite a bit more
5:44
substantial big spay
5:46
Terminals and the interface for this
5:48
turned out to be extremely simple it's
5:50
literally a 5V TTL level to enable each
5:54
color doesn't seem to be any tensity
5:56
control but there are some additional
5:58
pins on that connector that weren't used
5:59
in the through time thing so maybe
6:01
there's some extra lines that control
6:02
that I couldn't find any data on this uh
6:05
unit and the um their current product
6:07
range is quite significantly different
6:09
so we've got the uh blue these
6:13
might may well be saturating the camera
6:16
so they might look a bit weird so that's
6:17
the 430
6:18
blue the 575
6:24
yellow uh
6:26
475 light blue
6:29
the uh 520
6:31
green and the uh 630 red now one
6:36
interesting thing I noticed for the
6:39
575 it's actually it's actually using a
6:42
white lead and then filtering it rather
6:44
than using all the other ones are using
6:46
leads which are the fundamental colors
6:47
but uh this is actually doing white and
6:50
it's a combination of this filter and
6:52
the dichroic mirrors that are turning to
6:55
Yellow if we take the filter out and a
6:57
lot of the a lot of the um blue content
7:00
is going this way the red is going
7:02
straight through these two mirrors so
7:05
this is clearly not reflecting much of
7:08
that so we end up with the yellow coming
7:10
out of uh out of there which is a fairly
7:14
light yellow color which you don't
7:16
really see from high intensity leads so
7:19
that's clearly why they've used the
7:20
white to uh do this power consumption of
7:23
the white is pretty high so going up to
7:25
about 2 and 1 half amps on that color
7:27
whereas most of the other colors are
7:28
only drawing half an amp or so at 24
7:30
volts the uh the green is up to about
7:32
1.2 but say this thing is uh much
7:35
brighter and if you actually run all the
7:38
colors at the same time you get a fairly
7:41
reasonable um looking white coming out
7:43
of it and one thing you might just be
7:45
out to notice is there is some sort
7:46
color banding around here that's not
7:49
getting uh everything s completely
7:51
concentric and I think that's where this
7:53
fiber optic thing comes
7:58
in I'll
8:00
get a couple of Fairly accurately shaped
8:04
very sort of uniform color and looking
8:06
at What's um inside here we've basically
8:09
just got this Square Rod so this is
8:12
clearly yeah the lights just bouncing
8:13
off all the all the various sides to um
8:16
get a nice uniform illumination uh this
8:19
back bit looks like it's all potted so
8:21
nothing I really do to get in there I
8:24
think this is fiber so I have come
8:26
across um cables like this which are
8:27
liquid fill but just looking through the
8:30
end of this it's probably a bit hard to
8:31
see it does look like there fiber ends
8:34
going going on there and so there's this
8:36
feedback thing which is just obviously
8:39
compensating for the any light losses
8:41
through here to get an accurate
8:43
representation of uh the light that's
8:45
been launched out of these two
8:47
fibers and you see uh
8:49
these have got this sort of trapezium
8:54
shape light guides again it's like a
8:56
sort of acrylic or glass light guide
9:00
guess projected just to make the right
9:03
rectangular
9:04
shape and look at this Center assembly
9:07
um the light output doesn't uh change
9:10
whether you feed this in or not so it's
9:11
clear not doing any internal Clos Loop
9:14
control obviously there may well be some
9:16
facility for it to do that but it's not
9:17
being used in this
9:19
application and so this output just
9:21
produces a voltage on the uh outle
9:24
connector proportional to the amount of
9:26
light that's present so there's a little
9:28
diffuser in the back there
9:30
and then there's just some kind of uh
9:33
Optical sensor looks like a
9:35
chip looking at the lead it's a very
9:37
small package on the PCB with this lens
9:40
assembly over the top and these look
9:43
like they're actually on a copper
9:44
Metalized PCB for maximum thermal
9:47
performance and yeah it's a very small
9:49
package looks like it's a ceramic
9:51
package and there's a thermister there
9:53
for temperature monitoring this is the
9:56
475 blue one this is the 520 need to
9:59
Green which is uh rather different OB
10:02
it's a much bigger D with lots of bond
10:04
wise but also this looks like it's using
10:05
a phosphor if I shine a blue light at it
10:08
lights up green so this is actually a
10:10
phosphor conversion green lead which
10:12
I've I've come across before they want
10:15
that specific wavelength so they may be
10:17
easier to tune a phosphor than tune the
10:20
um semiconductor material to get the uh
10:23
right right wavelength from the lead
10:24
directly uh red 630 similar size to the
10:28
blue one or does seem to have a uh a
10:31
lens on top of it there is a sort of red
10:33
coloring to
10:35
the die but that doesn't appear to be
10:38
fluorescent as far as I can
10:39
tell and the white one again a little
10:41
bit different sort of much higher
10:43
current
10:46
connectors a makeer name on that
10:48
connector flot light not sure if that's
10:52
the connector or the lead
10:54
itself and obviously with the phosphor
10:56
and I'd imagine that phosphor may well
10:58
be tuned to get the maximum to the uh 5
11:01
cenm and actually this white one looks
11:04
like a St fairly standard product I just
11:06
found it in Mouse made by luminous
11:09
devices in fact actually I think all
11:11
these are based on various luminous
11:13
devices modules and they're you take
11:17
looks like they taking the nearest
11:18
wavelength and then just using these
11:19
filters to clean it up to get a precise
11:22
uh spectral line out of it so quite a
11:25
nice neat and um extreme
11:30
bright light source uh sure I've got any
11:33
particular use for it so I think this
11:35
might end up on
11:36
eBay but uh very pretty to look out and
11:40
without the uh risk of burning your eyes
11:43
out like you do with lasers so I thought
11:45
it would be interesting to try and
11:46
figure out the runtime of this things
11:48
like this we usually keep some sort
11:49
record of runtime cuz leads degrade over
11:51
time I couldn't get any software to work
11:52
through the USB face but then had a
11:54
thought probably going to be writing the
11:55
runtime periodically to the e s prom so
11:58
I just just scope up that and noticed it
12:00
was doing right every 5 minutes so I
12:02
just ran it for a while periodically
12:04
reading the E squ I just held the pick
12:05
in in reset and um put clip over to read
12:07
the square prom and found it was writing
12:10
one location per color every 5 minutes
12:12
so if one color was on it would write
12:14
that location every 5 minutes and just
12:16
increment it by one so after doing a few
12:18
tests with different colors of different
12:19
time periods it looked extremely
12:21
straightforward it's like a four bite
12:22
count for each color looking at the
12:24
original data that was in it all the
12:26
colors apart from Green were reading
12:28
zero and the green was reading four
12:30
indicating a total 20 minutes run time
12:32
ever if it was turned on run for a short
12:34
time then turned off that might not have
12:36
been counted but even so indicates this
12:37
thing wasn't used a great deal the whole
12:40
s process of doing a run can be several
12:42
hours but it'll only be doing probably
12:43
the Imaging at the end of that so you
12:46
wouldn't expect to be running for a long
12:47
time but say a single color for 20
12:50
minutes over its whole lifetime does
12:52
seem a little bit on the low side okay
12:55
let's look at the camera un fortunately
12:57
I managed to not record any sound when I
12:58
did this it's also a couple of months
13:00
ago so there's going to be a few details
13:02
that I've forgotten so I'm just going to
13:04
dub this over the original footage so um
13:07
take the lid off see this massive great
13:10
heat sink so this is a pel cool camera
13:12
we've got this blower fan producing a
13:14
fair amount of air flow through
13:16
it the connector here there's the ccds
13:19
mounted on the board on the
13:24
right this unplugs so we've got a bit of
13:27
power supply stuff on here
13:29
USB interface I think that's the Cyprus
13:32
microcontroller High speeded USB
13:34
interface there's a zyink spon fpga some
13:40
RAM and there's a couple of ATD
13:42
converters can't quite read what those
13:45
those are but anal
13:47
devices um little bit of bodgery around
13:51
here extra decoupling obviously they
13:53
have having some noise issues this is
13:55
around the ram chip quite a lot of extra
13:57
capacitors been added there
13:59
uh there's a couple of amplifiers prior
14:01
to the HD converter buffers or Andor
14:05
amplifiers taking the CCD
14:08
signal um bit more power spy stuff here
14:11
this is probably all to do with
14:12
generating the various CCD bias voltages
14:14
they uh need quite a lot of exotic
14:18
voltages next board down is just a
14:20
shield and an interconnect
14:24
boardly shielding the power supply stuff
14:26
from some the more sensitive an log
14:28
stuff
14:31
and this is the bottom board which is
14:32
just all power supply
14:34
stuff as you can see tons of capacitors
14:37
or Transformer in
14:42
there and this is the CCD which is a uh
14:47
very impressive thing this is a kf50 100
14:50
originally by true sense then codec
14:53
there ON
14:54
Semiconductor it's 50 megapixels uh the
14:58
only price I could find was this one
15:00
5,000 bucks and the architecture you can
15:03
see there actually two separate halves
15:04
which explains the Dual AZ converters
15:06
and two amplifiers it's literally split
15:08
down the middle and duplicated so it's
15:10
outputting two streams in parallel just
15:13
to keep the bandwidth sensible and it's
15:15
got this amazing um diffraction effects
15:18
it's got micro lenses over the pixel so
15:20
there's there's a bit more Optics going
15:22
on than on a normal
15:25
sensor few more bodges on the CCD board
15:28
including this wire which isn't really
15:29
tacked down very well which is a bit uh
15:32
bit of a mess quite a few bits around
15:34
this board where they've uh tacked
15:36
various bits on which is not super
15:38
impressive looks like CCD drivers on the
15:40
left with those 3 ohm um damping
15:43
resistors on the
15:47
output get a few more little bodges
15:50
around here some of
15:52
the and there's this separator the
15:54
silica gel to keep the moisture down but
15:56
there's this separator that actually
15:58
appears to be cut from piece of
15:59
antistatic
16:04
bag and this sort of thermal block on
16:06
top of this stack of three pel Cola
16:12
modules so as with any Stacks they get
16:16
um larger as they go back towards the
16:18
heat sink because each P's got to not
16:20
only take the heat from the previous but
16:21
also the waste heat which is quite
16:27
significant you see a little temperature
16:29
sensor here that copper block which
16:32
makes contact with the back of the
16:37
CCD and this's the back of the
16:40
pelas this then contacts the heat sink
16:44
on the uh rear there a few thermal pads
16:46
as well for some of the other power
16:47
components on this
16:51
PCB okay I've connected this uh camera
16:54
up I found some drivers on the disc that
16:56
seem to work under Windows 7 couldn't
16:58
get to install under Windows 11 though
17:01
um in the absence of any sort of lens or
17:03
being bothered to the proper amount I've
17:04
just put some f over it and put a little
17:06
pin in there to make a pinhole lens and
17:08
software gives a few options I'm not
17:11
entirely sure what all these are there's
17:12
obviously a clock frequency 22 MHz low
17:15
gain and with PFG no idea what that is
17:19
something something game programmable
17:20
Something game perhaps ver exposure
17:23
types I think focus is just like a
17:25
continuous grab until you tell it to
17:27
stop not entirely sure all these options
17:30
are obviously exposure time uh triggers
17:33
there ex external hardware trigger inut
17:35
you just trigger using a um thing on
17:37
screen so the resolution is 8176 by
17:40
6132 and you can actually bin those
17:42
where you combine multiple pixels to get
17:46
increased gain at the expense of lower
17:48
resolution down this is a 10sec exposure
17:51
obviously of the pin hole it's very uh
17:53
intensitive so we just stand still now
17:56
downloading it there's the uh exposure
17:59
so when it's
18:01
um there's a little status thing down
18:03
here so that tells you the um exposure
18:07
[Applause]
18:09
time it's this is just it
18:15
downloading um it is quite I'm seeing
18:18
quite a lot like smearing I think that I
18:20
don't know whether that's just due to
18:21
pixels overloading or something else I
18:24
mean yeah it's not it's not um out of
18:26
the question that there's something not
18:27
totally right about this camera
18:28
certainly was bodge wise on there um I
18:31
don't I'd imagine a camera like this
18:32
it's got a fairly narrow range of
18:34
intensities that it's happy with I'm not
18:36
going to spend a great deal of time on
18:38
this if you're interested in this camera
18:40
maybe for astronomy or something and
18:42
happy to sort of take the risk of it may
18:44
not be uh perfect I'll um I think I'll
18:47
stick this on eBay along with the
18:48
Illuminator I'll put a link down in the
18:50
description to the listing take your
18:52
chances to grab a bargain so for example
18:54
here we see this vertical streaking so
18:56
I'm not sure how normal that is this is
18:58
on fairly bright scene looking out the
19:02
window if I cut the exposure time down
19:04
on that it's now 1 second
19:07
exposure again most of the image
19:09
disappears again this is looks like it's
19:11
possibly over still overloading here go
19:14
that go down to say say quarter a
19:16
second so again I think there might be
19:19
some Auto gain control going on here um
19:21
this is with the PFG option let's try
19:23
turning that off and see what
19:25
happens so I'm not sure this is actually
19:27
more streaking or which just it's
19:29
cranked up the gain all the dis display
19:31
gray scale to show what um you know the
19:33
range of things that it's captured
19:36
there's one of one of 12 things in the
19:38
software there's um you can see of you
19:40
can't seem to read out the temperature
19:42
of the pelta cooler but you can set the
19:44
temperature and if you said it's a
19:46
different temperature you see the power
19:48
consumption jump up running the cooler
19:50
to get the temperature you requested but
19:52
I can't see anything anywhere that tells
19:54
you whether the cool is at the at the
19:56
temperature other than the power
19:57
consumption going down and there's no
19:59
temperature read out
20:03
here and just some yeah this is just
20:05
sort of very basic software I'm sure
20:07
there's like an API for more
20:09
sophisticated
20:10
applications but so if you know anything
20:12
more about these cameras please um stick
20:14
in the
20:15
comments um incidentally when I was
20:18
editing I didn't notice there was a bent
20:19
pin on the um CCD but I did fix that
20:22
before doing these tests and also
20:24
reactivated the um silica gel desicant
20:26
cuz I noticed it was uh I was getting
20:28
bit of condensation on the window but um
20:31
yeah so a couple of uh interesting but
20:34
maybe not particularly uh useful pieces
20:37
of Kit except for someone that's got a
20:38
very specific use so um I'll stick a
20:42
I'll stick these on eBay put a link in
20:44
the description and say hopefully
20:45
someone could actually make some uh good
20:47
use of these things
Example Output:
**Abstract:**
This video presents Part 2 of a teardown focusing on the optical components of a Fluidigm Polaris biotechnology instrument, specifically the multi-wavelength illuminator and the high-resolution CCD camera.
The Lumen Dynamics illuminator unit is examined in detail, revealing its construction using multiple high-power LEDs (430nm, 475nm, 520nm, 575nm, 630nm) combined via dichroic mirrors and filters. A square fiber optic rod is used to homogenize the light. A notable finding is the use of a phosphor-converted white LED filtered to achieve the 575nm output. The unit features simple TTL activation for each color, conduction cooling, and internal homogenization optics. Analysis of its EEPROM suggests extremely low operational runtime.
The camera module teardown showcases a 50 Megapixel ON Semiconductor KAF-50100 CCD sensor with micro-lenses, cooled by a multi-stage Peltier stack. The control electronics include an FPGA and a USB interface. Significant post-manufacturing modifications ("bodges") are observed on the camera's circuit boards. Basic functional testing using vendor software and a pinhole lens confirms image capture but reveals prominent vertical streaking artifacts, the cause of which remains uncertain (potential overload, readout artifact, or fault).
**Exploring the Fluidigm Polaris: A Detailed Look at its High-End Optics and Camera System**
* **0:00 High-End Optics:** The system utilizes heavy, high-quality lenses and mirrors for precise imaging, weighing around 4 kilos each.
* **0:49 Narrow Band Filters:** A filter wheel with five narrow band filters (488, 525, 570, 630, and 700 nm) ensures accurate fluorescence detection and rejection of excitation light.
* **2:01 Customizable Illumination:** The Lumen Dynamics light source offers five individually controllable LED wavelengths (430, 475, 520, 575, 630 nm) with varying power outputs. The 575nm yellow LED is uniquely achieved using a white LED with filtering.
* **3:45 TTL Control:** The light source is controlled via a simple TTL interface, enabling easy on/off switching for each LED color.
* **12:55 Sophisticated Camera:** The system includes a 50-megapixel Kodak KAI-50100 CCD camera with a Peltier cooling system for reduced noise.
* **14:54 High-Speed Data Transfer:** The camera features dual analog-to-digital converters to manage the high data throughput of the 50-megapixel sensor, which is effectively two 25-megapixel sensors operating in parallel.
* **18:11 Possible Issues:** The video creator noted some potential issues with the camera, including image smearing.
* **18:11 Limited Dynamic Range:** The camera's sensor has a limited dynamic range, making it potentially challenging to capture scenes with a wide range of brightness levels.
* **11:45 Low Runtime:** Internal data suggests the system has seen minimal usage, with only 20 minutes of recorded runtime for the green LED.
* **20:38 Availability on eBay:** Both the illuminator and camera are expected to be listed for sale on eBay.
Here is the real transcript. What would be a good group of people to review this topic? Please summarize provide a summary like they would:
TERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 21, NO. 5, MAY 1999
Direct Least Square Fitting of Ellipses
Andrew Fitzgibbon, Maurizio Pilu, and Robert B. Fisher
Abstract—This work presents a new efficient method for fitting ellipses
to scattered data. Previous algorithms either fitted general conics or
were computationally expensive. By minimizing the algebraic distance
subject to the constraint 4ac - b
2
= 1, the new method incorporates the
ellipticity constraint into the normalization factor. The proposed method
combines several advantages: It is ellipse-specific, so that even bad
data will always return an ellipse. It can be solved naturally by a
generalized eigensystem. It is extremely robust, efficient, and easy to
implement.
Index Terms—Algebraic models, ellipse fitting, least squares fitting,
constrained minimization, generalized eigenvalue problem.
———————— F ————————
1 INTRODUCTION
THE fitting of primitive models to image data is a basic task in
pattern recognition and computer vision, allowing reduction and
simplification of the data to the benefit of higher level processing
stages. One of the most commonly used models is the ellipse
which, being the perspective projection of the circle, is of great
importance for many industrial applications. Despite its importance, however, there has been until now no computationally efficient ellipse-specific fitting algorithm [14], [5].
In this paper, we introduce a new method for fitting ellipses,
rather than general conics, to segmented data. As we shall see
in the next section, current methods are either computationally
expensive iterative approaches, or perform ellipse fitting by
least-squares fitting to a general conic and rejecting nonelliptical fits. These latter methods are cheap and perform well
if the data belong to a precisely elliptical arc with little occlusion but suffer from the major shortcoming that under less
ideal conditions—nonstrictly elliptical data, moderate occlusion or noise—they often yield unbounded fits to hyperbolae.
In a situation where ellipses are specifically desired, such fits
must be rejected as useless. A number of iterative refinement
procedures [16], [8], [12] alleviate this problem, but do not
eliminate it. In addition, these techniques often increase the
computational burden unacceptably.
This paper introduces a new fitting method that combines the
following advantages:
1) ellipse-specificity, providing useful results under all noise
and occlusion conditions;
2) invariance to affine transformation of the data;
3) high robustness to noise; and
4) high computational efficiency.
After a description of relevant previous ellipse fitting methods,
in Section 3 we describe the method and provide a theoretical
analysis of the uniqueness of the elliptical solution. Section 4 contains experimental results, notably to highlight behavior with
0162-8828/99/$10.00 © 1999 IEEE
²²²²²²²²²²²²²²²²
A. Fitzgibbon is with the Department of Engineering Science, University of
Oxford, 19 Parks Road, Oxford, OX1 3BJ, England.
E-mail: awf@robots.ox.ac.uk.
M. Pilu is with Hewlett-Packard Research Laboratories, Filton Road, Stoke
Gifford, Bristol, BS12 6QZ, England. E-mail: mp@hplb.hpl.hp.com.
R.B. Fisher is with the Division of Informatics, University of Edinburgh, 5 Forrest
Hill, Edinburgh, EH1 2QL, United Kingdom.
E-mail: rbf@dai.edinburgh.ac.uk.
Manuscript received 4 Jan. 1999. Recommended for acceptance by R. Chin.
For information on obtaining reprints of this article, please send e-mail to:
tpami@computer.org, and reference IEEECS Log Number 107704.
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 21, NO. 5, MAY 1999 477
nonelliptical data, low-eccentricity bias, and noise resilience. We
conclude by presenting some possible extensions.
2 PREVIOUS METHODS AND THEIR LIMITATIONS
The literature on ellipse fitting divides into two broad techniques:
clustering (such as Hough-based methods [9], [19]) and leastsquares fitting.
Least-squares techniques center on finding the set of parameters that minimize some distance measure between the data points
and the ellipse. In this section, we briefly present the most cited
works in ellipse fitting and its closely related problem, conic fitting. It will be shown that the direct specific least-square fitting of
ellipses has, up to now, not been solved.
Before reviewing the literature on general conic fitting, we
will introduce a statement of the problem that allows us to unify
several approaches under the umbrella of constrained least
squares. Let us represent a general conic by an implicit second
order polynomial:
F(a, x) = a ¼ x = ax2
+ bxy + cy2
+ dx + ey + f = 0, (1)
where a = [a b c d e f]
T
and x = [x
2
xy y
2
x y 1]T
. F(a; xi
) is called the
“algebraic distance” of a point (x, y) to the conic F(a; x) = 0. The
fitting of a general conic may be approached by minimizing the
sum of squared algebraic distances
'A i
i
N
1 6 a x = F2 7 =
Ê 2
1
(2)
of the curve to the N data points xi
[7]. In order to avoid the trivial
solution a = 06, and recognizing that any multiple of a solution a
represents the same conic, the parameter vector a is constrained in
some way. Many of the published algorithms differ only in the
form of constraint applied to the parameters. For instance, many
authors suggest iai
2
= 1, Rosin [14] and Gander [5] impose a + c = 1
while Rosin also investigates f = 1 [14]. Taubin’s approximate
square distance [17] may also be viewed as the quadratic constraint iNai
2
= 1 where N is the Jacobian [¶F(a; x1) ¡ ¶F(a; xN)]T
.
Note that these constraints are all either linear, of the form c ¼ a
= 1 or quadratic, constraining aT
Ca = 1 where C is a 6 6 constraint
matrix.
In a seminal work, Bookstein [1] showed that if a quadratic
constraint is set on the parameters (e.g., to avoid the trivial solution a = 06) the minimization (2) can be solved by considering rankdeficient generalized eigenvalue system:
DT
Da = lCa, (3)
where D = [x1 x2 L xn]
T
is called the design matrix and C is the matrix that expresses the constraint.
A simple constraint is iai = 1 but Bookstein used the algebraic
invariant constraint a bc 2 1
2
2 2 + += 1; Sampson [16] presented an
iterative improvement to Bookstein method that replaces the algebraic distance (2) with a better approximation to the geometric
distance, which was adapted by Taubin [17] to turn the problem
again into a generalized eigensystem.
Despite the amount of work, direct specific ellipse fitting, however, was left unsolved. If ellipse fitting was needed, one had to
rely either on generic conic fitting or on iterative methods to
“nudge” the estimation towards ellipticity. For instance, Porrill
[12], Ellis et al. [2], and Rosin [14] use conic fitting to initialize a
Kalman filter that iteratively minimizes some error metric in order
to gather new image evidence and to reject nonellipse fits by testing the discriminant b
2
- 4ac < 0 at each iteration. Another iterative
algorithm is that of Haralick [7, Section 11.10.7], where the coefficients {a, b, c} are transformed into {p
2
, 2pq, q
2
+ r
2
} so as to keep the
conic discriminant always negative. A nonlinear minimization of
the algebraic error over the space {p, q, r, d, e, f} is performed.
In this journal, Rosin [15] reiterated this problem by stating that
ellipse-specific fitting is essentially a nonlinear problem and iterative methods must always be employed for this purpose. In the
following section, we show that this is no longer true.
3 DIRECT ELLIPSE-SPECIFIC FITTING
In order to fit ellipses specifically while retaining the efficiency
of solution of the linear least-squares problem (2), we would like
to constrain the parameter vector a so that the conic that it represents is forced to be an ellipse. The appropriate constraint is well
known, namely, that the discriminant b
2
- 4ac be negative. However, this constrained problem is difficult to solve in general as
the Kuhn-Tucker conditions [13] do not guarantee a solution. In
fact, we have not been able to locate any reference regarding the
minimization of a quadratic form subject to such a nonconvex
inequality.
Although the imposition of this inequality constraint is difficult
in general, in this case we have the freedom to arbitrarily scale the
parameters so we may simply incorporate the scaling into the constraint and impose the equality constraint 4ac - b
2
= 1 [4]. This is a
quadratic constraint which may be expressed in the matrix form
a
T
Ca = 1 as
a a T
0 02000
0 10000
2 00000
0 00000
0 00000
0 00000
1
-
!
"
$
#
#
#
#
#
= . (4)
Now, following Bookstein [1], the constrained ellipse fitting problem reduces to
minimizing E = iDai
2
subject to the constraint a
T
Ca = 1 (5)
where the design matrix D is defined as in the previous section.
Introducing the Lagrange multiplier l and differentiating, we
arrive at the system of simultaneous equations1
2 20
1
D Da Ca
a Ca
T
T
- =
=
l
(6)
This may be rewritten as the system
Sa = lCa (7)
a
T
Ca = 1 (8)
where S is the scatter matrix DT
D. This system is readily solved by
considering the generalized eigenvectors of (7). If (li
, ui
) solves (7),
then so does (li
, mui
) for any m and from (8) we can find the value
of mi
as mi i
T
i
2
u Cu = 1 , giving
mi
i
T
i i
T
i
= = 1 1
u Cu u Su . (9)
Finally, setting a u $i ii = m solves (6).
We note that the solution of the eigensystem (7) gives six eigenvalue-eigenvector pairs (li
, ui
). Each of these pairs gives rise to a
local minimum if the term under the square root of (9) is positive.
In general, S is positive definite, so the denominator u Su i
T
i is
positive for all ui
. Therefore, the square root exists if li > 0, so any
solutions to (6) must have positive generalized eigenvalues.
Now we show that the minimization of iDai
2
subject to 4ac - b
2
= 1 yields exactly one solution, which corresponds, by virtue of the
constraint, to an ellipse [11]. For the demonstration, we will require Lemma 1.
1. Note that the method of Lagrange multipliers is not valid when the
gradient of the constraint function becomes zero. In (5), this means Ca = 0,
but then a
T
Ca = 0, so the constraint is violated and there is no solution.
478 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 21, NO. 5, MAY 1999
LEMMA 1. The signs of the generalized eigenvalues of Su = lCu,
where S ³ Rnn is positive definite and C ³ Rnn is symmetric, are
the same as those of the constraint matrix C, up to permutation of
the indices.
PROOF. Let us define the spectrum s(S) as the set of eigenvalues of
S and, analogously, s(S, C) the set of generalized eigenvalues of
(7). Let the inertia i(S) be defined as the set of signs of s(S), and
let i(S, C) analogously be the inertia of s(S, C). Then, the lemma
is equivalent to proving that i(S, C) = i(C). As S is positive definite, it may be decomposed as Q2
for symmetric Q, allowing us
to write Su = lCu as Q2
u = lCu. Now, substituting v = Qu and
premultiplying by Q-1
gives v = lQ-1
CQ-1
v so that s(S, C) =
s(Q-1
CQ-1
)
-1
and thus i(S, C) = i(Q-1
CQ-1
). From Sylvester’s
Law of Inertia [18], we have that for any symmetric and nonsingular X, i(S) = i(XT
SX). Therefore, substituting X = XT
= Q-1
,
we have i(C) = i(Q-1
CQ-1
) = i(S, C). o
We can now state Theorem 1.
THEOREM 1. The solution of the conic fitting problem (5) subject to the
constraint (4) admits exactly one elliptical solution corresponding to
the single positive generalized eigenvalue of (7).
PROOF. Since the eigenvalues of C are {-2, -1, 2, 0, 0, 0}, from
Lemma 1 we have that (7) has exactly one positive eigenvalue li
> 0, giving the unique solution a u $ = mi i to (6). As DT
D is positive semidefinite, the constrained problem has a minimum,
which must satisfy (6), and we conclude that a$ solves the constrained problem. o
This unique solution has also some desirable properties in ellipse fitting:
low eccentricity bias: An eigenvector of the eigensystem (7)
is a local minimizer of the Rayleigh quotient a Sa
a Ca
T
T . In this case,
the implicit normalization by b
2
- 4ac turns singular for b
2
- 4ac = 0, which is a parabola. Since the minimization
tends to “pull” the solution away from singularities [14],
the unique elliptical solution tends to be biased towards
low eccentricity.
affine invariance: Let us represent the conic as x
Á
Ax + xÁ
b +
c = 0. Under an affine transform H the leading form becomes
A = HÁ
AH, so that |A| = |H|2
|A|. Being the Rayleigh
quotient that we minimize a Sa T
A , the new error measure is a
Fig. 1. (a) Fits to hand-input data to illustrate the ellipse specificity of the method. (b) Experiments with noisy parabolic data (after Sampson). Encoding is BOOK: dotted; GAND: dashed; TAUB: dash-dot; B2AC: solid.
Fig. 2. (a) Variation of center position for increasing noise level when fitting to a whole ellipse. (b) Fits to arc of ellipse with increasing noise level.
Notice how B2AC presents a much more graceful degradation with respect to noise.
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 21, NO. 5, MAY 1999 479
scalar multiple of the original one and thus the new minimizer is transformed by H, which proved the affine invariance of the method.
4 EXPERIMENTAL RESULTS
This section describes some experiments that illustrate the interesting features of the new method and its noise performance compared to some of the least-squares fitting method reviewed in Section 2. In this short paper, we are not able to present a large body
of results—which can be found in abundance in [3]—so we limited
ourselves to those that are the most representative.
All experiments were conducted using the Matlab system [10].
Eigensystems are solved using the underlying EISPACK routines.
We shall use the following abbreviations:
LIN = linear method;
BOOK = Bookstein method [1];
TAUB = Taubin method [17];
GAND = Gander method [5];
BIAS = Kanatani bias-corrector method [8]; and finally
B2AC = our new method.
4.1 Ellipse-Specificity
Despite the theoretical proof of the algorithm’s ellipse specificity, it is
instructive to observe its performance on some real data, of which Fig.
1a provides examples with hand-drawn datasets. The results of the
method are superimposed on those of Bookstein and Gander. Dataset
A is almost elliptical and indistinguishable fits were produced. The
other sets exhibit varying degrees of nonellipticity and illustrate the
potential of the method for coarse bounding of generic 2D data.
4.2 Low-Eccentricity Bias
Fig. 1b shows three experiments designed after Sampson [16] (following [6]) and basically consists of the same parabolic data but
with different realizations of added isotropic Gaussian noise (s =
7% of data spread). Sampson’s iterative fit produced an ellipse
with low eccentricity that was qualitatively similar to the one produced by our direct method (solid lines) but the total cost of our
method is the same as that of acquiring his initial estimate. As
anticipated in the previous section, the low eccentricity bias of our
method is most evident in Fig. 1b when compared to Bookstein’s,
Taubin’s, and Gander’s results. It must be again remarked that this
is not surprising, because those methods are not ellipse-specific,
whereas ours is.
4.3 Noise Sensitivity
In this section, we describe some experiments concerning the noise
performance of our method compared to others.
The first experiment is concerned with the stability of the estimated ellipse center with increasing noise levels. We consider a
whole ellipse centered at the origin of semi-axis 1 and 0.5 and rotated by 40 degrees. The sampled ellipse was corrupted with noise
(from 2-3
to 23
) for 100 runs and the distance between the true ellipse center and the center of the conic returned by the fitting algorithm was recorded. Returned hyperbolae were included for the
other algorithms. Fig. 2a shows the 90th percentile error in the
centers as a function of noise level. At low noise levels (s < 0.5), all
algorithms can be seen to perform similarly, while at high levels,
only the new (B2AC) algorithm degrades gracefully.
The good performance of the presented method is more evident
when the data is occluded. In the second experiment, shown in
Fig. 2b, increasing level of isotropic Gaussian was added to points
on a given elliptical arc. The standard deviation of the noise varies
from 3% in the leftmost column to 20% of data spread in the
rightmost column; the noise has been set to a relatively high level
because the performance of the three algorithms is substantially
the same at low noise level of precise elliptical data. The top row
shows the results for the method proposed here. Although, as expected, the fitted ellipses shrink with increasing levels of noise [8]
(in the limit, the elliptical arc will look like a noisy line), it can be
noticed that the ellipse dimension decreases smoothly with the
noise level: This is an indication of well-behaved fitting. This
shrinking phenomenon is evident also with the other methods but
presents itself more erratically. Many more quantitative experiments on performance with occluded data can be found in [3].
The last experiment that we show here is perhaps the most interesting (although we have not seen it in related papers) and is
concerned with assessing stability to different realizations of noise
with the same variance. It is very desirable that an algorithm’s performance be affected only by the noise level, and not by a particular realization of the noise. Fig. 3 shows five different runs for s =
0.1, and the results of our method, Gander’s method, and Taubin’s
method are given. This and similar experiments (see [11], [3]) show
that our method has a greater stability to noise than the other
Fig. 4. Simple six-line Matlab implementation of the ellipse fitting method. methods.
Fig. 3. Stability experiments for different runs with same noise variance (10% of data spread). The ellipse-specific method shows a remarkable stability.
480 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 21, NO. 5, MAY 1999
5 CONCLUSION
In this paper, we have presented a least squares fitting method
which is specific to ellipses and direct at the same time. Previous
methods were either not ellipse-specific or were iterative.
We have theoretically demonstrated that our method uniquely
yields elliptical solutions that, under the normalization 4ac - b
2
= 1,
minimize the sum of squared algebraic distances from the points
to the ellipse.
Experimental results illustrate the advantages conferred by the
ellipse-specificity in terms of occlusion and noise sensitivity. The
stability properties widen the scope of application of the algorithm
from ellipse fitting to cases where the data are not strictly elliptical
but need to be minimally represented by an elliptical “blob.”
In our view, the method presented here offers the best trade-off
between speed and accuracy for ellipse fitting, and its uniqueness
property makes it also extremely robust to noise and usable in
many applications, especially in industrial vision. In cases where
more accurate results are required, this algorithm provides an excellent initial estimate.
Its simplicity is demonstrated by the inclusion in Fig. 4 of a complete six-line implementation in Matlab. (An interactive Java demonstration is available at http://vision.dai.ed.ac.uk/maurizp/ElliFitDemo/demo.html.)
Future work includes the incorporation of the algorithm into a
bias-correction algorithm based on that of Kanatani [8]. We note
also that the algorithm can be trivially converted to a hyperbolaspecific fitter, and a variation may be used to fit parabolae.
REFERENCES
[1] F.L. Bookstein, “Fitting Conic Sections to Scattered Data,” Computer Graphics and Image Processing, no. 9, pp. 56-71, 1979.
[2] T. Ellis, A. Abbood, and B. Brillault, “Ellipse Detection and
Matching With Uncertainty,” Image and Vision Computing, vol. 10,
no. 2, pp. 271-276, 1992.
[3] A.W. Fitzgibbon, “Stable Segmentation of 2D Curves,” PhD thesis, Dept. of Artificial Intelligence, Univ. of Edinburgh, 1998.
[4] A.W. Fitzgibbon and R.B. Fisher, “A Buyer’s Guide to Conic Fitting,” Proc. British Machine Vision Conf., Birmingham, England,
1995.
[5] W. Gander, G.H. Golub, and R. Strebel, “Least-Square Fitting of
Circles and Ellipses,” BIT, no. 43, pp. 558-578, 1994.
[6] R. Gnanadesikan, Methods for Statistical Data Analysis of Multivariate Observations. New York: Wiley, 1977.
[7] R. Haralick and L. Shapiro, Computer and Robot Vision. Reading,
Mass.: Addison-Wesley, 1992.
[8] K. Kanatani, “Statistical Bias of Conic Fitting and Renormalization,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16,
no. 3, pp. 320-326, 1994.
[9] V.F. Leavers, Shape Detection in Computer Vision Using the Hough
Transform. New York: Springer-Verlag, 1992.
[10] The MathWorks. The Matlab Package.
[11] M. Pilu, “Part-Based Grouping and Recogntion: A Model-Guided
Approach,” Dept. Artificial Intelligence, Univ. of Edinburgh, PhD
thesis, Aug. 1996.
[12] J. Porrill, “Fitting Ellipses and Predicting Confidence Envelopes
Using a Bias Corrected Kalman Filter,” Image and Vision Computing, vol. 8, no. 1, pp. 37-41, Feb. 1990.
[13] S.S. Rao, Optimization: Theory and Applications, 2nd ed. New York:
Wiley Estern, 1984.
[14] P.L. Rosin, “A Note on the Least Squares Fitting of Ellipses,” Pattern Recognition Letters, no. 14, pp. 799-808, Oct. 1993.
[15] P.L. Rosin and G.A. West, “Nonparametric Segmentation of
Curves Into Various Representations,” IEEE Trans. Pattern Analysis
and Machine Intelligence, vol. 17, no. 12, pp. 1,140-1,153, Dec. 1995.
[16] P.D. Sampson, “Fitting Conic Sections to Very Scattered Data: An
Iterative Refinement of the Bookstein Algorithm,” Computer
Graphics and Image Processing, no. 18, pp. 97-108, 1982.
[17] G. Taubin, “Estimation of Planar Curves, Surfaces and NonPlanar Space Curves Defined by Implicit Equations, With Applications to Edge and Range Image Segmentation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 11, pp. 1,115-
1,138, Nov. 1991.
[18] J.H. Wilkinson, The Algebraic Eigenvalue Problem. Oxford, England:
Clarendon Press, 1965.
[19] H.K. Yuen, J. Illingworth, and J. Kittler, “Shape Using Volumetric
Primitives,” Image and Vision Computing, vol. 1, no. 7, pp. 31-37,
1989.