Evolutionary design of robust noise-specific image filters

Abstract

Evolutionary design has shown as a powerful technique in solving various engineering problems. One of the areas in which this approach succeeds is digital image processing. Image filtering represents a wide topic in 2D signal processing. In this case different types of noise are considered in the filtering process to restore the image quality that has been decreased by changing values of some pixels in the image (e.g. due to the transmission through unreliable lines or in the process of acquiring the image). Impulse noise represents a basic type of non-linear noise typically affecting a single pixel in different regions of the image. In order to eliminate this type noise median filters have usually been applied. However, for higher noise intensity or wide range of the noise values this approach leads to corrupting non-noise pixels as well which results in images that are smudged or lose some details after the filtering process. Therefore, advanced filtering techniques have been developed including a concept of noise detection or iterative filtering algorithms. In case of the high noise intensity, a single filtering step is insufficient to eliminate the noise and obtain a reasonable quality of the filtered image. Therefore, iterative filters have been introduced. In this paper we apply an evolutionary algorithm combined with Cartesian Genetic Programing representation to design image filters for the impulse noise that are able to compete with some of the best conventionally used iterative filters. We consider the concept of noise detection to be designed together with the filter itself by means of the evolutionary algorithm. Finally, it will be shown that if the evolved filter is applied iteratively on the filtered image, a high-quality results can be obtained utilizing lower computational effort of the filtering process in comparison with the conventional iterative filters.

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Evolutionary Design of Robust
Noise-Specific Image Filters
Zdenek Vasicek
Brno University of Technology
Faculty of Information Technology
Boˇ zetˇ echova 2, 61266 Brno, Czech Republic
Email: vasicek@fit.vutbr.cz
Michal Bidlo
Brno University of Technology
Faculty of Information Technology
Boˇ zetˇ echova 2, 61266 Brno, Czech Republic
Email: bidlom@fit.vutbr.cz
Abstract—Evolutionary design has shown as a powerful tech-
nique in solving various engineering problems. One of the areas in
which this approach succeeds is digital image processing. Image
filtering represents a wide topic in 2D signal processing. In this
case different types of noise are considered in the filtering process
to restore the image quality that has been decreased by changing
values of some pixels in the image (e.g. due to the transmission
through unreliable lines or in the process of acquiring the image).
Impulse noise represents a basic type of non-linear noise typically
affecting a single pixel in different regions of the image. In order
to eliminate this type noise median filters have usually been
applied. However, for higher noise intensity or wide range of the
noise values this approach leads to corrupting non-noise pixels
as well which results in images that are smudged or lose some
details after the filtering process. Therefore, advanced filtering
techniques have been developed including a concept of noise
detection or iterative filtering algorithms. In case of the high
noise intensity, a single filtering step is insufficient to eliminate
the noise and obtain a reasonable quality of the filtered image.
Therefore, iterative filters have been introduced. In this paper
we apply an evolutionary algorithm combined with Cartesian
Genetic Programing representation to design image filters for
the impulse noise that are able to compete with some of the best
conventionally used iterative filters. We consider the concept of
noise detection to be designed together with the filter itself by
means of the evolutionary algorithm. Finally, it will be shown that
if the evolved filter is applied iteratively on the filtered image, a
high-quality results can be obtained utilizing lower computational
effort of the filtering process in comparison with the conventional
iterative filters.
Index Terms—evolutionary design; image filter; impulse noise
I. INTRODUCTION
Linear filters became the most popular image signal pro-
cessing systems in the past years. The reason of their popular-
ity is caused by the existence of robust mathematical models
which can be used for their analysis and design. However,
there exist many areas in which the nonlinear filters provide
significantly better results. The advantage of nonlinear filters
lies in their ability to preserve edges and suppress the noise
without loss of details. The success of nonlinear filters is
caused by the fact that image signals as well as existing noise
types are usually nonlinear.
The impulse noise is the most frequently referred type of
noise. While it is very difficult to remove the impulse noise
using linear filters as they tend to smudge the resulting images,
the nonlinear filters are able to provide filtered images with a
reasonably good quality.
In most cases, the impulse noise is caused by malfunc-
tioning pixels in camera sensors, faulty memory locations in
hardware, or errors in the data transmission. We distinguish
two common types of impulse noise: salt&pepper noise (com-
monly referred as intensity spikes) and random-valued shot
noise. For images corrupted by the salt&pepper noise, the
noisy pixels can take only the maximum or minimum values.
In case of grayscale images, the noisy pixels are represented
by the value 255 or 0 (no other values are considered).
However, this kind of noise is a theoretical (model) instance
rather than a corruption frequently occurred in praxis. In
case of the random-valued impulse noise, the noisy pixels
can have an arbitrary value (from 0 to 255 in case of
grayscale images) uniformly distributed in this range. It is
evident that this type of noise is more difficult to remove in
comparison with salt&pepper noise because the noisy pixels
may actually possess any value representing the uncorrupted
pixels. Therefore, the noisy pixels are difficult to detect which
represents a problem especially in the images containing many
details because the filtering process may tend to remove the
details assuming their pixels being corrupted by the noise.
On the other hand, if the noise occurs in a smooth grade
transition, where the difference between the noise pixel and the
neighbouring pixels is low, the noise may not be recognized
and remains in the filtered image which degrades its quality. A
serious problem represents the case of random-valued noise if
occurred in a high intensity in which the common single-step
filters do not usually provide a reasonable quality. Thus the
iterative filtering may be the solution.
The common median filter (MF) [1], initially introduced
to eliminate impulse noise, continues to be the most used
approach even if the quality of the filtered images is poor
in comparison with the other advanced techniques. The main
advantage of the common median filter lies in its simple
and effective software as well as hardware implementation
which is straightforward and does not require many resources.
The median-based filtering approach has been intensively
studied and extended to promising approaches such as Center
Weighted Median Filter (CWMF) [2], more general Weighted
Median Filter (WMF) [3] or weighted order statistic filter

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[4]. Nevertheless, all these median-based methods tend to
smudge the image since applying the median filtering to the
entire image would inevitably remove details presented in the
image. In order to overcome this drawback, a switching-based
median filtering concept has been proposed [5]. This concept
splits the filtering process into two parts – noise detection
and noise replacement. The noise detector determines which
pixels are affected by the impulse noise and only these
pixels are replaced (e.g. using MF). Noise detection can be
based on various concepts: a median-based filter [5], fuzzy
techniques [6] or neural networks [7]. The common problem
of the before-mentioned detection mechanism is the necessity
to predetermine the value of a threshold parameter which
significantly influences the filtering quality. Therefore these
approaches are rarely used in practice. The adaptive median
filter (AMF) proposed in [8] is a robust approach which tries
to identify and replace the affected pixels only. In contrast with
the previous approaches, the detection method is based on the
statistical ordered filters with gradually increasing kernel size
[9]. In contrast with the median-based methods, AMF provides
significantly better results even for the images corrupted with
high intensity impulse noise. Apart from the non-iterative
algorithms, the iterative algorithms such as Pixel-Wise Median
of the Absolute Deviations from the median (PWMAD) [10]
or Directional Weighted Median filter (DWM) outlined in [11]
have been introduced. These approaches provide very good
results if the random valued impulse noise is considered, do
not contain any varying parameters and require no previous
training or optimization. The main disadvantage is apparent –
the iterative approach place higher requirements to the memory
resources especially in case of hardware implementation.
Two different approaches can be identified to perform image
filtering using evolutionary techniques: (1) An existing filter is
considered for a given noise type and the evolution is applied
to find suitable coefficients of the filter (i.e. it is a case of filter
optimization). An instance of this approach for the filtering
of color images can be found in [12]. (2) The evolutionary
algorithm is used to create entirely new filter considering a
set of suitable functions and limitations specified by human
designer (it is a case of evolutionary design of a new filtering
algorithm) [13]. In this paper we focus on the second approach.
The goal of this paper is to design robust impulse-noise
filters by means of Cartesian Genetic Programming and to
show that the evolutionary algorithm is able to design filters of
at least the same or better performance in comparison with the
conventional solutions. The proposed results will be compared
with the conventional median filters (common MF and AMF)
and the iterative filters DWM and PWMAD that provide very
good results in filtering images corrupted by a higher noise
intensity. We show that the filters designed by the evolution
are able to compete with these solutions from the point of
view of the filtering quality with respect to the computational
effort (i.e. the number of iterations) that is needed to filter an
image. Two sets of experiments will be presented considering
salt&pepper noise and random-valued impulse noise.
II. SWITCHING BASED FILTERING CONCEPT
The main disadvantage of the common median-based filters
is that the filtering transformation is applied on all the pixels
of the image regardless of the pixel represents the noise or
not. Thus this approach results in the loss of the image details
and causes the degradation of the image quality especially
if a larger filter kernel is used. In order to improve the
filtering quality, the switching-based median filter has been
proposed. The switching-based approach outlined in [5] can
be considered as a general process of filtering that operates in
two steps. In the first step, the noisy pixels are detected using
a detection algorithm. Then, the new values of the corrupted
pixels are estimated using an estimation algorithm.
Let xijand yijdenote pixels with coordinates i,j in a noisy
and a filtered image respectively. If the estimated value of the
corrupted pixel xijis zij, the switching filter concept can be
defined as
yij= sij· zij+ (1 − sij) · xij
where sij is a binary noise map – an output produced by
the estimation algorithm. Noise map sij contains ones at the
positions of pixels detected as noisy pixels.
In general, sij is determined by comparing the absolute
difference between the original pixel value xij and some
local statistics Ω(xij) with a threshold T. Statistics Ω(xij)
can be produced by common median filter, weighted median
filter, adaptive median filter or using a complex detection
mechanism, e.g. DWM or PWMAD. Since the value of T
is highly correlated to the image contents, noise probability
and distribution, T has to be calculated for each filtered
image. This is unpractical since the problem of finding the
optimal threshold is a complex task. While setting T too
high leaves a lot of the noisy pixels unfiltered, too low T
causes that image details will be treated as noise and the
overall image quality will be degraded. In order to avoid
setting of this parameter, the process of noise map estimation is
usually applied iteratively with varying threshold (e.g. DWM).
Estimated value of the filtered pixel zij is usually based on
common median filter or its variants (e.g. weighted median
filter).
Fig. 1.
kernel
The concept of the switching-based filtering using a 3 × 3 filter
The concept the switching-based filtering is shown in Fig-
ure 1. In this paper the task is to design a circuit with two 8-bit
outputs; the first output represents the image filter output and

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the second one is the output of the noise detector. If the most
significant bit of the noise detector output equals 1, then the
filtered value is taken from the image filter (i.e. the processed
pixel was detected as noise), otherwise the original value is
used.
III. CARTESIAN GENETIC PROGRAMMING
Cartesian Genetic Programming (CGP) has been introduced
by J. F. Miller and P. Thompson in [14]. CGP was originally
intended for the gate-level evolution, however, it has been
extended for the functional level evolution and successfully
applied in many areas. In its basic version, a candidate solution
is directly represented in the chromosome which is represented
by a sequence of integers.
A. Representation
In contrast with the common Genetic Programming, which
encodes a candidate solution using a tree representation,
CGP utilizes a matrix consisting of the fixed number of
programmable nodes. More precisely, a candidate solution
is modeled as an array of nr (rows) ×nc (columns) of
programmable nodes. Each programmable node can have
several inputs ni and outputs no and can implement one of
the predefined functions; usually ni = 2 and no = 1. The
number of the primary inputs pi, primary outputs poas well
as the number of programmable nodes and their parameters
are fixed. Each node input can be connected either to the
output of a node placed in the previous l columns or to one
of the primary inputs. The parameter called as l-back, in fact,
defines the level of connectivity and thus reduces or extends
the search space. For example, if l = 1 only neighboring
columns may be connected; if l = nc, full connectivity is
enabled. Feedback is not allowed in the basic version of CGP;
in order to avoid a feedback, the inputs are not allowed to
connect to elements in the same and any consequent column.
The primary outputs can be connected usually to the output
of any programmable node since the direct connection to the
primary inputs implements the identity function only. Each
programmable node can perform one of the fi-input functions
predefined in a set Γ (0 ≤ fi≤ ni). The Γ is chosen according
to the application requirements; while in case of the gate-level
evolution of digital circuits Γ consists of two-input boolean
functions only, in case of functional level evolution Γ contains
complex building blocks such as adders, multipliers etc.
A candidate solution is encoded using a sequence of integers
that encodes the interconnection of the programmable nodes,
their configuration (i.e. index of the function that a certain
node implements) and the connection of the primary outputs.
In order to encode a candidate solution, the primary inputs
as well as the outputs of the programmable nodes have to be
assigned a unique index. The primary inputs have assigned
the indices 0,...,pi− 1. Each output of each programmable
node has assigned an integer starting by piand increasing by
one and numbered row by row, column by column. Every ni-
input programmable node is fully encoded by a ei+1-tuple of
integers. The first niintegers encodes the connection of each
OR
0
1
2
3
4
5
6
7
8
9
10
1
2
2
0
111
1
2
OR
ORORANDAND
ANDAND
Fig. 2. An example of a candidate circuit in Cartesian Genetic Programming.
The CGP parameters are as follows: nr = 2, nc= 4, l = 3, ni= 2 (for all
the nodes), no= 1 (for all the nodes), pi= 3, po= 2, Γ = {AND (0), OR
(1)}. Nodes 5,7 and 9 are not utilized. The corresponding chromosome is:
1,2,1, 2,0,0, 1,3,1, 3,4,0 1,6,0, 1,6,1, 1,7,0, 2,8,1, 6, 10. The last two integers
indicate the outputs of the program. The function code of a gate is typed in
bold.
node input. The last integer in the tuple encodes the function
of the programmable node. The chromosome is encoded into a
sequence of (ni+1)·nc·nr+pointegers. The last pointegers
encode the connection of the primary outputs.
Figure 2 shows an example of a candidate circuit and the
corresponding chromosome in the CGP encoding. It can be
seen that while the size of the chromosome is fixed, the size
of phenotype is variable (i.e. some nodes may not be used).
B. Search Algorithm
CGP usually operates with a small population of 1 + λ
individuals (typically, λ = 4 − 20). The initial population is
generated randomly. Every new population consists of the best
individual and its variants (mutants). The variants are created
using a point mutation operator. In case when two or more
individuals have received the same fitness score in the previous
population, the individual which did not serve as a parent in
the previous population will be selected as a new parent. This
strategy is used to ensure the diversity of the population.
The fitness function usually takes one of two forms. For the
symbolic regression problems, a training set is used. The goal
is to minimize the difference between the output of a candidate
program and the required output. For the evolution of logic
circuits, all the possible input combinations are applied at the
candidate circuit inputs, the outputs are collected and the goal
is to minimize the difference between the obtained truth table
and the required one. The evolution is stopped when the best
fitness value stagnates or the maximum number of generations
is exhausted.
IV. IMAGE FILTER DESIGN USING CGP
Evolutionary design of image filters at the functional level
has been introduced in [13]. In case of the evolutionary design
of image filters we are addressing the following problem.
Given an image corrupted by a certain type of noise that needs
to be removed. Let us denote it as Ic. A reference image Ir
is available that represents the uncorrupted version of Ic. The
task is to construct a filter (using an evolutionary algorithm)
working with a filter kernel k × k pixels that suppresses
the given type of noise from Icaccording to the knowledge
contained in this image.

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TABLE I
THE LIST OF FUNCTIONS THAT CAN BE IMPLEMENTED IN EACH
PROGRAMMABLE NODE
code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
function
255
x
255 − x
max(x,y)
min(x,y)
x ? 1
x ? 2
x + y
x +Sy
(x + y) ? 1
y if (x > 127) else x
|x − y|
x ? 1
x ? 2
description
constant
identity
inversion
maximum
minimum
division by 2
division by 4
addition
addition with saturation
average
conditional assignment
absolute difference
multiplication by 2 with saturation
multiplication by 4 with saturation
In order to evolve an image filter, the CGP at the functional
level can be utilized since a particular image filter can be
considered as a digital circuit having pi= k·k 8-bit inputs and
a single 8-bit output where k corresponds to the dimensions
of the filter kernel (e.g. pi= 9 in case of the 3 × 3 kernel).
The Fig. 2 shows the concept of evolutionary design of
a image filter. Every pixel value of the filtered image is
calculated using a corresponding pixel and its eight neighbors
in the processed image (for the simplicity 3 × 3 kernel is
considered).
In order to evolve an image filter which removes a given
type of noise from a corrupted image, we need (a) a set of
suitable functions (building blocks of the filter circuit) and (b)
a training data to measure the fitness of the candidate filters
(i.e. their quality). The goal of the evolutionary algorithm is
to minimize the difference between the original image and the
filtered image. The generality of the evolved filters (i.e., the
ability to operate sufficiently also for other images containing
the same type of noise the filters have not been trained for) is
tested by means of a test set.
An image filter can be encoded using a CGP matrix consist-
ing of nr×ncprogrammable nodes, pi= k·k and po= 1. Note
that the inputs and outputs of the nodes as well as the primary
inputs and primary outputs ate considered as 8-bit ports. Each
node can implement one of the predefined set of functions. For
the purpose of the evolutionary design of non-linear filters
we are using the functions listed in Table I. The utilized
function set contains the standard non-linear operations such
as minimum, maximum or absolute difference.
A. Evaluation of Candidate Filters
Every 8-bit pixel value of the filtered image is calculated
using the value of the corresponding pixel in the corrupted
image and its neighbouring pixels. Since it is impossible
to evaluate all the input test vectors in the fitness function
(for example, there are in total 25625different combinations
for the 5x5-pixel kernel in 256 degrees of grayscale), it is
necessary to select a representative training data to calculate
the quality of the candidate filters. Commonly the training data
is created as a corruption of a reference image by applying a
noise model to this image. The training data represents an
important information for evaluating the results of candidate
filters during the evolutionary process. The corrupted image
provides training vectors that are applied as the input for
the candidate filter. The output of the filter (i.e. the filtered
image – If) is compared to the reference image in order to
evaluate the filter quality. For the 5x5-pixel filter kernel the
training vectors are generated as a set of 25-tuples (from the
corrupted image) and the corresponding correct values (from
the reference image).
The goal of the evolutionary algorithm is to design a
filter which minimizes the difference between If and Ir. The
evolved filter is required to be robust, i.e. it provides a good
filtering quality not only on the training data but also on other
real images for which the reference version is not known. Note
that this ability can usually be achieved only in case of the
type of noise the filter was trained for.
If the corrupted image is of the size R × C pixels and we
are using a square-shaped input mask of 5x5-pixel filter kernel,
then the number of filtered pixels is equal to the (R−4)×(C−
4). The quality of the evolved image filter is expressed by the
fitness function (the value of this function is to be minimized)
shown in Eguation 1. In this equation, we consider that the
pixels are indexed from 0 to R − 1 or C − 1.
R−3
?
B. Experimental Setup
The following setup of the CGP was utilized for the
experiments presented in this paper. The CGP array consists
of nc× nr= 7 × 9 nodes. The l-back parameter has been set
to nc(i.e. the full connectivity), however, only the elements
situated in the first four columns can be connected directly
to the primary inputs. The evolutionary algorithm works with
the population of λ = 8 individuals. Up to 15 genes in an
individual can be mutated. The initial population is generated
randomly. The results were obtained from 100 independent
runs of the CGP system. Each single experiment takes 200,000
generations. The goal of each evolutionary experiment is to
find a filter working with 5x5-pixel kernel for the filtering the
given type of noise.
The training data are provided by an artificial 256x256-pixel
image corrupted by 20% noise which is illustrated in Figure
3. The training vectors are generated for the evolution of the
salt&pepper noise and random-valued impulse noise filters.
There are 62,061 unique training vectors for the salt&pepper
noise in Fig. 3b and 63,437 unique training vectors for the
random-valued impulse noise in Fig. 3c. The utilization of
artificial image for generating the training data is in virtue of
our experience in evolving filters for different types of noise.
The image contains crucial features that should be preserved
in the filtered image – smooth gradients of different types
combined with sharp edges. These components showed as
important for the filter to be trained on. The experiments
showed that if the noise intensity is low during the training
fit(If,Ir) =
i=2
C−3
?
j=2
|If(i,j) − Ir(i,j)|.
(1)

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process (≤ 10%), the training data does not provide a sufficient
amount of information for the filter to be robust (i.e. in most
cases the resulting filter is not able to remove noise of a higher
intensity). On the other hand, if the training noise intensity is
high (≥ 30%), substantial amount of (training) image data is
lost and the resulting filters do not exhibit reasonable filtering
quality.
Fig. 3. The training data utilized in the experiments: (a) the reference image,
(b) the image corrupted by 20% salt&pepper noise, (c) the image corrupted
by 20% random-valued impulse noise. The noise intensity 20% means that
value of 20% of pixels of the reference image is was changed (corrupted by
the noise).
C. Principles of Conventional Iterative Filters
This section briefly summarizes the principles of the con-
ventional iterative filters that were chosen for the comparison
with our evolved solutions. The filters DWM and PWMAD
were selected because they are considered as state-of-the-art
nonlinear filters designed to suppress the impulse noise
The DWM filter has introduced an impulse detector which
makes use of the difference between the current pixel and its
neighbors aligned with four main directions. After the impulse
detection, the filter does not simply replace the noisy pixels
identified by the outputs of the median filter but continues to
use the information of the four directions to weight the pixels
in the window in order to preserve the details as removing
the noise. A threshold, that is decreased in each iteration,
is utilized to identify the noisy pixels. The DWM filter is
supposed to perform much better than the other median-based
filters in removing random-valued impulse noise, especially
for higher noise intensity. Furthermore, it can preserve more
detail features, even thin lines [11].
The PWMAD filter calculates (five-times iteratively and for
each pixel in the image) the median and the absolute deviation
of the median and the original pixel value. Then the median
of the absolute deviations is calculated to determine the value
of the filtered pixel.
V. EXPERIMENTAL RESULTS
Two sets of experiments were performed. The first set
considered the evolution of salt&pepper noise filters and
the second set was devoted to the design of random-valued
impulse noise filters. This section summarizes the obtained
results and discusses their properties in comparison with some
conventionally designed filters for each type of noise.
The quality of the evolved filters will be compared to
several conventional single-step and iterative filters providing
TABLE II
COMPARISON OF THE SALT&PEPPER NOISE FILTERS IN TERMS OF PSNR
(DB). THE SIZE OF THE KERNEL IS ALSO SPECIFIED FOR EACH FILTER.
noise intensity in percent
10 15filter15 2025 30
F18 5x5
39.0
38.1
36.4
35.9
33.7
34.0
30.7
31.2
32.5
27.7
28.5 25.923.4
F18, 2 iter. 31.330.1 29.0
PWMAD 3x3 33.032.424.521.6 19.1
PWMAD 5x5 29.028.928.8 28.427.726.2 24.0
DWM 5x528.828.327.8 27.226.6
31.4
25.8
25.9
30.5
25.6
25.0
29.5
25.3
AMF 5x5 34.333.9 33.232.2
MF 5x5 26.526.4 26.226.0
unfiltered 25.1 18.115.113.312.1 11.110.3
the best results in removing the impulse noise. In order to
show the ability of the evolved solutions to improve the filtered
image using the iterative processing, one and two iterations of
these filters will be performed and the results compared to
the images filtered by DWM and PWMAD. Moreover, the
filtering results will also be compared to standard median
filter (MF) and adaptive median filter (AMF). It will be shown
that the evolved filters are able to provide the output quality
that is very close to (or even better than) the results of the
conventional filters and in less iterations in comparison with
the conventional filters.
The resulting filters obtained by means of the CGP system
were evaluated using a set of 30 different images, each of
which was corrupted by 1%–30% noise. The filtering quality
(expressed as the Peak Signal to Noise Ratio – PSNR) is
calculated as the average of the PSNR for each image in the
evaluation set and the noise intensity.
TABLE III
COMPARISON OF THE RANDOM VALUED NOISE FILTERS IN TERMS OF
PSNR (DB). THE SIZE OF THE KERNEL IS ALSO SPECIFIED FOR EACH
FILTER.
noise intensity in percent
10 15filter15 20 2530
F17 5x5
36.0
34.4
33.4
32.5
30.928.7
29.6
29.5
26.6
28.4
27.4
24.7
27.2
25.3
23.0
F17, 2 iter. 30.9
31.2
28.7
25.9
PWMAD 3x333.132.523.3
26.0
25.7
PWMAD 5x529.1 29.028.3 27.827.0
DWM 5x5 28.928.427.827.326.826.2
AMF 5x533.930.026.0 23.3 21.219.6 18.3
MF 5x526.6 26.526.3 26.125.9 25.525.2
unfiltered28.521.5 18.516.715.5 14.513.7
A. Salt&Pepper Noise Filters
Table II summarizes the quality of the salt&pepper noise
filters. The evolved filter is denoted as F18. As the results show
this filter provides the best results for lower noise intensity