Edge detection: gradients and derivatives Two possible approaches to edge detection: Detection of the maxima of the gradient Detection of the zero crossings of second derivative Most methods are gradient based.The edge pixels need to be connected together to get to a sequence of edge points! (“edge linking”)
The gradient The gradient is a vector with a norm and a direction: The partial derivatives can be computed by convolution with appropriate linear filter masks:
Gradient: alternatives Different possible norms: Taking the discrete nature of an image into account:
Theorem of Taylor LaGrange applied to an image z=f(x,y) Numerical approximation of the gradient
The Roberts filter
Differentiation with 1 pixel allows to detect fast transitions (sharp edges) but not slower transition (blurred edges) dt f(x) df(x) Problem of edge width f(x) df(x)
The gradient can be smoothed by averaging the pixel in the neighborhood. Prewitt gradient filter Increasing the differentiation step
Prewitt filter: the principal edges are better detected. Prewitt filter: example Roberts filter Prewitt filter
Differentiation: noise suppression Sobel filter
Gradient masks: summary Roberts Prewitt Sobel Mask size A bigger mask means less sensitivity to noise A bigger mask means higher computational complexity A bigger mask means less localization precision
The Canny/Deriche gradient operator In 1983 Canny proposed 3 criteria for edge detection:
Detection quality (maximum signal to noise ratio) Localization precision Uniqueness (one response per edge) + Gaussian noise Maximization of these criteria leads to the solution of a differential equation. The solution can be approximated by the derivative of a Gaussian:
The Deriche solution Canny’s solution has been developed for an finite impulse response filter (FIR). Deriche developed an infinite impulse response filter (IIR) from the same equations and different initial conditions: Scale parameter The filter is implemented as a recursive filter, i.e. the filter result of one pixel depends on the results of the preceding pixel.
?=1 ?=0.5 ?=0.25 Original image ?=5 ?=2 A higher value of ? means a higher sensitivity to detail
Zero-Crossings (the Laplacian filter) Instead of the maxima of the gradient we search the zero crossings of the second derivative. The laplacian operator Usually, the image smoothed (e.g. with a Gaussian filter) before calculating the derivative. Using the properties of convolution, this can be done in one step: The “mexican hat” filter:
The Laplacian filter: properties Advantages: Closer to mechanisms of visual perception (ON/OFF cells) One parameter only (size of the filter) No threshold Produces closed contours
Disadvantages: Is more sensitive to noise (usage of second derivative) No information on the orientation of the contour
Combination of gradient and contour Search of zero-crossings of the Laplacian in the neighborhood of local maxima of the gradient
Laplacian filter: examples
Zero crossings: examples ?=1 ?=2 ?=3
Mejoramiento de la nitidez
Mejoramiento de la Nitidez – 1/2 Corresponde a la mejora de la calidad visual de una imagen
Se basa en los filtros “ unsharp masking ” o filtros de enmascaramiento de imagen borrosa
Principio :
Añadir detalles ( frecuencias altas ) a una imagen borrosa ( frecuencias bajas ) imagen mejorada = (A–1) imagen original + imagen filtrada paso-altas
A > 1
Mejoramiento de la Nitidez – 2/2 imagen mejorada = (A–1) imagen original + imagen filtrada paso-altas Filtro Laplaciano ( filtro paso-altas ) A=1 : Filtro Laplaciano estándar A>1 : una parte de la imagen original se añade a la paso-alta A=2 ¡ añade ruido !
Filtro Unsharp Masking – 1/3 Se tiene una imagen borrosa ( pendiente pequeña ) = f Se le resta con una pendiente aún más pequeña = fLPF Lo anterior se multiplica por un factor = k ( entre 1 y 3 ) La señal de la diferencia anterior se suma a la original (Gp:) Señal de la diferencia (Gp:) Señal original (Gp:) Señal con menos Nitidez o borrosa
Filtro Unsharp Masking – 2/3 Se tiene una imagen borrosa ( pendiente pequeña ) = f Se le resta con una pendiente aún más pequeña = fLPF Lo anterior se multiplica por un factor = k ( entre 1 y 3 ) La señal de la diferencia anterior se suma a la original
señal con mayor resolución ( pendiente grande )
Nótese que :
con k = 1 : = imagen mejorada
Filtro Unsharp Masking – 3/3 De lo anterior se obtiene la definición del filtro Unsharp Masking hUM(x, y) :
La forma del filtro Unsharp Masking hUM(x, y) depende de la forma del filtro paso-bajas hLPF(x, y)
Ejemplo : si entonces
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