Frist and second derivatives of image

Notes

• Derivative-based Operations
  
  - First derivative (Gradient): basic, Prewitt (basic*uniform), Sobel (basic*triangular), Gaussian (basic+gaussian). I.e.: h_1 = [1 0 -1]
  - Second derivative (Laplacian): basic, frequency domain, Second-Derivative-in-the-Gradient-Direction (SDGD). I.e.: h_2 = h_1 (x) h_1 = [1 0 -2 0 1]

Conditional Random Field

Notes

• [Blog] Introduction to Conditional Random Fields

Watershed transformation on image segmentation

Notes

• Image segmentation and mathematical morphology

Level set

Notes

• What is level set?

• What is gradient flow?

Fourier Transform

Notes

• Fourier Transform, Spectral Analysis, Frequency Analysis

• INTRODUCTION TO FOURIER TRANSFORMS FOR IMAGE PROCESSING

Kalman Filter

Notes

• Object tracking using a Kalman filter (MATLAB)

Visual system

Notes

• Eye
  
  When light enters the eye, it is refracted after passing through the cornea (角膜), then passes through the pupil (瞳孔) that is controlled by the iris (虹膜) and is further refracted by the lens. An inverted image is projected onto the retina (视网膜) by a compond lens based on the corea and lens.

• Retina
  
  In the retina, the photoreceptors (光感受器) synapse onto bipolar cells (双极细胞) and then ganglion cells (节细胞), which conduct action potential (动作电位) to the brain.
  
  The retina consists of a large number of photoreceptor cells with opsins (视蛋白), which absorbs a photon (光子) and transmits a signal to the cell. There are two types of opsins, rod (视杆) ones and cone (视锥) ones. Rod opsins are used to see at low levels of light whereas cone opsins are used to see at normal levels of light. There are three types of cones that differ in the wavelengths of light they absorb: short or blue, middle or green, long or red.
  
  The processing in the retina includes convergence and divergence from photoreceptors to bipolar cells, as well as the formation of center-surround receptive fields (RF) of bipolar and ganglion cells and more complex receptive field (i.e.: sensitive to either color or motion) by transmitting information laterally (in the same layer). The release of neurotransmitters between bipolar and ganglion cells is controlled by the rhodopsin (视紫质), brightness detectors that change from violet to colorless after light presents.
  
  Finally, five populations of ganglion cells send action potential (visual information) to brain: K cells (with very large center-only RF's that are sensitive to color), M cells (with large center-surround RF's that are sensitive to depth and rapidly adapt to a stimulus), P cells (with smaller center-surround RF's that are sensitive to color and shape),

• Optic nerve
  
  Through the optic nerve, K, M and P cells send action potential to the lateral geniculate nucleus (外侧膝状核) in the thalamus (丘脑) through different axons (轴突) parallelly so as to reconstruct the visual world.
  
  The forth population above sends information to the pretectum (前顶盖) for the contol of the biological clock and sleep regulation.
  
  The last population sends information to the superior colluculus (上丘脑) for the control of eye movements and other motor responses.

• Optic chiasm (视交叉)
  
  Through optic chiasm at the base of the hypothalmus (下丘脑), the primary visual cortex on the right is able to deal with the left half of the field of view from both eyes, and so is the left brain.

• Optic tract (视束)
  
  Both end in the lateral geniculate nucleus (LGN).

• Lateral geniculate nucleus (外侧膝状核)
  
  It is a sensory relay nucleus in the thalamus of the brain that relay the cisual image to the primary visual cortex (V1) , which is located at the back of the brain in the occipital lobe (枕骨脑叶). Meanwhile, it also processes information from the visual cortex.

• Optic radiation (视辐射)
  
  It sends information from LGN to layer 4 of the visual cortex.

• Visual cortex (视皮层)
  
  A human brain contains five layers of visual cortex, V1~V5, where V1 receives information directly from LGN. V1 and V2 are believed to provide edge/corner detection and basic color/motion processing. The complexity of the neural representations increases with the layer in the visual hierarchy. Whereas neurons in V1 may respond to a line segment with particular location and orientaion, neurons in higher levels and visual association cortex may selectively respnd to complte object or even a particular object. Regarding the increasing complexity of neural representation in the visual hierarchy, two interconnected streams hypothesis was proposed: the "where" stream (involved in spatial attention and eye/hand movements) and the "what" stream (involved in recognition and categorization).

Bayes' Theorem

Notes

• Most common form:

  • P(A|B) = P(A,B)/P(B)
    

• Chain rule: 
  
  P(A,B,C,D) = P(A,B,C|D)P(D) = P(A,B|C,D)P(C|D)P(D)
  
  = P(A|B,C,D)P(B|C,D)P(C|D)P(D)
  

• P(A,B|C) = P(A,B,C)/P(C) = P(A|B,C)P(B,C)/P(C) = P(A|B,C)P(B|C)P(C)/P(C) 
  
  = P(A|B,C)P(B|C)
  

• If B and C are independent (1) and conditional independent given A (2), then:

  • P(A|B,C) = P(A,B,C)/P(B,C) 
    
    = P(B,C|A)P(A)/P(B)P(C)      ...(1)
    
    = P(B|A)P(C|A)P(A)/P(B)P(C)  ...(2)
    
    = P(B|A)P(A|C)/P(B)
    

Matrix Computations

Notes

• QR decomosition (QR factorization):
  • Given m-by-n matrix A,

    		A = QR

    where Q is m-by-m orthogonal and R is m-by-n upper triangular. If A has full column rank then the first n columns of Q form an orthonormal basis for range(A). Thus, calculation of the QR factorization is one way to compute an orthonormal basis for a set of vectors.
  • Orthogonal matrix -- QQ'= I. Then each column vector of Q form an orthonormal basis.
  • Orthonormal basis:
    The vectors v_1,...,v_k form an orthonormal basis for a subspace S if they are orthonormal and span S.
  • Span of {v_1,...v_n}:
    the set of all linear combinations of these vectors
  • IDEA: dimension <-- basis for span <-- maximal linearly independent subset + span <-- linear combination --> linear independent --> orthonormal --> orthonormal basis
  • Range -- range(A) = {y: y = Ax}
  • Null space -- null(A) = {x: Ax = 0}
  • Rank -- rank(A) = dim(range(A)) which means the maximal number of independent rows or columns.
  • Singular -- if inverse matrix A^(-1) exists, then A is nonsingular. OW A is singular.
  • Inverse matrix -- AA^(-1)=1

• SVD decomosition (Singular Value Decomposition):
  • Description: if A is a real m-by-n matrix, then there exist orthogonal matrices (singular vectors):

    		m-by-m matrix U		and	n-by-n matrix V

    such that

    		U'AV = diag(d_1,d_2,...d_p)

    , where p = min{m,n} and singular values d_i >= 0 are the lengths of the semi-axes of the hyperellipsoid E defined by E = {Ax: ||x||_2 = 1}.

Unsupervised segmentation

Notes

• LOCUS

Deep learning background knowledge

Notes

• Understanding Convolution in Deep Learning

Medical image datasets

MRI

• OASIS (Brain)

• Lung Cancer Datasets