Robust Histogram Construction from
Color Invariants for Object
Recognition
Authors : Theo Gevers and Harro Stokman
Presented By:
Neha Sisodiya (201111032)
Introduction

Color is a useful cue for object identification.
1.
An object's color is not an intrinsic property of the object itself.
2.
3.
Depends on confounding imaging condition (e.g., illumination,
shading, highlights, and interreflections)
Color invariants :Transformations of ``color'' which are invariant


Normalized color rgb (c1c2c3) color model
Hue color space (l1,l2,l3) color model
Image Formation
Figure : A simple model of image formation: light from a
source (e.g. the sun) is incident upon and reflected from a
surface. The reflected light enters the eye leading eventually
to our perception of colour.

The response R,G,B is given by
Qx(λ) is characterize the response of incident light to color signal
C(λ) is the color signal product of the light incident upon a surface E(λ)
and surface reflectance function S(λ)

the intensity of the reflected light depends on the cosine of the angle
between the surface normal (a vector denoted ) and the angle of
incidence of the light (a vector denoted
)

The equation above surface/lighting geometry changes, sensor
responses change by the same scale factor

It is thus straightforward to remove the dependence of sensor
responses on lighting geometry.

So we can have rgb co-ordinates as below
referred to as chromaticity co-ordinates .
 invariant to a change in camera viewpoint, object pose, and the
direction and intensity of the incident light.
 But normalized color rgb is is unstable (or undefined at the black
point (R = G = B = 0)
Hue, Saturation ,and Value (HSV)

Hue refers to a pure color i.e linear mixtures between adjacent pairs

Model is cylindrical rearrange the geometry of RGB in an attempt to
be more intuitive and perceptually relevant than the cartesian (cube)
representation

.
Saturation can be defined
as either the ratio of
colorfulness to brightness

The hue θ is below,
Effect of Sensor Noise



rgb and H(hue) become very unstable at singularities.
Traditionally is suppressed by ad hoc thresholding of the
transformed values prior to color histogram construction for object
identification
Proposed technique is to estimate the noise propagation through
color invariants.

They use Variable Kernel Density estimators to suppress the effect

of noise to construct color invariant histograms.
Associated uncertainty is known for each color invariant value and
this is used to is used to derive the optimal parameterization of the
variable kernel used during histogram construction.
Photometric Color Invariance

The reflection from inhomogeneous dielectric materials under white,
spectrally smooth illumination is given by
,
k є(R,G,B), Fk(λ) are spectral sensitivities , B(λ) is surface albedo
E denotes the white light source and S is the Fresnel reflectance.
 Normalized color rg is color invariant by substituting r and g in (1)


Above equation only dependent on the sensors and the surface
albedo.
The opponent color space defined by


The opponent color space o1o2 is independent of highlights as
follows from substituting (1) in (5) and (6)
Hue can be computed by substituting the reflection term of (1) in θ we
will get
Noise Propagation

Additive Gaussian noise is widely used.

Assumption sensor noise is normally distributed.

True value of a measurand u is related to its N arguments,
denoted by uj as follows

We estimate û of u we will get û1,.,.,.,.,.,.,.,...ûN measured with
standard deviation

Approximation by taylor series w.r.t noise given by
where û1=u1+Є1 and û2=u2+Є2

The error of an indirect measurement is in general

In general, only the first linear term is used to compute the
error

If the uncertainties in û1,.,.,.,.,.,.,.,...û N are independent, random, and
relatively small, the predicted uncertainty in q is given by eq(16)

Substitution of (2) and (3) in (16) gives the uncertainty for the
normalized coordinates

The uncertainties of o1 and o2 are given by eq(18)

Substitution of (8) in (16) gives the uncertainty for the hue
HISTOGRAM CONSTRUCTION BY VARIABLE
KERNEL DENSITY ESTIMATION

The (one-dimensional) histogram is defined as eq(20)

Problem is Bin-width parameter and the position of the bin
edges to be choosen

The kernel density estimator is insensitive to the placement of
the bin edges

Variable kernel density estimator eq(22)

Assuming normally distributed noise, the distribution is
approximated well by the Gauss distribution

the variable kernel method estimating the univariate,directional hue
density is as follows eq(24)
where σθ is derived according to (19), with corresponding value θ.

The variable kernel method for the bivariate normalized rg kernel is
given by
where σr,σg are derived according to (17).

Similarly bivariate normalized o1 o2 kernel is given by
where σo1,σo2 are derived according to (18).
Experimental Results
Experimental Results …..
Fig. 2. Various images which are included in the image database of 500
images. The images are representative for the images in the database.
Objects were recorded in isolation (one per image).
Robustness Against Noise: Simulated
Data
Fig. 3. Two images generating together 10 images by adding noise
with   (8,16,32,64,128)
Robustness Against Noise: Realistic Data
Fig. 6. Two objects under varying illumination intensity generating each four images with SNR
(24,12, 6, 3).
Comments

Kernel density estimator overcome the problem of ad hoc thresholding
at unstable color invariants.

The variable kernel density estimation is derived from a theoretical
framework for noise propagation through color invariants.

Proposed method is less sensitive to Gaussian noise than traditional
histogram construction schemes.

It is computationally more expensive.
Thanks