public interface IHttpHandlerBase : IHttpHandler
{
    void RequestContext { get; set; }
}

public class GenericRouteHandler<T> : IRouteHandler
    where T : IHttpHandlerBase, new()
{
    public IHttpHandler GetHttpHandler(RequestContext requestContext)
    {
        var retVal = new T();
        retVal.RequestContext = requestContext;
        return retVal;
    }
}

public class ImageHandler : IHttpHandlerBase
{
   ...
}

// Это прописать в инициализации правил маршрутизации
RouteTable.Routes.Add(new Route("Image/{ImageID}", new GenericRouteHandler<ImageHandler>()));