A Fast and Robust Max- C Projection Fuzzy Autoassociative Memory with Application for Face Recognition Alex Santana dos Santos and Marcos Eduardo Valle* Department of Applied Mathematics Institute of Mathematics, Statistics, and Scientific Computing University of Campinas - Brazil 5 October 2017 Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 1 / 22
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A Fast and RobustMax- C Projection Fuzzy Autoassociative Memory
with Application for Face Recognition
Alex Santana dos Santos andMarcos Eduardo Valle*
Department of Applied MathematicsInstitute of Mathematics, Statistics, and
Scientific ComputingUniversity of Campinas - Brazil
5 October 2017
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 1 / 22
Fuzzy Autoassociative Memories
The human brain ability to recall information by associationinstigated researches on associative memory models.
In an associative memory, an stimulus is associated to a response insuch a way that the presentation of the stimulus gives rise to theresponse.
We speak of an autoassociative memory if the stimulus and theresponse coincide.
An autoassociative memory can be used, for instance, to recognizea face occluded by sunglasses or scarf.
A fuzzy associative memory (FAM) is a memory designed for thestorage and recall of fuzzy sets!
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 2 / 22
Outline of this Talk
Basic Concepts on Fuzzy Sets and Fuzzy Logic
Max-C Projection Autoassociative Fuzzy Memories
Zade max-C PAFM
Noise Masking Strategy
Preliminary Computational Experiments for Face Recognition
Concluding Remarks
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 3 / 22
Fuzzy Logic
The symbols “∨” and “∧” represent the supremum (maximum) andinfimum (minimum) operations.
Definition (Fuzzy conjunction)
An increasing mapping C : [0,1]× [0,1] −→ [0,1] is a fuzzyconjunction if C(0,0) = C(0,1) = C(1,0) = 0 and C(1,1) = 1.
Example
• CM(x , y) = x ∧ y , (minimum)
• CG(x , y) =
{0, x = 0,y , otherwise.
(Gaines’ conjunction)
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 4 / 22
Fuzzy Implication
Definition (Fuzzy Implication)
A mapping I : [0,1]× [0,1] −→ [0,1] decreasing in the first argumentand increasing in the second argument is a fuzzy implication ifI(0,0) = I(0,1) = I(1,1) = 1 and I(1,0) = 0.
Example
• IM(x , y) =
{1, x ≤ y ,y , x > y .
(Gödel implication)
• IG(x , y) =
{1, x ≤ y ,0, x > y ,
(Gaines implication)
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 5 / 22
Adjunction
Fuzzy conjunctions and implications are not inverse operations butthey can be linked by an adjunction.
Definition (Adjunction)
A fuzzy implication I and a fuzzy conjunction C form an adjunction if
C(x , y) ≤ z ⇐⇒ x ≤ I(y , z), ∀x , y , z ∈ [0,1].
Example
The pairs (IM ,CM) and (IG,CG) are adjunctions.
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 6 / 22
Fuzzy Sets
A fuzzy set x on U = {u1,u2, . . . ,un} can be identified with a vectorx = [x1, x2, . . . , xn]
T ∈ [0,1]n, where xj = x(uj) denotes the degree ofmembership of uj in the fuzzy set x.
Like linear combinations, z ∈ [0,1]n is a max-C combinations ofA =
{a1, . . . ,ak} ⊆ [0,1]n if
z =k∨ξ=1
C(λξ,aξ) ⇐⇒ zi =k∨ξ=1
C(λξ,aξi ), ∀i = 1, . . . ,n,
where λξ ∈ [0,1] for all ξ = 1, . . . , k .
The family of all max-C combinations of A ={
a1, . . . ,ak} is
C(A) =
z =k∨ξ=1
C(λξ,aξ) : λξ ∈ [0,1]
.
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 7 / 22
Fuzzy Inclusion Measures
A fuzzy inclusion measure IncF (a,b) yields the degree of inclusionof a fuzzy set a in a fuzzy set b.
Example (Zadeh fuzzy inclusion measure)
IncZ(a,b) =
{1, aj ≤ bj , ∀j = 1, . . . ,n,0, otherwise.
Example (Inf-I inclusion measure)
IncF (a,b) =n∧
j=1
I(aj ,bj).
Zadeh fuzzy inclusion measure is obtained if I ≡ IG.Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 8 / 22
Fuzzy Similarity Measure
A fuzzy similarity measure yields the degree of similarity betweentwo fuzzy sets a,b.
Example
SH(a,b) = 1− 1n
N∑i=1
|ai − bi | .
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 9 / 22
Max-C Projection Autoassociative Fuzzy Memories
Given a set A ={
a1, . . . ,ak} ⊆ [0,1]n, called fundamental memoryset, an autoassociative fuzzy memory (AFM)M should satisfy
M(aξ) = aξ, ∀ξ ∈ K = {1,2, . . . , k} .
The memoryM is also expected to exhibit some noise tolerance:
M(x) = aξ, for a noisy version x of aξ.
A max-C projection autoassociative fuzzy memory (max-C PAFM)projects the input x on the max-C combinations of A = {a1, . . . ,ak}:
V(x) =∨{z ∈ C(A) : z ≤ x} .
The output V(x) is the largest max-C combination of thefundamental memories less than or equal to the input x.Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 10 / 22
Noise Tolerance and Convergence
TheoremA max-C PAFM satisfies V(x) ≤ x and V(V(x)) = V(x) for allx ∈ [0,1]n.
Consequences:• A max-C PAFM converges in a single iteration if employed with
feedback.• A max-C PAFM exhibits tolerance only with respect to dilative
noise:◦ a distorted version x of aξ has undergone a dilative change if x ≥ aξ.◦ x has undergone an erosive change x ≤ aξ.
• A max-C PAFM is extremely sensitive to either erosive or mixed(dilative and erosive) noise; aξ cannot be retrieved if x ≥ aξ!
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 11 / 22
Storage Capacity and Implementation
Theorem (Optimal Absolute Storage Capacity)
A max-C PAFMs satisfy V(aξ) = aξ, ∀ξ ∈ K, if the fuzzy conjunctionC has a left identity.
Theorem (Implementation)
Let (I,C) be an adjunction. The output of the max-C PAFM satisfies
V(x) =k∨ξ=1
C(λξ,aξ) where λξ =n∧
j=1
I(aξj , xj),∀ξ ∈ K.
Note that λξ is the degree of inclusion of aξ in x:
λξ = IncF (aξ,x), ∀ξ ∈ K.
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 12 / 22
Example
Let (IM ,CM) and consider
A =
a1 =
0.40.30.70.2
,a2 =
0.10.70.50.8
,a3 =
0.80.50.40.2
.
Given the input
xd =
0.40.30.80.7
,we obtain
λ1 = 1.0, λ2 = 0.3, and λ3 = 0.3.
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 13 / 22
Example
The output of the max-CM PAFM is
VM(xd) = CM(λ1,a1) ∨ CM(λ2,a2) ∨ CM(λ3,a3)
=
1.0 ∧
0.40.30.70.2
∨
0.3 ∧
0.10.70.50.8
∨
0.3 ∧
0.80.50.40.2
=
0.40.30.70.2
∨
0.10.30.30.3
∨
0.30.30.30.2
=
0.40.30.70.3
6= a1.
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 13 / 22
Zadeh max-C PAFM
We can improve the noise tolerance of a max-C PAFM by defining
λξ = IncZ(aξ,x) =
{1, aξj ≤ xj , ∀j = 1, . . . ,n,0, otherwise.
and
VZ(x) =k∨ξ=1
CG(λξ,aξ).
Equivalently, we have
VZ(x) =∨ξ∈I
aξ, where I = {ξ : aξj ≤ xj ,∀j = 1, . . . ,n}.
No arithmetic operation is performed! We only perform comparisons!
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 14 / 22
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 15 / 22
Noise Tolerance
TheoremThe identity VZ(x) = aη holds true if there exists an unique η ∈ Ksuch that aη ≤ x.
Consequence:• The Zadeh max-C PAFM is extremely robust to the dilative noise
but it is extremely sensitive to erosive noise!
The noise tolerance of a max-C PAFM can be significantly improvedby masking the noise contained in a corrupted input.
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 16 / 22
Noise Masking Strategy
In some sense, noise masking aims to remove the erosive noisefrom the input x.
1. Select an index η such that
S(x,aη) =k∨ξ=1
{S(x,aξ)
},
where S is a fuzzy similarity measure.2. Define the novel memory
VM(x) = V(x ∨ aη).
If we use the fuzzy similarity measure SH , VMZ performs O(nk)
operations during the retrieval phase.Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 17 / 22
Computational Experiments
Face recognition problem using the AR database:• Gray-scale face images of size 50× 40 from 120 individuals.• Eight frontal face images used for training.• Two testing scenarios:
a) Sunglasses plus illumination – 4 images.b) Scarf plus illumination – 4 images.
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 18 / 22
Computational Experiments
Face recognition problem using the AR database:• Gray-scale face images of size 50× 40 from 120 individuals.• Eight frontal face images used for training.• Two testing scenarios:
a) Sunglasses plus illumination – 4 images.b) Scarf plus illumination – 4 images.
Training images from one individual:
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 18 / 22
Computational Experiments
Face recognition problem using the AR database:• Gray-scale face images of size 50× 40 from 120 individuals.• Eight frontal face images used for training.• Two testing scenarios:
a) Sunglasses plus illumination – 4 images.b) Scarf plus illumination – 4 images.
Testing images from one individual:
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 18 / 22
Max-C PAFM Classifier
• We identify a face image of an individual i with aξ,i ∈ [0,1]2000.• We formed the fundamental memory sets Ai = {a1,i , . . . ,a8,i}, for
i = 1, . . . , c = 120.
Independent max-C PAFM model for each individual:
Mi (= V iM or V i
Z).
An unknown face image x ∈ [0,1]n is assigned to η such that
SH(x,Mη(x)
)≥ SH
(x,Mi(x)
), ∀i = 1, . . . , c.
In words, x belongs to an individual such that the recalled vector isthe most similar to the input!Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 19 / 22
• p and q denote the height and width of the face images images,• c is the number of individuals,• k is the number of training images per individual,• T = ck , and β denotes the number of blocks in which the face
images are partitioned.Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 20 / 22
Concluding Remarks
We introduced the Zadeh max-C projection autoassociative fuzzymemory (max-C PFAM):• Optimal absolute storage capacity.• Excellent tolerance to dilative noise.• Uses only comparisons (no operations)!
We also proposed the noise making strategy based on a fuzzysimilarity measure to improve the tolerance with respect to noise.
Preliminary experiments suggested that the novel memory may becompetitive with other classifiers by providing fairly recognition rateswith low computational cost.
Thank you!Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 21 / 22
Have an idea?
Submit a paper to 2018 WCCI(FUZZ-IEEE or IJCNN)!
Marcos Eduardo Valle (Unicamp) Fuzzy Associative Memory 5 October 2017 22 / 22