From BMVA summer school 2014

1. Low-level vision tutorial 1
© E.R. Davies, 2014
Low-level vision – a tutorial
Professor Roy Davies
Overview and further reading
Abstract
This tutorial aims to help those with some
experience of vision to obtain a more in-depth
understanding of the problems of low-level
vision. As it is not possible to cover everything
in the space of 90 minutes, a carefully chosen
series of topics is presented. This section of the
notes is a commentary to accompany the slides
in the presentation.
1. The nature of vision
Vision is a complex process and has long been
modelled, rather loosely, as a cascade of low,
intermediate and high-level stages following
image acquisition. However, this model is
somewhat impoverished in that it ignores the
possibility of downward as well as upward flow
of information which can help with interpretation.
In addition, there is need for considerable
complexity and sophistication – so much so that
the problems of low-level vision are often felt to
be unimportant and are forgotten. Yet it remains
the case that information that is lost at low level
is never regained, while distortions that are
introduced at low level can cause undue trouble
at higher levels [1]. Furthermore, image
acquisition is equally important. Thus, simple
measures to arrange suitable lighting can help to
make the input images easier and less ambiguous
to interpret, and can result in much greater
reliability and accuracy in applications such as
automated inspection [1]. Nevertheless, in
applications such as surveillance, algorithms
should be made as robust as possible so that the
vagaries of ambient illumination are rendered
relatively unimportant.
2. Low-level vision
This tutorial is concerned with low-level vision,
and aims to indicate how some of its problems
and limitations can be solved. This is a huge
subject covering image acquisition, noise
suppression, segmentation, colour and texture
analysis, shape analysis, object recognition, and
many other topics. Hence it is not possible to do
justice to it in the space of one and a half hours.
It is therefore assumed that participants have a
reasonable concept of the subject area, and the
tutorial therefore focusses on a number of topics
that have been chosen because they involve
important issues. The topics fall into five broad
categories:
• Feature detection and sensitivity
• Image filters and morphology
• Robustness of object location
• Validity and accuracy in shape analysis
• Scale and affine invariance.
In what follows, references are given for the
topics falling under each of these categories, and
for other topics that could not be covered in
depth in the tutorial.
3. Feature detection and sensitivity
General [1].
Edge detection [2, 3, 4].
Line segment detection [5, 6, 7].
Corner and interest point detection [1, 8, 9].
General feature mask design [10, 11].
The value of thresholding [1, 12, 13].
4. Image filters and morphology
General [1].
Effect of applying rank-order filters [1, 14].
Mathematical morphology [1, 15].
5. Robustness of object location
General [1, 16].
Centroidal profiles v. Hough transforms [1, 17–
19].
Fast object location by selective scanning [1, 20].
Robust statistics [1, 21, 22].
The RANSAC approach [23].

2. Low-level vision tutorial 2
© E.R. Davies, 2014
6. Validity and accuracy in shape analysis
Shape analysis [1, 24].
Object labelling [1].
Distance transforms and skeletons [1, 25, 26].
Boundary distance measures [27, 28].
Symmetry detection [29, 30].
7. Scale and affine invariant features
At this point it will be useful to consider a topic
that has been developing for just over a decade,
starting with papers [31, 32]. It arose largely
because of difficulties in wide baseline stereo
work, and with tracking object features over
many video frames – because features change
their character over time and correspondences
are easily lost. To overcome this problem it was
necessary first to eliminate the relatively simple
problem of features changing in size, thereby
necessitating ‘scale invariance’ (it being implicit
that translation and rotation invariance have
already been dealt with). Later, improvements
became necessary to cope with ‘affine
invariance’ (which also covers shear and skew
invariance). Lindeberg’s pioneering theory [31]
was soon followed by Lowe’s work [32, 33] on
the ‘Scale Invariant Feature Transform’ (SIFT).
This was followed by affine invariant methods
[34–37]. In parallel with these developments,
work was proceeding on maximally stable
extremal regions [38] and other extremal
methods [39, 40] (the latter concept has also
been followed in a different context [13]).
Much of this work employs the interest point
methods of Harris and Stephens [9], and is
underpinned by careful in-depth experimental
investigations and comparisons [37, 41, 42].
Finally, there are signs that the tide may be
turning in other directions, specifically the
design of feature detectors that concentrate on
especially robust forms of scale invariance – as
in the case of the ‘Speeded Up Robust Features’
(SURF) approach [43, 44]. See also the fuller
review article [45].
Note that reference [1] contains a full and up-to-
date appraisal of this important area.
8. Concluding remarks
The topics presented in this tutorial necessarily
give an incomplete picture because of the short
span of time available. Nevertheless, they
provide interesting lessons, and demonstrate
important factors – specifically, the need for
sensitivity, accuracy, robustness, reliability and
validity. Further factors enter into the equation
when practical systems are being designed –
speed and cost of real-time hardware being
amongst the most relevant.
Overall, it is clear that low-level vision is an
essential ingredient of the vision hierarchy and
that its problems are fundamental to the whole of
vision. Although the focus of attention in the
subject may have moved on to more modern
topics, it is definitely not the case that this side
of the subject is played out and that everything is
now known about it. While data sets change and
specifications for low-level algorithms remain
incomplete, workers will have to go on
developing algorithms to cope with the specific
idiosyncrasies of their data to make the most of it
in all the ways outlined above.
9. Further reading
Participants may wish to further explore the
material covered in this tutorial. The author’s
book [1] covers many of the topics in fair depth;
particular attention is drawn to the following
chapters:
• Chapter 3, for median and mode filters,
including edge shifts;
• Chapter 5, for edge detection;
• Chapters 9 and 10, for shape analysis –
especially thinning and centroidal profiles;
• Chapter 12, for Hough-based methods of
circular object location;
• Appendix on robust statistics.
The book also contains a considerable amount of
detailed information on 3D interpretation,
invariants, texture analysis, automated inspection
and practical issues.
Acknowledgements
The author would like to credit the following
sources for permission to reproduce material
from his earlier publications:

3. Low-level vision tutorial 3
© E.R. Davies, 2014
• Elsevier for permission to reprint text and
figures from [1, 2, 20, 26];
• IEE/IET for permission to reprint text and
figures from [5, 7, 8, 11, 13, 14, 16, 28];
• IFS Publications Ltd. for permission to
reprint figures from [18];
• Woodhead Publishing Ltd, for permission to
reprint figures from [46].
References
1. Davies, E.R. (2012) Computer and Machine
Vision: Theory, Algorithms, Practicalities,
Academic Press (4th
edition), Academic
Press, Oxford, UK
2. Davies, E.R. (1986) Constraints on the
design of template masks for edge detection.
Pattern Recogn. Lett. 4, no. 2, 111–120
3. Canny, J. (1986) A computational approach
to edge detection. IEEE Trans. Pattern Anal.
Mach. Intell. 8, 679–698
4. Petrou, M. and Kittler, J. (1988) On the
optimal edge detector. Proc. 4th Alvey
Vision Conf., Manchester (31 Aug.–2 Sept.),
pp. 191–196
5. Davies, E.R. (1997) Designing efficient line
segment detectors with high orientation
accuracy. Proc. 6th IEE Int. Conf. on Image
Processing and its Applications, Dublin (14–
17 July), IEE Conf. Publication no. 443, pp.
636–640
6. Davies, E.R. (1997) Vectorial strategy for
designing line segment detectors with high
orientation accuracy. Electronics Lett. 33, no.
21, 1775–1777
7. Davies, E.R., Bateman, M., Mason, D.R.,
Chambers, J. and Ridgway, C. (1999)
Detecting insects and other dark line features
using isotropic masks. Proc. 7th IEE Int.
Conf. on Image Processing and its
Applications, Manchester (13–15 July), IEE
Conf. Publication no. 465, pp. 225–229
8. Davies, E.R. (2005) Using an edge-based
model of the Plessey operator to determine
localisation properties, Proc. IEE Int.
Conference on Visual Information
Engineering (VIE 2005), University of
Strathclyde, Glasgow (4–6 April), pp. 385–
391
9. Harris, C. and Stephens, M. (1988) A
combined corner and edge detector. Proc.
Alvey Vision Conf., Manchester University,
UK, pp. 147–151
10. Davies, E.R. (1992) Optimal template masks
for detecting signals with varying
background level. Signal Process. 29, no. 2,
183–189
11. Davies, E.R. (1999) Designing optimal
image feature detection masks: equal area
rule. Electronics Lett. 35, no. 6, 463–465
12. Hannah, I., Patel, D. and Davies, E.R. (1995)
The use of variance and entropic
thresholding methods for image
segmentation. Pattern Recogn. 28, no. 8,
1135–1143
13. Davies, E.R. (2008) Stable bi-level and
multi-level thresholding of images using a
new global transformation. IET Computer
Vision, 2, no. 2, Special Issue on Visual
Information Engineering, Ed. Valestin, S.,
pp. 60–74
14. Davies, E.R. (1999) Image distortions
produced by mean, median and mode filters.
IEE Proc. – Vision Image and Signal
Processing 146, no. 5, 279–285
15. Bangham, J.A. and Marshall, S. (1998)
Image and signal processing with
mathematical morphology. IEE Electronics
and Commun. Journal 10, no. 3, 117–128
16. Davies, E.R. (2000) Low-level vision
requirements. Electron. Commun. Eng. J. 12,
no. 5, 197–210
17. Hough, P.V.C. (1962) Method and means
for recognising complex patterns. US Patent
3069654
18. Davies, E.R. (1984) Design of cost-effective
systems for the inspection of certain
foodproducts during manufacture. In Pugh,
A. (ed.), Proc. 4th Conf. on Robot Vision
and Sensory Controls, London (9–11 Oct.),
pp. 437–446
19. Ballard, D.H. (1981) Generalizing the
Hough transform to detect arbitrary shapes.
Pattern Recogn. 13, 111–122
20. Davies, E.R. (1987) A high speed algorithm
for circular object location. Pattern Recogn.
Lett. 6, no. 5, 323–333
21. Meer, P., Mintz, D., Rosenfeld, A. and Kim,
D.Y. (1991) Robust regression methods for
computer vision: a review. Int. J. Comput.
Vision 6, 59–70
22. Kim, D.Y., Kim, J.J., Meer, P., Mintz, D.
and Rosenfeld, A. (1989) Robust computer
vision: a least median of squares based
approach. Proc. DARPA Image

4. Low-level vision tutorial 4
© E.R. Davies, 2014
Understanding Workshop, Palo Alto, CA
(23–26 May), pp. 1117–1134
23. Fischler, M.A. and Bolles, R.C. (1981)
Random sample consensus: A paradigm for
model fitting with applications to image
analysis and automated cartography. Comm.
ACM 24, no. 6, 381–395
24. Hu, M.K. (1962) Visual pattern recognition
by moment invariants. IRE Trans. Inf.
Theory 8, 179–187
25. Rosenfeld, A. and Pfaltz, J.L. (1968)
Distance functions on digital pictures.
Pattern Recogn. 1, 33–61
26. Davies, E.R. and Plummer, A.P.N. (1981)
Thinning algorithms: a critique and a new
methodology. Pattern Recogn. 14, 53–63
27. Koplowitz, J. and Bruckstein, A.M. (1989)
Design of perimeter estimators for digitized
planar shapes. IEEE Trans. Pattern Anal.
Mach. Intell. 11, 611–622
28. Davies, E.R. (1991) Insight into operation of
Kulpa boundary distance measure.
Electronics Lett. 27, no. 13, 1178–1180
29. Kuehnle, A. (1991) Symmetry-based
recognition of vehicle rears. Pattern Recogn.
Lett. 12, 249–258
30. Cho, M. and Lee, K.M. (2009) Bilateral
symmetry detection via symmetry-growing.
Proc. British Machine Vision Conf. (BMVC),
paper 286, pp. 1–11
31. Lindeberg, T. (1998) Feature detection with
automatic scale selection. Int. J. Computer
Vision 30, no. 2, 79–116
32. Lowe, D.G. (1999) Object recognition from
local scale-invariant features. Proc. 7th Int.
Conf. on Computer Vision (ICCV), Corfu,
Greece, pp. 1150–1157
33. Lowe, D. (2004) Distinctive image features
from scale-invariant keypoints. Int. J.
Computer Vision 60, 91–110
34. Tuytelaars, T. and Van Gool, L. (2000)
Wide baseline stereo matching based on
local, affinely invariant regions. Proc.
British Machine Vision Conf. (BMVC),
Bristol University, UK, pp. 412–422
35. Mikolajczyk, K. and Schmid, C. (2002) An
affine invariant interest point detector. Proc.
European Conf. on Computer Vision
(ECCV), Copenhagen, Denmark, pp. 128–
142
36. Mikolajczyk, K. and Schmid, C. (2004)
Scale and affine invariant interest point
detectors. Int. J. Computer Vision 60, no. 1,
63–86
37. Mikolajczyk, K., Tuytelaars, T., Schmid, C.,
Zisserman, A., Matas, J., Schaffalitzky, F.,
Kadir, T. and Van Gool, L. (2005) A
comparison of affine region detectors. Int. J.
Computer Vision 65, 43–72
38. Matas, J., Chum, O., Urban, M. and Pajdla,
T. (2002) Robust wide baseline stereo from
maximally stable extremal regions. Proc.
British Machine Vision Conf. (BMVC),
Cardiff University, UK, pp. 384–393
39. Kadir, T. and Brady, M. (2001) Scale,
saliency and image description. Int. J.
Computer Vision 45, no. 2, 83–105
40. Kadir, T., Brady, M. and Zisserman, A.
(2004) An affine invariant method for
selecting salient regions in images. Proc. 8th
European Conf. on Computer Vision
(ECCV), pp. 345–457
41. Schmid, C., Mohr, R. and C. Bauckhage
(2000) Evaluation of interest point detectors.
Int. J. Computer Vision 37, no. 2, 151–172
42. Mikolajczyk, K. and Schmid, C. (2005) A
performance evaluation of local descriptors.
IEEE Trans. Pattern Anal. Mach. Intell. 27,
no. 10, 1615–1630
43. Bay, H., Tuytelaars, T. and Van Gool, L.
(2006) SURF: speeded up robust features.
Proc. 9th European Conf. on Computer
Vision (ECCV), Springer LNCS volume
3951, part 1, pp. 404–417
44. Bay, H., Ess, A., Tuytelaars, T., Van
Gool, L. (2008) Speeded-up robust features
(SURF). Computer Vision Image
Understanding 110, no. 3, 346–359
45. Tuytelaars, T. and Mikolajczyk, K. (2008)
Local invariant feature detectors: a survey.
Foundations and Trends in Computer
Graphics and Vision 3, no. 3, 177–280
46. Davies, E.R. (2012) Computer vision for
automatic sorting in the food industry.
Chapter 6 In: Sun, D.-W. (ed.), Computer
Vision Technology in the Food and
Beverage Industries, Woodhead Publishing
Ltd, Cambridge, UK, ISBN 978-0-85709-
036-2, pp. 150–180.
http://www.woodheadpublishing.com/en/bo
ok.aspx?bookID=2323

5. 1
Low-level vision – a tutorial
by
Professor Roy Davies
of
Royal Holloway, University of London
2
The role of low-level vision*
Image acquisition
Feature detection
Intermediate level vision
High-level vision
*A somewhat simplified and naïve schema.
Low-level
vision
Image preprocessing
3
An original image
4
Result of applying a median filter
5

6. 5
Result of applying a Harris corner detector
6
Result of applying a Sobel edge detector
7
Edge detection
Objectives:
To demonstrate the design of efficient edge detectors.
This section contains copyright material reproduced from [1,2] with permission of
Elsevier.
8
Theory of 3 3 template operators
First we assume that eight masks are to be used, for
angles differing by 45°. Four of the masks differ from
the others only in sign, and we are left with two
paradigm masks:
The all-too-ready assumption that C = A and D = B
is by no means confirmed by theory, as we shall see.
6

7. 9
Let us apply these masks to the window:
then estimation of the 0°, 90° and 45° components of
gradient by the earlier general masks gives:
10
If vector addition is to be valid:
Equating coefficients of a, b, c, d, e, f, g, h, i leads to the
self-consistent pair of conditions:
Insisting that the masks give equal responses at 22.5°
leads to the final formula:
11
We have now obtained the following template masks for
edge detection:
It is also possible to use the first two of these masks for
detecting and orientating edge segments, taking them
to provide vector components of edge intensity: gx, gy .
Thus we have derived the Sobel operator masks in a
principled, non-ad hoc way.
12
Basic theory of edge detection and orientation
edge intensity:
edge orientation:
7

8. 13
The Canny operator
The aim of the Canny edge detector is to be far more
accurate than basic edge detectors such as the Sobel. To
achieve this, a number of processes are applied in turn:
Smooth image using a 2D Gaussian
Differentiate using 1D derivative functions, e.g. Sobel
Perform non-maximum suppression to thin the edges
(resample to achieve sub-pixel accuracy)
Perform hysteresis thresholding.
14
Canny in action
© Elsevier 2012
15
Line segment detection
Objectives:
To demonstrate the design of efficient line segment
detectors.
To show how high orientation accuracy can be
achieved.
This section contains copyright material reproduced from [5] and [7] with
permission of the IET.
16
Insects can be modelled as dark rectangular bars:
Arbitrary orientation means that detection
could involve considerable computation.
8

9. 17
Applying the vectorial approach to line
segment detection
This is difficult as line segments have a rotational period
of , apparently ruling out the vectorial approach.
The key is to refrain from considering
line segments as local lines of the form:
Instead we consider line segments
as intensity maps in the form of
alternating quadrants:
18
The result of this strategy leads to masks of the forms:
However, the last two masks are sign-inverted versions
of the first two: this gives just two masks – just the
number required for a vectorial computation.
19
This leads to an effective gradient magnitude:
g = [g0
2 + g45
2]1/2
and apparent orientation:
= arctan (g45 /g0)
and finally to an actual line orientation of:
= = arctan (g45 /g0)
where is defined only within the range 0 .
20
0
45 o
45o
1
3
2
thin line segments (w = 0) wide line segments
1 w = 1.0
2 w = 1.4
3 w = 2.0
Estimated v. actual orientation for wide line segments
© IEE 1997
0
45 o
45o
9

10. 21© IEE 1999
Line segment detector locating insects
22Mask design geometry
23
Corner and interest point detection
Objectives:
To demonstrate the principle of the Harris corner
detector.
To show how the signal depends on corner geometry.
This section contains copyright material reproduced from [1,8] with permission of
Elsevier and the IET.
24
Note:
1. The need for averaging
2. The built-in rotational invariance
Mathematical definition of the Harris operator
10

11. 25
General corner in a
circular window
Single straight edge in a
circular window
© IEE 2005
26
det = l1 l2 g4 sin2
and trace = (l1 + l2)g2
det = 0, so C = 0.
Case of a corner:
Case of a straight edge:
27
Possible geometries for sharp corners
(c) is case of maximum signal (corner shift = a).
© IEE 2005
28
Harris interest points + non-maximum suppression
© Elsevier 2012
11

12. 29
Harris interest points + non-maximum suppression
© Elsevier 2012
30
Harris interest points + non-maximum suppression
31
Harris interest points + non-maximum suppression
© Elsevier 2012
32
Harris interest points + non-maximum suppression
© Elsevier 2012
12

13. 33
General feature mask design
Objectives:
To show how sensitivity can be optimised using the
spatial matched filter concept.
To demonstrate a limitation on the sizes of template
masks.
This section contains copyright material reproduced from [11] with permission of
the IET.
34
Optimisation calculation
When performing feature detection in saturated grey-
scale images, we shall take the object feature being
detected as region 2, and its background as region 1.
© IEE 1999
35
On this model we have to calculate optimal values for
the mask weighting factors w1 and w2 and for the region
areas A1 and A2.
First we write the total signal from a template mask as:
S = w1A1S1 + w2A2S2
and the total noise power as:
N2 = w1
2A1N1
2 + w2
2A2N2
2
36
Thus we obtain a power SNR:
It is easy to see that if both mask regions are increased in size by
the same factor , 2 will also be increased by this factor. This
makes it interesting to optimise the mask by adjusting the relative
values of A1, A2, leaving the total area A unchanged. Let us first
eliminate w2 using the zero-mean condition:
w1A1 + w2A2 = 0
The signal and noise equations now become:
S = w1A1(S1– S2)
N2 = w1
2A1N1
2 + w1
2A2 (A1/A2)2 N2
2
13

14. 37
Optimising the SNR now leads to:
A1/A2 = N1/N2
Taking N1 = N2, we obtain an important result – the equal area rule:
A1 = A2 = A/2
Finally, when the equal area rule applies, the zero mean rule takes
the form:
w1 = – w2
Note that many cases, such as those arising when the foreground
and background have different textures, can be modelled by
writing N1 N2, In that case the equal area rule will not apply.
38
Applying the equal area rule. (a) Small feature, such as a hole.
(b) Thick line segment. (c) Large blob. (d) Corner of a large
object. (e) Corner detected by a specially shaped mask.
(f) Sharp corner detected away from its apex.
© IEE 1999
39
The value of thresholding
Here the aims are to note that:
Where applicable, thresholding can be extremely effective.
Considerable advances are still being made in this area.
It is silly to regard thresholding as ‘old hat’: the old adage
‘horses for courses’ should be borne in mind.
This section contains copyright material reproduced from [1,13] with permission of
Elsevier and the IET.
40
Selecting the most suitable thresholds
When considering whether a minimum is ‘significant’,
it seems best to judge it in relation to global rather
than local (noisy) maxima.
dark foreground objects light background
peaks from irrelevant ‘clutter’
f(I)
I
Distribution of intensities for dark objects on a light background
14

15. 41
Use the Arithmetic Mean or the Geometric Mean for
finding significant minima?
AM
GM
orig y1
y2
42
smoothed GM
GM
original
Intensity
matching to
locate the road
surface
© IET 2007
43
smoothed GM
GM
original
Finding cars
from their
shadows,
within the road
region
© IET 2007
44
Intensity
matching to
locate ergot
contaminants
amongst wheat
grains
smoothed GM
GM
original
© IET 2007
15

16. 45
Image filters and morphology
Objectives:
To model the effect of applying rank-order filters.
To highlight the shifts produced by median filters.
To show how rank-order filters can be used for
morphology.
This section contains copyright material reproduced from [14,46] with permission of
the IET and Woodhead Publishing Ltd.
46
Edge shifts D obtained on applying rank-order filters with
various rank-order coefficients at locations of various
curvatures relative to the window radius a.
1–1
– a
a
D
0
0
= 0.3a
= 0
47
Geometry for estimation of contour shifts. This
figure shows idealised intensity variations within a
circular neighbourhood C of radius a.
© IEE 1999
48
Optimal segmentation of rat droppings
original
opened
thresholded
closed
© Woodhead 2012
16

17. 49
closed and
eroded
median,
eroded
best
option
median
opened and
eroded
© Woodhead 2012
Optimal segmentation of rat droppings
50
Centroidal profiles v. Hough
transforms
Objectives:
To show that the centroidal profile approach is non-
robust.
To demonstrate the robustness of the Hough
transform.
This section contains copyright material reproduced from [1,16,18] with permission
of Elsevier and IFS.
51
(a) A broken circular
object, with centroid
indicated.
(b) Centroidal
profile of an ideal
round object.
(c) Centroidal
profile of broken
object (a).
Note the
difficulties
caused in (c)
by the shift of
the centroid –
the reference
point for the
computation.
© IEE 2000
52
Robust circle detection by the Hough transform:
candidate circle centre locations are shown being
accumulated in parameter space.
© IEE 2000
17

18. 53
Locating three biscuits from incomplete edges
© IFS 1984
54
Location of broken and overlapping biscuits, showing
the robustness of the Hough transform. Location
accuracy is indicated by the black dots.
© IFS 1984
55
Iris location using the Hough transform
56
Fast object location by selective
scanning
Objectives:
To show that selective scanning locates objects
rapidly.
To show that selective scanning is subject to graceful
degradation.
This section contains copyright material reproduced from [20] with permission of
Elsevier.
18

19. 57
Object location using the chord bisection algorithm, using
a step size of 8 pixels. The black dots show the positions
of the horizontal and vertical chord bisectors, and the
white dot shows the position found for the centre.
© Elsevier 1987
58
Location of a broken object using the chord bisection
algorithm when about quarter of the boundary is missing:
This gives a good indication of the limitations of the
technique.
© Elsevier 1987
59
Robust statistics
Objectives:
To demonstrate the problems of line fitting.
To outline M-estimator and LMedS techniques.
This section contains copyright material reproduced from [1] with permission of
Elsevier.
60
(b) shows the ‘averaging in’ effect that results from
application of a mean filter.
(c) shows that it is not evident for a median filter.
© IEE 2000
19

20. 61
Fitting data points to straight lines. Rogue (‘outlier’)
points make the problem far from straightforward.
© Elsevier 2005
62
The idea of influence functions such as these is to limit the
influence of outlier residuals ri by applying a variable
weighting which tends to zero for large values.
© Elsevier 2005
Tukey
biweight
Hampel
3-part re-
descending
63
The least median
of squares
technique finds
the narrowest
parallel-sided strip
that includes half
the population of
the distribution.
The RANSAC
technique applies
a number of
narrow parallel-
sided strips,
searching for the
one that best
matches the data.
© Elsevier 2005Random Sampling Consensus
64
RANSAC in action
20

21. 65
Use of RANSAC to locate lines in a medical application.
Video of laparoscopic tool location
66
Shape analysis
Objectives:
To demonstrate the complexities of object labelling.
To indicate the problems of using simple shape
descriptors.
To show that skeleton formation should be
mediated by global operations.
This section contains copyright material reproduced from [1,26] with permission of
Elsevier.
67
Object labelling. This diagram shows how a basic labelling
algorithm copes with a simple scene: two objects have label
clashes which may be corrected by a reverse scan.
© IEE 2000
68
Distance function of a simple shape. © Elsevier 1981
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22. 69
Guided thinning: here a thinning operation is applied in which the
local maxima of the distance function must not be removed.
Finally, excess local maxima points are eliminated.
© Elsevier 1981
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Boundary distance measures
Objectives:
To show that boundary distance measurement has
limited accuracy.
To calculate a boundary distance correction factor.
This section contains copyright material reproduced from [28] with permission of
the IET.
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Introduction
Many recognition schemes involve tracking around the boundaries
of objects and measuring boundary distance s or perimeter P. To
do this the most obvious method is to assume that adjacent pixels
are unit distance from the current pixel:
Clearly, a better measure assumes the following distances:
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Near 0° and 45°, boundary coding leads to an over-
estimate of the boundary length by a factor of 2.
© IEE 1991
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23. 73
Geometry for calculating the error in boundary length
estimation. Take a boundary segment to consist of
horizontal and diagonal sections, with overall horizontal
and vertical displacements a and b, as shown above.
© IEE 1991
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Summary
Optimisation of accuracy involves rigorous estimation, rather than
the all too ready assumption that quick fixes will lead to good
results. Note also that rigorous estimation of boundary length
involves at its heart the estimation of the length of the original
analogue curve rather than some chance version that results when
the boundary is digitised.
Finally, note that the final form:
L = (nh + 2 nd) where
still has limited accuracy for individual orientations – a factor that
can only be improved by shape modelling over more than adjacent
pixel separation distances.
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Symmetry detection
Objectives:
To note the wide applicability of this approach to
object detection.
To note that it is in principle achievable by using 1D
or 2D Hough transforms.
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Potential applications of symmetry detection
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24. 77
Scale and affine invariance
Objectives:
To consider the need for scale-invariant and affine-
invariant feature detectors.
To outline recent approaches to the problem.
(This section will merely aim to provide a lead-in to the lecture of Krystian Mikolajczyk.)
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Recent important feature detectors
SIFT = Scale Invariant Feature Transform (Lowe 1999, 2004)
Similarity transform (4 DoF):
Affine transform (6 DoF):
Affine transforms have the properties:
They convert parallel lines to parallel lines.
They preserve ratios of distances on straight lines.
They cover the transforms produced by Weak Perspective
Projection (WPP).
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The 4th edition of this book was published in 2012. It covers:
the whole of this talk
detailed discussions of scale- and affine-invariant features
new chapters on surveillance and in-vehicle vision systems
over 1000 references, including ~500 since 2000.
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