Image Segmentation with Level Sets Group reading

Outline 1.Active Contours 2.Level sets in Malladi Prior knowledge in Leventon 2000

1. Active Contours

1.1 Introduction to AC 1.2Types of curves 1.3Types of influences 1.4Summary

1.1 Introduction to AC AC = curve fitted iteratively to an image –based on its shape and the image values –until it stabilises (ideally on an objects boundary). curve:polygon = parametric AC continuous = geometric AC parametric geometric

1.1 Introduction to AC Example: cell nucleus Initialisation: external Fitting: iterative –Using the current geometry (Contour influence) and the image contents (Image influence) –Influence: scalar = energy vector= force –Until the energy is minimum, or the forces are null Image: Ian Diblik 2004

1.2 Types of curves Parametric AC: –Stored as vertices –Each vertex is moved iteratively Geometric AC: –Stored as coefficients –Sampled before each iteration –Each sample is moved –New coefficients are computed (interpolation)

1.3 Types of influences Each vertex /sample v i is moved: in a direction that minimises its energy: E(v i ) = E contour (v i ) + E image (v i ) or, along the direction of the force: F(v i ) = F contour (v i ) + F image (v i ) ( Linear combinations of influences )

1.3 Types of influences Contour influence: –meant to control the geometry locally around v i –corresponds to the prior knowledge on shape Energy: example: E contour (v i ) =. ||v i || 2 +. ||v i || 2 Force: example: F contour (v i ) = v i + v i + n i where n i is the normal of the contour at v i and is a constant pressure (balloon)

1.3 Types of influences Image influence: –meant to drive the contour locally –corresponds to the prior knowledge on the object boundaries Energy: example: E image (v i ) = ( I (v i ) - I target ) 2 E image (v i ) =. 1 / ( 1 + || I (v i ) || ) Malladi 1995 E image (v i ) = -. || I (v i ) || Leventon 2000 Force: example: F image (v i ) = E image (v i ) n i F image (v i ) = - I (v i )

1.3 Types of influences Exotic influences: E contour (v i ) proportional to the deformations of a template: E image (v i ) based on region statistics: I outside I inside Bredno et al Hamarneh et al. 2001

1.4 Summary Active Contours: –function v : N P ( [ 0, I width ] x [ 0, I height ] ) t { (x,y) } –v(0) is provided externally –v(t+1) is computed by moving sample points v i of v(t) –moved using a linear combination of influences –on convergence, v(t end ) is the border of the object Issues: –fixed topology –possible self-crossing

2. Level sets in Malladi 1995

2.1Introduction to LS 2.2Equation of motion 2.3Algorithm 1 2.3Narrow-band extension 2.4Summary

2.1 Introduction to LS Transition from Active Contours: –contour v(t) front (t) –contour energy forces F A F C –image energy speed function k I Level set: –The level set c 0 at time t of a function (x,y,t) is the set of arguments { (x,y), (x,y,t) = c 0 } –Idea: define a function (x,y,t) so that at any time, (t) = { (x,y), (x,y,t) = 0 } there are many such has many other level sets, more or less parallel to only has a meaning for segmentation, not any other level set of

2.1 Introduction to LS Usual choice for : signed distance to the front (0) - d(x,y, ) if (x,y) inside the front (x,y,0) = 0 on d(x,y, ) outside (t) (x,y,t)

2.1 Introduction to LS (x,y,t) (x,y,t+1) = (x,y,t) + (x,y,t) no movement, only change of values the front may change its topology the front location may be between samples

2.1 Introduction to LS Segmentation with LS: Initialise the front (0) Compute (x,y,0) Iterate: (x,y,t+1) = (x,y,t) + (x,y,t) until convergence Mark the front (t end )

2.2 Equation of motion Equation 17 p162: link between spatial and temporal derivatives, but not the same type of motion as contours! constant force (balloon pressure) (x,y,t+1) - (x,y,t) extension of the speed function k I (image influence) smoothing force depending on the local curvature (contour influence) spatial derivative of product of influences

2.2 Equation of motion Speed function: –k I is meant to stop the front on the objects boundaries –similar to image energy: k I (x,y) = 1 / ( 1 + || I (x,y) || ) –only makes sense for the front (level set 0) –yet, same equation for all level sets extend k I to all level sets, defining –possible extension: ^ kIkI ^ k I (x,y) = k I (x,y) where (x,y) is the point in the front closest to (x,y) ^ ( such a k I (x,y) depends on the front location )

2.3 Algorithm 1 (p163) compute the speed k I on the front extend it to all other level sets 2. compute (x,y,t+1) = (x,y,t) + (x,y,t) 3. find the front location (for next iteration) modify (x,y,t+1) by linear interpolation (x,y,t)

2.4 Narrow-band extension Weaknesses of algorithm 1 –update of all (x,y,t): inefficient, only care about the front –speed extension: computationally expensive Improvement: –narrow band: only update a few level sets around –other extended speed: k I (x,y) = 1 / ( 1 + || I(x,y)|| ) ^

2.4 Narrow-band extension Caution: –extrapolate the curvature at the edges –re-select the narrow band regularly: an empty pixel cannot get a value may restrict the evolution of the front

2.5 Summary Level sets: –function : [ 0, I width ] x [ 0, I height ] x N R ( x, y, t ) (x,y,t) –embed a curve : (t) = { (x,y), (x,y,t) = 0 } – (0) is provided externally, (x,y,0) is computed – (x,y,t+1) is computed by changing the values of (x,y,t) –changes using a product of influences –on convergence, (t end ) is the border of the object Issue: –computation time (improved with narrow band)

3. Prior knowledge in Leventon 2000

3.1Introduction 3.2Level set model 3.3Learning prior shapes 3.4Using prior knowledge 3.5Summary

3.1 Introduction New notations: function (x,y,t) shape u(t) ( still a function of (x,y,t) ) front (t) parametric curve C(q) (= level set 0 of u(t) ) extended speed k I stopping function g(I) Malladis model: choose: F A = -c (constant value) F G ( ) = - (penalises high curvatures), ^

3.1 Introduction u(t) use a level set model (based on current geometry and local image values): u(t+1) = u(t) + u(t) use prior knowledge and global image values to find the prior shape which is closest to u(t): call it u*(t) combine linearly these two new shapes to find the next iteration: u(t+1) = u(t) + 1 u(t) + 2 ( u*(t) - u(t) )

3.2 Level set model u(t) use a level set model (based on current geometry and local image values): u(t+1) = u(t) + u(t) use prior knowledge and global image values to find the prior shape which is closest to u(t): call it u*(t) combine linearly these two new shapes to find the next iteration: u(t+1) = u(t) + 1 u(t) + 2 ( u*(t) - u(t) )

3.2 Level set model not Malladis model: its Caselles model: use it in: u(t+1) = u(t) + 1 u(t) + 2 ( u*(t) - u(t)) Thats all about level sets. Now its all about u*. ( implicit geometric active contour ) ( implicit geodesic active contour )

3.3 Learning prior shapes From functions to vectors: u(t) (continuous) sample it: put the values in one column = vector u(t) (discrete)

3.3 Learning prior shapes Storing the training set: –set of n images manually segmented: { C i, 1 i n } –compute the embedding functions : T = { u i, 1 i n } –compute the mean and offsets of T: = 1/n i u i û i = u i - –build the matrix M of offsets: M = ( û 1 û 2 û n ) –build the covariance matrix 1/n MM T, decompose as: 1/n MM T = U U T U orthogonal, columns: orthogonal modes of shape variations diagonal, values: corresponding singular values

3.3 Learning prior shapes Reducing the number of dimensions: –u: dimension N d (number of samples, e.g. I width x I height ) –keep only first k columns of U: U k (most significant modes) –represent any shape u as: = U k T ( u - ) = shape parameter of u, dimension k only –restore a shape from : u = U k + –other parameter defining a shape u: pose p (position) –prior shape: u* (, p) ~

3.3 Learning prior shapes At the end of the training: most common variation least common variation The most common shapes have a higher probability.

3.4 Using prior knowledge u(t) use a level set model (based on current geometry and local image values): u(t+1) = u(t) + u(t) use prior knowledge and global image values to find the prior shape which is closest to u(t): call it u*(t) combine linearly these two new shapes to find the next iteration: u(t+1) = u(t) + 1 u(t) + 2 ( u*(t) - u(t) )

3.4 Using prior knowledge Finding the prior shape u* closest to u(t) –Given the current shape u(t) and the image I, we want: u* map = ( map, p map ) = arg max P(, p | u(t), I) –Bayes rule: P(,p | u,I) = P(u |,p). P(I |u,,p). P( ). P(p) / P(u,I) has to be max = P (u | u*) = exp( - V outside ) compares u and u* = P (I | u, u*) P ( I | u*) = exp ( - | h(u*) - | I| | 2 ) compares u* with the image features measures how likely u* is as a prior shape = U(–,) uniform distribution of poses (no prior knowledge) constant

3.4 Using prior knowledge P( u | u* ) : P(u | u*) = exp ( - V outside ), V outside : volume of u outside of u* u* u V outside = Card { (x,y), u(x,y) < u*(x,y) } – meant to choose the prior shape u* most similar to u – only useful when C(q) gets close to the final boundaries

3.4 Using prior knowledge P(u(t) | u* 1 ) =1 P(u(t) | u* 2 ) =1 P(u(t) | u* 3 ) =1 u* 1 u* 2 u* 3 P(u(t) | u* 2 ) =0.6 P(u(t) | u* 2 ) =0 P(u(t) | u* 3 ) =0.6 P(u(t) | u* 1 ) =1 P(u(t) | u* 3 ) =1 u(t)

I(x,y) 3.4 Using prior knowledge P( I | u, u* ) : –meant to choose the prior shape u* closest to the image –depends on the boundary features (image influence) P( I | u*) = exp ( - | h(u*) - | I| | 2 ) u*=0 u* I 0-22 u*=2 u*=-2 h(u*)

3.4 Using prior knowledge u* 1 u* 2 P(I|u* 1 ) = 0.9 P(I|u* 2 ) = 0.6 u(t) u(t) + 2 (u* 1 - u(t))

3.5 Summary Training: –compute mean and covariance of training set –reduce dimension Segmentation: –initialise C, compute u(0) with signed distance –to find u(t+1) from u(t), compute: a variation u(t) using current shape and local image values a prior shape u* so that: u* is a likely prior shape u* encloses u(t) u* is close to the object boundaries –combine u(t) and u* to get u(t+1)

Conclusions Active contours: –use image values and a target geometry –topology: fixed but little robust Level sets: –also use image values and a target geometry –but in higher dimension, with a different motion –topology: flexible and robust Prior knowledge: –prior shapes influence the segmentation –not specific to level sets –ideas can be adapted to other features (boundaries, pose)

Thats all folks!