A Fast and Robust Max- C Projection Fuzzy …valle/PDFfiles/Talk17bracisB.pdf · A Fast and Robust...

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!


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


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)


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)


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.


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]

.


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 | .


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!


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.


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.


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.


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!


Example

Consider

A =

a1 =

0.40.30.70.2

,a2 =

0.10.70.50.8

,a3 =

0.80.50.40.2

.

Given the inputxd =

[0.4 0.3 0.8 0.7

]T,

we obtainI = {ξ : aξj ≤ xj ,∀j = 1, . . . ,n} = {1}.

The output of the Zadeh max-C PAFM is

VZ(xd) =∨ξ∈I

aξ = a1.


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.


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.





Training images from one individual:





Testing images from one individual:


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

Classifier Sunglasses Scarf ComplexityLRC 90.2 30.4 O(pqT (k + 1) + (pq)3)

SRC 95.6 54.8 O(T 2(T + pq)1.3)

NMR 94.6 70.4 O((pq)1.5 + pqT )

SNL2R2 95.7 70.2 O(pq2 + pqT + T 2)

SNL1R1 96.1 71.2 O(pq2 + pqT + T 2)

DNL2R2 95.8 70.4 O(pq2 + pqT + T 2)

DNL1R1 96.7 72.3 O(pq2 + pqT + T 2)

VMM 92.3 46.5 O(pqT + T )

VMZ 92.7 47.3 O(pqT + T )

VMZ (partitioned) 95.8 80.8 O(pqT + βT )

• 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)!


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